What's Your Philosophy of Mathematics?

[apeiron]

Your leading statement that:
I take to mean that such machinery is constructed by us to describe (among much else) the constraints we discover that exist, among the contingent circumstances we find ourselves in.

Not just to describe but also to make. We look at the world mathematically so as to be able to take control of it better. So we actually want to construct those constraints that produce control, such as when we use maths in any applied way.

Of course, a big part of maths self-image is that it is non-utilitarian. It is a pure exercise in thought that just happens also to be unreasonably effective.

But what I am arguing is that it is a way of viewing the world that in evolution already has proved generally effective. If you can atomise the description of constraints, then you can also build them. And the ability to construct constraints is a fantastically powerful trick in itself.

This is a point of view based on emerging disciplines like infodynamics for example...
http://www.harmeny.com/twiki/pub/Main/SaltheResearchOnline/ssaltheinfodynamics_update.pdf

I guess that much of what you say can be summarised as: Evolution is the Name of the Game.

But just how, as you say, “Nature constructs its own constraints via material dissipation”, we don’t understand all that well — yet.

Yes, this has been a very active field of research the past 20 years. You can see it becoming mainstream now with movements like evo-devo and dissipative structure theory.

 

[Ken G]

I put "other", because I believe that mathematics involves elements of all the things on the list, but none of them summarize the complete picture very well. To see that, I'd like to offer a quick critique of each of the options:
1) Logicism - Mathematics is reducible to logic, and mathematical truths are just tautologies.
It is true that math makes tautological connections between theorems and postulates, but mathematics has another important element: that of an axiom. An axiom is treated no differently, in a formal sense, from a postulate, but the meaning of an axiom is quite a bit different-- it is something that is expected to be, or seems to be, true based on our experience. If mathematics were only logicism, there would never be any reason to have two words, axiom and postulate, when one word would do fine.
2) Formalism - Mathematics is just a meaningless symbolic game that happens to be useful.
This is close to #1, so a similar objection obtains. The added problem here is that if math is meaningless and symbolic, then we have little expectation for it to be useful. Indeed, it is not a requirement that math be useful, but it quite often is anyway. For anyone who is unhappy to say this "just happens" to be the case, we need to dig deeper than choice #2.
3) Intuitionism/Constructivism - Mathematics is an arbitrary invention of the human mind/brain.
We can demonstrate that math is a product of the human brain, whether we should call that an "invention" immediately gets us into debate. If I invent a mousetrap, then I can catch mice in a way that no mouse has ever been caught before. But if I "invent" an axiom, we can attempt to judge its validity applied to times prior to my birth-- suggesting that the axiom was as true before I "invented" it as afterward, and that questions the applicability of the term "invent." Even more clear is that if I prove a theorem based on some axioms, then that theorem was as true before I proved it as after, so I can hardly claim to have "invented" that theorem. Another troublesome word here is "arbitrary"-- axioms can be arbitrary, but would have to be considered postulates instead if they did not seem to contain any self-evident truthfulness. What's more, no mathematician would invoke postulates that could prove contradictory things, nor would they use axioms that could do that unless they seemed to be extremely self-evident and the contraction seemed harmless, though of course even then it would call into question the whole meaning of a "proof". So neither axioms, nor postulates, are "arbitrary"-- they have reasons for being what they are.
4)Platonism - Mathematical truths are truths ABOUT something objectively real, like "Platonic heaven".
The problem here is that the words "objectively real" are having a little fight with the words "mathematical truths." I think Einstein said it well when he said words to the effect that, to the extent that we can know something is true, it can't be real, and to the extent that something is real, we can't know it to be true. So I think this characterization of mathematics is internally inconsistent if typical meanings of the words are used, and it only becomes consistent if the meanings are chosen to make it tautological. But one can still hold this view if one rejects the idea that knowing is epistemologically different from observing-- I would say that stance implies that the mind is in some sense "more perfect" than the senses, which is fundamentally rationalistic.
5) Physism - Mathematics is based on the patterns humans gleam from studying the physical world.
This is the counterpart to #4, and is based in empiricism. I think it must be true that some of the skills a mathematician uses, including the rules of logic, "make sense" because of studies of the physical world. But it begs the question to notice this and conclude that the physical world is where math comes from, expressly because if one holds that the physical world comes from math, then it would be natural to find math in the physical world. What's more, math clearly extends beyond the physical world, because we can prove theorems that we already know don't hold in the physical world, and yet it is still math. So it's not really meeting the challenge to say what math is "based on", our goal is to say what it is.
6) Fictionalism - Mathematics is just a made-up story that has its own internal logic.
This one is hard to parse from #1 and #2, so if we are to give it the status of a separate possibility, we must stress the "made up story" part. This seems to imply that we use math in the way we use fictional stories, as essentially an entertainment for our imagination that can convey some life lessons by embedding "morals" into the stories. But this again overlooks the role of axioms and postulates, and it just tacks on "has its own internal logic" as if that was a detail of little importance. But the role of axioms and postulates in math go way beyond the desire to tell a fantasy story-- they are a means to assess the validity, usefulness, or even just aesthetic appeal of a set of axioms by assessing the set of theorems they lead to. The purpose seems quite a bit different than the purpose of a fictional story, though the similarity to #1 and #2 means we cannot completely discount this element of what mathematics is and does.

So I see significant failings in all the above views of what math is, though I don't think any miss the boat completely. What's not clear to me is that it needs to be just one of those things, any more than I need to be just my career, or just my family status, or my age or height. I am a lot of things at once, and so is mathematics. So I put "other", because "all of the above" was not an option.

 

[Paulibus]

Apeiron: Thanks for the link to Salthe's introduction to Infodynamics. I've never been quite clear on the inverse relation between entropy and information. This will enhance my understanding.

About the philosophy of mathematics: I'm still not quite clear on what you mean by "constructing constraints". A specific example of what you call a mathematical constraint that has been constructed would help me. And an explanation of how it produces control, and of what it controls would be appreciated. You're not just talking of human invention ... or are you?

You also mentioned that "the ability to construct constraints is a fantastically powerful trick". Talking of tricks, I'm of the opinion that the essence of nature's most fantastically powerful tricks is that they have an self-perpetuating flavour. Having asked you for an example, I'm bound to justify this claim by giving examples myself. Here are two: First trick: Fluvial erosion; the more erosion, the bigger becomes the catchment area; which enhances erosion... Hence rivers. Second trick: the still-mysterious genesis of a self-perpetuating molecular replication mechanism in the form of an unzippable pair of coded molecular spirals ... Hence Life.

Success breeds success is a very powerful motto.

 

[Pythagorean]

I put "other", because I believe that mathematics involves elements of all the things on the list, but none of them summarize the complete picture very well. To see that, I'd like to offer a quick critique of each of the options:
1) Logicism - Mathematics is reducible to logic, and mathematical truths are just tautologies.
It is true that math makes tautological connections between theorems and postulates, but mathematics has another important element: that of an axiom. An axiom is treated no differently, in a formal sense, from a postulate, but the meaning of an axiom is quite a bit different-- it is something that is expected to be, or seems to be, true based on our experience. If mathematics were only logicism, there would never be any reason to have two words, axiom and postulate, when one word would do fine.
2) Formalism - Mathematics is just a meaningless symbolic game that happens to be useful.
This is close to #1, so a similar objection obtains. The added problem here is that if math is meaningless and symbolic, then we have little expectation for it to be useful. Indeed, it is not a requirement that math be useful, but it quite often is anyway. For anyone who is unhappy to say this "just happens" to be the case, we need to dig deeper than choice #2.
3) Intuitionism/Constructivism - Mathematics is an arbitrary invention of the human mind/brain.
We can demonstrate that math is a product of the human brain, whether we should call that an "invention" immediately gets us into debate. If I invent a mousetrap, then I can catch mice in a way that no mouse has ever been caught before. But if I "invent" an axiom, we can attempt to judge its validity applied to times prior to my birth-- suggesting that the axiom was as true before I "invented" it as afterward, and that questions the applicability of the term "invent." Even more clear is that if I prove a theorem based on some axioms, then that theorem was as true before I proved it as after, so I can hardly claim to have "invented" that theorem. Another troublesome word here is "arbitrary"-- axioms can be arbitrary, but would have to be considered postulates instead if they did not seem to contain any self-evident truthfulness. What's more, no mathematician would invoke postulates that could prove contradictory things, nor would they use axioms that could do that unless they seemed to be extremely self-evident and the contraction seemed harmless, though of course even then it would call into question the whole meaning of a "proof". So neither axioms, nor postulates, are "arbitrary"-- they have reasons for being what they are.
4)Platonism - Mathematical truths are truths ABOUT something objectively real, like "Platonic heaven".
The problem here is that the words "objectively real" are having a little fight with the words "mathematical truths." I think Einstein said it well when he said words to the effect that, to the extent that we can know something is true, it can't be real, and to the extent that something is real, we can't know it to be true. So I think this characterization of mathematics is internally inconsistent if typical meanings of the words are used, and it only becomes consistent if the meanings are chosen to make it tautological. But one can still hold this view if one rejects the idea that knowing is epistemologically different from observing-- I would say that stance implies that the mind is in some sense "more perfect" than the senses, which is fundamentally rationalistic.
5) Physism - Mathematics is based on the patterns humans gleam from studying the physical world.
This is the counterpart to #4, and is based in empiricism. I think it must be true that some of the skills a mathematician uses, including the rules of logic, "make sense" because of studies of the physical world. But it begs the question to notice this and conclude that the physical world is where math comes from, expressly because if one holds that the physical world comes from math, then it would be natural to find math in the physical world. What's more, math clearly extends beyond the physical world, because we can prove theorems that we already know don't hold in the physical world, and yet it is still math. So it's not really meeting the challenge to say what math is "based on", our goal is to say what it is.
6) Fictionalism - Mathematics is just a made-up story that has its own internal logic.
This one is hard to parse from #1 and #2, so if we are to give it the status of a separate possibility, we must stress the "made up story" part. This seems to imply that we use math in the way we use fictional stories, as essentially an entertainment for our imagination that can convey some life lessons by embedding "morals" into the stories. But this again overlooks the role of axioms and postulates, and it just tacks on "has its own internal logic" as if that was a detail of little importance. But the role of axioms and postulates in math go way beyond the desire to tell a fantasy story-- they are a means to assess the validity, usefulness, or even just aesthetic appeal of a set of axioms by assessing the set of theorems they lead to. The purpose seems quite a bit different than the purpose of a fictional story, though the similarity to #1 and #2 means we cannot completely discount this element of what mathematics is and does.

So I see significant failings in all the above views of what math is, though I don't think any miss the boat completely. What's not clear to me is that it needs to be just one of those things, any more than I need to be just my career, or just my family status, or my age or height. I am a lot of things at once, and so is mathematics. So I put "other", because "all of the above" was not an option.

All of the above would be contradictory. It's like choosing between:

(A and not B)
or
(A and B)

you seem to choose A and B, which explicitly rules out 1), 2), and 6). let's say 1) 2) and 6) are:

1) Just A = A and not (B,C,D, or F)
2) Just B = B and not (A, C, D, or F)
6) Just F = F and not (A,B,C, or D)

wheras:

3) At least C
4) At least D
5) At least E

so I think the view that they're all valid (ABCDEF) is most consistent with 3), 4), and 5).

I think one of the problems with asking this question about mathematics is people have to clarify their definition of mathematics: do they mean the field of study, or some ideal construct within the field? Does the question assume that there's a single congruent set of axioms that can be called mathematics, or is mathematics a patchwork of logical clay?

If you were a purist 3), you may think mathematics is only a field. A really lucky field:
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
(apologies if that's already been posted)

A purist 4) might think that their is a human discipline called mathematics, but that the question (and the discipline) are really about the natural phenomena being studied: math, something that exists independently of our discovering it.

5) without 4) doesn't seem much more than 3) to me, but regardless, all three of them aren't exclusive statements, so they're somehow compatible.

I think it's a rational stance that mathematics is a discipline that involves a little bit of discovery and a little bit of invention. For me, that's 4) and 5). 3) too, I guess, but I'm not able to understand how that's different from 5). 5) seems to be a specific case of 3) to me.

 

[Hurkyl]

I might as well chime in with my opinions.

Logicism - Mathematics is reducible to logic, and mathematical truths are just tautologies​

While I believe this particular statement to be true, I don't believe logicists have the same notion of what 'logic' is that I do.

IMO, Zermelo-Fraenkel set theory -- or similarly, Topos theory -- is logic. Not reducible to logic, but actually is a form of logic. In particular, it is a systematic way to reduce higher-order logic to a first-order theory.

Furthermore, I believe in the importance of model theory, even in foundational issues. I get the impression a logicist who reduces Peano arithmetic to logic would say "and *that* is what arithmetic is". However, the only importance I ascribe to the reduction is as a relative consistency proof and an example of a model -- and we typically want to consider many different models of arithmetic.

Additionally, I believe that foundations need to be coherent rather than reductionist -- we can reduce computation with strings to arithmetic, and arithmetic to set theory. We can develop set theory in formal logic. We can develop formal logic from the theory of computation with strings. We can navigate the whole circle, and IMO we must navigate the whole circle: even if we were to take formal logic as foundational, we still need to walk around the circle and then study the formal logic developed by the computation of strings which is internal to the set theory that we developed from foundational logic.

(and really, computation with strings is probably more foundational than formal logic. )

Formalism - Mathematics is just a meaningless symbolic game that happens to be useful​

I definitely claim to be a formalist. But there are two aspects that need to be paid attention to, and kept separate as appropriate.

The first aspect is syntax -- i.e. of "form" -- and is, unfortunately, the only part that people tend to think about. I believe the ideal in mathematical argument is that we have a game with all of the rules laid out beforehand, and mathematical arguments are just following the rules of the game from a starting point to a desired result.

The second aspect is semantics -- i.e. of "meaning". The act of interpreting the components of the game as referring to "something".

I think many of these philosophical issues are simply because either don't recognize the second aspect at all, or fail to see the value in the mental process of abstract thought.

Intuitionism/Constructivism - Mathematics is an arbitrary invention of the human mind/brain​

While the summary I agree with as a formalist, the intuitionist goes further than this statement. Honestly, when I see an intuitionist argue, it really just looks like he's studying the theory of computation with arrogant disdain for other fields of study.

Platonism - Mathematical truths are truths ABOUT something objectively real, like "Platonic heaven"​

This seems completely silly to me. I think it's more of a psychological failure: an error in the process of mental abstraction.

It is somewhat common in mathematics that one would have (or is developing) an abstract concept / intuitive idea, and then is seeking to lay out the rules of the game that capture it. I think Platonism is the failure to recognize the abstract concept as an abstraction, instead transplanting it to "reality"

To a lesser extent, it could be language restricts thought, and they have difficulty mentally separating the mathematical symbol $\exists$ from the common English word "exist".

Physism - Mathematics is based on the patterns humans gleam from studying the physical world​

This is true! To some extent. Much mathematics is developed to describe the things we see in the physical world. But there are two problems:

• This philosophy overlooks the fact that much mathematics is developed to describe things we see in other places too, such as mathematics itself.
• I believe the philosophy goes further and believes that mathematical truth is not derived from logic, but instead true statements about the physical world. Logic is just a trick that has a good success rate, but you can't always trust it.

I think this philosophy can be dangerous, as it leads to a vicious cycle. The physicalist doesn't see the value in taking a claim from "physical reasoning" and adding it to the mathematical game. Then, they turn around and point out that this very important claim is not an aspect of the mathematical game, and thus we cannot place too much value on the mathematical game.

So it creates a sort of self-reinforcing loop where the person devalues mathematics, causing them to do mathematics "improperly", which in turn gives them more reason to devalue mathematics.


That said, invoking "physical reasoning" is important, and so is its analog in other disciplines as well. Mathematicians invoke their intuition all the time.


I've run out of time so I have to stop here.

 

[apeiron]

Having asked you for an example, I'm bound to justify this claim by giving examples myself. Here are two: First trick: Fluvial erosion; the more erosion, the bigger becomes the catchment area; which enhances erosion... Hence rivers.

Exactly. And this intuitive view of how boundary conditions arise in nature is the basis of dissipative structure theory.

If you want to check out the current state of modelling, Bejan's constructal theory is probably a good place to start.

http://en.wikipedia.org/wiki/Constructal_theory

http://www.constructal.org/en/art/Phil.%20Trans.%20R.%20Soc.%20B%20(2010)%20365,%201335%961347.pdf

About the philosophy of mathematics: I'm still not quite clear on what you mean by "constructing constraints". A specific example of what you call a mathematical constraint that has been constructed would help me. And an explanation of how it produces control, and of what it controls would be appreciated. You're not just talking of human invention ... or are you?

Human inventions are where it is obvious. Simple examples would be the way an engine cylinder traps and directs the explosion of a fuel/air mixture, or the way a NAND logic gate is designed. The information bound up in the structure is a constraint that organises a dissipative flow to achieve some end.

And then the broader claim is that all of nature is based on dissipative structure, so divides into local degrees of freedom in interaction with global constraints.

Maths is then a language that is very good for describing constraints in terms that make them easy to build. As the science of patterns, it tells us how to create desired patterns.

Maths more traditionally thinks it deals in abstract objects - an integer or polygon is something that Platonically exists. And in our minds, this is certainly an easy way to treat the elements of maths. They can be pictured as just objects with sets of properties - the sets of properties that then imply the kinds of operations that these objects will participate in. A number has addibility. A square has tileability.

But I am saying this is misleading. This is a reification, or as Whitehead would have it, the fallacy of misplace concreteness.

Nature itself has no objects, just processes. And a process in turn is the result of global constraints in interaction with local degrees of freedom. Or form in interaction with substance if you want to be more classical about it.

So maths focuses on the question of form, of global pattern or organisation - on constraints. For humans, or any kind of life, to make something happen, they need to find some energy gradient and then construct some kind of structure to channel the flow in useful fashion.

This really was not that obvious at the beginning of maths. It took a while for maths to become invaluable in this way. And it could even be said there is not much wrong with considering a triangle to be an abstract object. That is still the easiest way to think about it.

However if the question is about the basis of maths, then a problem arises because an abstract object clearly does not exist in the outside world, and yet it seems that it must exist somewhere - because existence is the most fundamental property of any object.

Once you step back to seeing that maths describes forms - the shapes or relationships that constrain things to be what they are - then this existence dilemma vanishes. Constraints are always something that actually have to be built materially to really exist. Otherwise they just are ideas. So the idea of a triangle describes a process for restricting material reality in some definite manner.

 

[apeiron]

So I see significant failings in all the above views of what math is, though I don't think any miss the boat completely. What's not clear to me is that it needs to be just one of those things, any more than I need to be just my career, or just my family status, or my age or height. I am a lot of things at once, and so is mathematics. So I put "other", because "all of the above" was not an option.

You are right in all the criticisms. But it seems too extreme to claim that maths is then so unsystematic that it must be treated as a mereological bundle.

In your analogy, your height, age, family status, etc, are "just properties". They are bound by belonging to a common object, but they share no necessary connections.

Yet we wouldn't even be debating this if we didn't have a strong feeling that maths has a systematic basis.

And again, in metaphysics, the systematic basis of things is always going to be dual - to be definitely something is to be definitely not everything else that it isn't.

So for instance, as with axioms, we can say maths is ultimately subjective, but it attempts to limit that subjectivity as much as possible. If something has to be simply assumed to get the game started, well we will make that plain and then go on from there.

In this way, axioms are A and not-A. They are subjective truths, but treated as objective ones. So both, say, constructivism and Platonism are correct, even if mutually contradictory. It is all subjective, but as little subjective as possible. It is not objective, but so near as damn it that any imaginable knower would come to the same truths.

A philosophy of maths would then just want to come up with a better way of capturing the essential dynamic than this tired old list of -isms that want to accept only one side of the story.

 

[Ken G]

you seem to choose A and B, which explicitly rules out 1), 2), and 6). let's say 1) 2) and 6) are:

1) Just A = A and not (B,C,D, or F)
2) Just B = B and not (A, C, D, or F)
6) Just F = F and not (A,B,C, or D)

wheras:

3) At least C
4) At least D
5) At least E

I would say that formal logic is inappropriate for assessing degrees of truth. For example, I see no problem in someone saying "the truth is a combination of (1) and (3)". That would be the case if, for example, they held that logic itself was an arbitrary invention of the human brain. They would argue that (1) is "partially true", because it focuses on the importance of logic but misses where logic comes from, and (3) is "partially true", because it focuses on where the rules are coming from but fails to recognize the importance of a particular non-arbitrary set of rules. That's how I feel about the entire list. It as though each said "an elephant is an animal with a trunk", or, "with tusks", and so on-- the truth is in the combination of them all. They only become false if they assert that an elephant possesses only those qualities, at which point they become formally false, but still "partially" true in that they do identify aspects of an elephant.

I think one of the problems with asking this question about mathematics is people have to clarify their definition of mathematics: do they mean the field of study, or some ideal construct within the field? Does the question assume that there's a single congruent set of axioms that can be called mathematics, or is mathematics a patchwork of logical clay?

Yes, different assumptions about what the question is asking can lead to different choices. In a way, that might be a feature, not a bug-- how we interpret the question "what is mathematics" speaks as much about our perspective as our answer does.

5) seems to be a specific case of 3) to me.

I'd say the difference between (3) and (5) is the contrast between the words "arbitrary" and "physical world." If we hold that the world is not arbitrary, we must take a stance and choose between (3) and (5). However, we can also have them both, if we simply recognize that the justification for doing mathematics depends on (5), which then affords it some "leeway" to become (3). There are plenty of mathematicians who try not to do anything that has any connection to reality, but the reason they get paid is that oftentimes they don't succeed.

 

[Ken G]

Yet we wouldn't even be debating this if we didn't have a strong feeling that maths has a systematic basis.

I would say it's not so much that we believe that, it's that we would like to find a way that let's us believe that. But we just don't succeed, if we are honest to ourselves we are forced to admit that math really is, as you put it, a kind of "bundle" of various different motivations and goals. We can certainly tell it is math if it chooses axioms and postulates and connects them to theorems using logic, but there are still many different things that process could be described as. In your human analogy, we can say that we know a human when we see one, but we would find it rather difficult to settle on a consensus definition of what a human fundamentally is. (Genes? Appearance? Behavior? Attitude? Intelligence? No, too much variation in all of those, they are each a continuous spectrum that we can bundle only because we don't encounter "borderline cases".)

And again, in metaphysics, the systematic basis of things is always going to be dual - to be definitely something is to be definitely not everything else that it isn't.

But how can we hold that as our standard? If I were to try and define what a human is, perhaps in some pangalactic population of aliens of all kinds, could I really make such a dual description work?

It is all subjective, but as little subjective as possible. It is not objective, but so near as damn it that any imaginable knower would come to the same truths.

A philosophy of maths would then just want to come up with a better way of capturing the essential dynamic than this tired old list of -isms that want to accept only one side of the story.

I agree with that, except I would say that any such attempt will merely devise yet one more "ism", which will end up being shown to be just as "tired" as the rest in time. Math is a bundle of different things that all share a basic structure, so we can define what math is in terms of being able to recognize it (using logic to prove from axioms and postulates), but we don't get a philosophy about what that "fundamentally is" without noticing all the goals and motivations that go into that bundle. If the Platonic view could really hold water, one could just settle on that and it would underpin all the rest, but that view comes with its own internal inconsistencies, so I would argue cannot stand alone as the whole truth.

 

[bohm2]

But we just don't succeed, if we are honest to ourselves we are forced to admit that math really is, as you put it, a kind of "bundle" of various different motivations and goals. We can certainly tell it is math if it chooses axioms and postulates and connects them to theorems using logic, but there are still many different things that process could be described as. In your human analogy, we can say that we know a human when we see one, but we would find it rather difficult to settle on a consensus definition of what a human fundamentally is. (Genes? Appearance? Behavior? Attitude? Intelligence? No, too much variation in all of those, they are each a continuous spectrum that we can bundle only because we don't encounter "borderline cases".) But how can we hold that as our standard? If I were to try and define what a human is, perhaps in some pangalactic population of aliens of all kinds, could I really make such a dual description work?I agree with that, except I would say that any such attempt will merely devise yet one more "ism", which will end up being shown to be just as "tired" as the rest in time.

Some authors have used that type of argument for suggesting that math ability like other cognitive abilities are biologically-given, innate structures. This is a long quote but it kind of summarizes this type of argument/point:

Crucially, even the simplest words and concepts of human language and thought lack the relation to mind-independent entities that appears to be characteristic of animal communication. The latter is held to be based on a one-one relation between mind/brain processes and “an aspect of the environment to which these processes adapt the animal's behavior,” to quote cognitive neuroscientist Randy Gallistel, introducing a major collection of papers on animal communication (Gallistel, 1990).

According to Jane Goodall, the closest observer of chimpanzees in the wild, for them “the production of a sound in the absence of the appropriate emotional state seems to be an almost impossible task” (Goodall, cited in Tattersall, 2002). The symbols of human language and thought are sharply different. Their use is not automatically keyed to emotional states, and they do not pick out mind-independent objects or events in the external world. For human language and thought, it seems, there is no reference relation in the sense of Frege, Peirce, Tarski, Quine, and contemporary philosophy of language and mind.

What we understand to be a river, a person, a tree, water, and so on, consistently turns out to be a creation of what 17th century investigators called the human “cognoscitive powers,” which provide us with rich means to refer to the outside world from intricate perspectives. As the influential neo-Platonist Ralph Cudworth put the matter, it is only by means of the “inward ideas” produced by its “innate cognoscitive power” that the mind is able to “know and understand all external individual things,” articulating ideas that influenced Kant. The objects of thought constructed by the cognoscitive powers cannot be reduced to a “peculiar nature belonging” to the thing we are talking about, as David Hume summarized a century of inquiry. In this regard, internal conceptual symbols are like the phonetic units of mental representations, such as the syllable [ba]; every particular act externalizing this mental object yields a mindindependent entity, but it is idle to seek a mind-independent construct that corresponds to the syllable. Communication is not a matter of producing some mind-external entity that the hearer picks out of the world, the way a physicist could. Rather, communication is a more-or-less affair, in which the speaker produces external events and hearers seek to match them as best they can to their own internal resources. Words and concepts appear to be similar in this regard, even the simplest of them. Communication relies on shared cognoscitive powers, and succeeds insofar as shared mental constructs, background, concerns, presuppositions, and so on, allow for common perspectives to be (more or less) attained. These properties of lexical items seem to be unique to human language and thought, and have to be accounted for somehow in the study of their evolution. How, no one has any idea. The fact that there even is a problem has barely been recognized, as a result of the powerful grip of the doctrines of referentialism.

Human cognoscitive powers provide us with a world of experience, different from the world of experience of other animals. Being reflective creatures, thanks to the emergence of the human capacity, humans try to make some sense of experience. These efforts are called myth, or
religion, or magic, or philosophy, or in modern English usage, science. For science, the concept of reference in the technical sense is a normative ideal: we hope that the invented concepts photon or verb phrase pick out some real thing in the world. And of course the concept of reference is just fine for the context for which it was invented in modern logic: formal systems, in which the relation of reference is stipulated, holding for example between numerals and numbers. But human language and thought do not seem to work that way, and endless confusion has resulted from failure to recognize that fact.

The Biolinguistic Program: The Current State of its Evolution and Development
http://www.punksinscience.org/klean...L/material/Berwick-Chomsky_Biolinguistics.pdf

 

[Ken G]

Yes, that's an interesting parallel between mathematics and linguistics. We could as easily be having a discussion on our "philosophy of what language is", and we would probably meet many of the same concepts. I think it's hard not to end up concluding that language, and mathematics, are a kind of intellectual behavior, above all, and the behavior can be lumped together under the headings of language or mathematics by noticing certain defining attributes, which only tell us how to recognize them-- not what they actually are. To understand what the behaviors are, we may need to understand better the creature doing them, and the complete context of what they are being used for.

So I might summarize my "other" category as, "mathematics is a bundle of human behaviors, recognizable by certain well-known attributes, that succeeds at and has the purpose of, elements of all 6 categories listed in the poll." What's more, I would argue that efforts at identifying a more "pure" form of what mathematics is, would be no more successful than trying to identify a more "pure" form of something like "love", rather than thinking of it as a bundle of different behaviors that are recognizable by virtue of certain common elements.

 

[apeiron]

We can certainly tell it is math if it chooses axioms and postulates and connects them to theorems using logic, but there are still many different things that process could be described as.

Well, here you are already accepting that maths is a systematic process rather than a mere bundle of properties (although there is no reason that a process can't generate an entity or state with many properties of course).

And what I was arguing is that in philosophy, any claim to universality is only ever understood in terms of its complementary. Thesis and antithesis, dialectics, etc.

So the process you are describing is deduction. And its complementary process is induction.

Thus if the basis of maths is deduction, then it is not induction. But then of course, axioms are the result of induction. And in fact, deduction is inverse induction. So even the logical method is the same, just reversed.

Induction procedes by generalisation - the successive or hierarchical relaxation of constraints. And deduction procedes from generalisations or axioms by the addition of constraints. Particular truths are derived by showing how they are contained within the more general truths, by demonstrating that a particular instance is a variety of the general instance.

So as I say, you end up with the usual philosophical tale of things seeming both the same and different. Maths is maths to the extent it is deduction and not induction. But it cannot help but also be about induction in avoiding being induction. Which is why it becomes so hard to catergorise in terms of any monadic -ism such as constructivism or Platonism. As soon as you assert the dominance of one pole of its being, you draw attention to the contradictory pole of its being - the context or antithesis necessary to make it definitely that thing.

Even your bundle approach can't help but fall into this mould. You are arguing that if maths cannot be just one kind of thing, then it must be many things. The classic dichotomy of the one and the many. So you are just again trying to place maths at one definite pole of existence by appealing to an argument based on what it is not.

In your human analogy, we can say that we know a human when we see one, but we would find it rather difficult to settle on a consensus definition of what a human fundamentally is. (Genes? Appearance? Behavior? Attitude? Intelligence? No, too much variation in all of those, they are each a continuous spectrum that we can bundle only because we don't encounter "borderline cases".)

A human is certainly a complex instance. But in science/philosophy, we still would break a human down into a nested hierarchy of dichotomous statements rather than treat a human as a property bundle.

So for example, we class humans as ape vs non-ape, mammal vs non-mammal, living vs non-living, etc. So we start with the general idea of a universal category (A is a kind of...) and add a hierarchy of increasingly specified constraints that contain what it means to be human.

As you note, there is still a lot of grey in any such hierarchy of constraint - local degrees of freedom still exist. There are borderline cases because something is always left indeterminate. Though when faced with such cases, we can always - via further dichotomous symmetry breakings - narrow down our descriptions still further.

Are humans defined by having brown hair? Are two legs essential? To make sense of such questions, we would have to step back to some clearly dichotomous basis of judgement. So for instance, we tend to ignore all individual variety because we believe that it is randomly or arbitrarily derived from a common gene pool. We do have some concrete reason based on a distinction between populations and individuals, or genetics and environment.

Math is a bundle of different things that all share a basic structure, so we can define what math is in terms of being able to recognize it (using logic to prove from axioms and postulates), but we don't get a philosophy about what that "fundamentally is" without noticing all the goals and motivations that go into that bundle.

OK, there is no problem with maths resulting in some great variety of outcomes if, as you seem to agree, there is a single shared basis in a process. And my argument in turn is that this process is sharply defined, philosophically based, to the extent that it is not some complementary process. If you say deduction, I say induction

Now you raise the further question of goals and motivations. Are these also a necessary part of the philosophical basis?

I certainly would say so. And so would look for the dichotomy that would be at their root (causing the usual confusion about "what is fundamental").

The rival poles so far as goals/motivations go would seem to be knowledge vs control. Or maximum information vs minimum information.

Is maths true because it knows everything or because it refines information about the world to its least principles? Do humans create maths because it is pure truth, or powerfully useful?

Of course, there are a spectrum of positions that can be taken once the complementary poles of possibility are defined. And as soon as anyone heads over towards one extreme, they draw attention to the counter point of view. If your goal is truth, then my equally valid goal is control.

However, the basis to the philosophising is then the underlying dichotomy, not the spectrum of divisions that it so happily supports.

 

[NewtonianAlch]

Might be a combination, but if I had to pick one it would be something close to Platonism. It's part of something that's objectively real and something far greater than what we know right now.

 

[Paulibus]

Thanks for your clarification, Apieron (Post # 76). The only quibble I have is that mathematics is just one of the many "machineries" for constructing "constraints" that evolution has endowed human creatures with. But liked your classification of mathematics as a describer of forms:

... maths describes forms - the shapes or relationships that constrain things to be what they are ...Constraints are always something that actually have to be built materially to really exist. Otherwise they just are ideas. So the idea of a triangle describes a process for restricting material reality in some definite manner.

I'd argue that description is what a language does and that languages are inventions created under the forcing rule of evolution, as it were (e.g. birdsong, baboon leopard alarms, French and Swahili) and not eternal Platonic stuff (and nonsense, in my view).

Why on Earth Maths as a language was not part of the O.P. list is hard to understand. Perhaps only because this option is not a philosophy-jargon ".. ism"?

 

[lugita15]

Why on Earth Maths as a language was not part of the O.P. list is hard to understand.

That would be formalism.

 

[Ken G]

I think we can parse the difference between language and formalism. Formalism says that math is syntactic, whereas language is both syntactic and semantic. Formalism expressly uses the word "meaningless", which differentiates it clearly from language. In my view, "meaning" (that which is "semantic") implies connections between what is unfamiliar to what is familiar. That is the job of a dictionary, to make those connections, but what graduates it to the level of "meaning" is the necessity that there actually be common familiarities. If I shout so loud in your ear it causes you pain, that isn't language, that's just the effects of sound. To be language, you have to mentally process my input, by assessing a grid of familiar experiences, and drawing semantic connections. That's "meaning."

So I would say that language is also a combination of every element on the list-- it is logical and formal (because of its connection to syntax, though it is not completely either one because the syntax of language is very sloppy), it is intuitive because we clearly invented it, it is Platonic because we like to imagine the words we use correspond to real things, it is physical because of its reliance on familiarity of experience, and it is fictional because it is capable of combining words in purely inventive ways. So that's why I think we should see that list, not as alternatives, but as building blocks, each imperfectly evidenced in any intellectual endeavor in which human cognition is involved-- including both language and mathematics. Since language and mathematics are built from similar types of building blocks, it's not surprising that we can see parallels between them as well.

 

[Paulibus]

Lugita 15: taxonomy. I've had a look at the entry on Formalism re
Mathematics in the Stanford Encycolpedia of Philosophy. This long and erudite entry doesn't seem to use the simplicity of calling maths, as a language; formalism. Apparently formalism "is often the position to which philosophically naïve respondents will gesture towards, when pestered by questions as to the nature of mathematics." I therefore stand chastised, but not further informed!

 

[bohm2]

While there are criticisms for this position, there are some linguistics/psychologists who believe that mathematics is derivative/parasitic from our language ability:

The classic illustration is the system of natural numbers. That brings up a problem posed by Alfred Russell Wallace 125 years ago: in his words, the “gigantic development of the mathematical capacity is wholly unexplained by the theory of natural selection, and must be due to some altogether distinct cause,” if only because it remained unused. One possibility is that it is derivative from language. It is not hard to show that if the lexicon is reduced to a single element, then unbounded Merge will yield arithmetic. Speculations about the origin of the mathematical capacity as an abstraction from linguistic operations are familiar, as are criticisms, including apparent dissociation with lesions and diversity of localization. The significance of such phenomena, however, is far from clear; they relate to use of the capacity, not its possession. For similar reasons, dissociations do not show that the capacity to read is not parasitic on the language faculty.

Some simple evo-devo theses: how true might they be for language?
https://docs.google.com/viewer?a=v&q...gO76OQ4A&pli=1

 

[Ken G]

I feel the danger is overgeneralization. It's true that we need to see connections, and we need to idealize to improve simplicity, but statements like "mathematics is a subset of language" or "we can do mathematics because we evolved to do language" just seem too oversimplified. We can find similarities between elephants and walruses, like they both have tusks, and end up calling both "mammals", without claiming that elephants are examples of walruses or stem from the same evolutionary channel that gave us walruses. They are what they are, and to understand them, we choose various different angles from which to look at them, but every angle tells us various different attributes, and a combination of all the angles and all the attributes is how we know what these things actually are.

 

[bohm2]

I feel the danger is overgeneralization. It's true that we need to see connections, and we need to idealize to improve simplicity, but statements like "mathematics is a subset of language" or "we can do mathematics because we evolved to do language" just seem too oversimplified.

The argument (at least with those who view mathematics as a cognitive module of our mind/brain) is that both language and mathematics have the property of "discrete infinity" and since this property may be unique in the biological world, perhaps our mathematical ability may have developed as a by-product of the language faculty. Some authors like Butterworth question this, however:

Cognitive development reflects neural organization in separating language from number. Indeed, the ontogenetic independence of the number domain has been argued vigorously by the authors of many previous publication looking at both normal and abnormal development of numerical abilities. It would be surprising if there were no effects of language on numerical cognition, but it is one thing to hold that language facilitates the use of numerical concepts and another that it provides their causal underpinning.

Number and language: how are they related?
http://www.mathematicalbrain.com/pdf/GELMANTICS05.PDF

 

[Ken G]

Yes, Butterworth's final sentence seems to echo the concern against oversimplification. And come to think of it, we've all seen people who were terrific in letters but horrible in math, and the converse, so that would seem to suggest some significantly different qualities.

 

[apeiron]

...both language and mathematics have the property of "discrete infinity" and since this property may be unique in the biological world...[/url]

And also genes. You can spin an unlimited number of proteins from combinations of amino acids. So actually, this is general to life and mind.

Yes, Butterworth's final sentence seems to echo the concern against oversimplification. And come to think of it, we've all seen people who were terrific in letters but horrible in math, and the converse, so that would seem to suggest some significantly different qualities.

Yes, the developmental disorder of dyscalculia has become well recognised over the past decade - http://en.wikipedia.org/wiki/Dyscalculia

But brains aren't evolved to do maths, any more than they are to do writing or play musical instruments. So dyscalculia is about a more general deficit in visuospatial imagination. The kind of intuitive feel for complex groupings and temporal relationships that is needed to make maths easy to learn.

That is, it is more a semantic than a syntatic issue for those with dyscalculia. Syntax handling happens in a quite different part of the brain, the frontal premotor cortex, or Broca's area.

 

[Paulibus]

Bohm2, @88: What Russel said long ago about mathematics shows how conservative even such a heterodox thinker could be. It’s ironic that the co-founder of evolutionary theory should be so impressed by our supposed “gigantic development of the mathematical capacity” that he would overlook the possibility that such capacity might be humbler than he imagined. But when communicated as a language represented by squiggles on paper, even as mundane an invention as natural numbers unexpectedly turned out to rise and rise, as it were, into today’s mathematical complexity. Perhaps the key trick here was the invention of recorded communication, starting with tally scratches on one’s arm and moving on through Roman numerals to Pauli (alas, not Paulibus) spin matrices.

Thanks also for the interesting link to the Munduruku´ Indian stuff. They have interesting sexual practices too.

 

[John Creighto]

Yes, Butterworth's final sentence seems to echo the concern against oversimplification. And come to think of it, we've all seen people who were terrific in letters but horrible in math, and the converse, so that would seem to suggest some significantly different qualities.

We tend to process language automatically (see automaticity), were in math we tend to consciously apply rules (e.g. axioms). This isn’t a black and white distinction as the more we do math the more automatic it comes. Math tends to deal with a much smaller set of ideas at a time. For instance, consider the number of rules you would apply in a typically proof vs say the amount of different words used in a book.

In language the words directly relate to something in our intuition, whereas in math we often address problems denotationally (that is we abstract away the meaning). Math requires us to consciously, construct representations, of ideas (for instance as a line in a graph) whereas in language our internal representation of worldly things is done instinctually through sensory induction. Math is very consistent, whereas in language we must learn to handle many exceptions to the rules.

For males our, semantic understanding, seems to be usually highly tied up with our sensory processing. For instance some people think in terms of how words sound while others think better in terms of how words are spelled. Because of this men often need to hear and read something to learn it well, where many women only need to do one or the other because for most women their brain separates the semantics better from the sense data.

For math it is not clear if this separation is a benefit or a hindrance because abstracting away the meaning is important for math but at the same time visual intuition can help gain understanding of such things as: functions and principles of geometry. Additionally relating equations to things you know like sounds could possibly help with remembering them.

So perhaps while there are a lot of similarities between the two but when looking at each in the concrete there are lots of qualitative differences.

 

[Ken G]

Yes, the developmental disorder of dyscalculia has become well recognised over the past decade - http://en.wikipedia.org/wiki/Dyscalculia

I hadn't heard of that before, that's interesting. Makes sense, if there's dyslexia, there should be dyscalculia as well.

That is, it is more a semantic than a syntatic issue for those with dyscalculia. Syntax handling happens in a quite different part of the brain, the frontal premotor cortex, or Broca's area.

That would gibe with the fact that there appears to be two flavors of math disability, one centered more around abstract thinking we might associate with mathematical semantics, and one centered more around simple calculations that we might associate with mathematical syntax. Since one person can apparently suffer from one but not the other, I think this also provides a neurological take on the idea that "math" is not just a single thing the brain is doing, but rather a complex combination of different skills. There's no "math bone" in there anywhere, and that is also why we cannot pick a single descriptor for what "math is" from the poll list.

 

[bohm2]

That would gibe with the fact that there appears to be two flavors of math disability, one centered more around abstract thinking we might associate with mathematical semantics, and one centered more around simple calculations that we might associate with mathematical syntax. Since one person can apparently suffer from one but not the other, I think this also provides a neurological take on the idea that "math" is not just a single thing the brain is doing, but rather a complex combination of different skills. There's no "math bone" in there anywhere, and that is also why we cannot pick a single descriptor for what "math is" from the poll list.

I don't have a clue about brain injury and effects on particular math abilities but in language one can find such dissociation but it may be that effects relate more to use of the capacity versus it possession (performance not competence). I'm not sure how strong the evidence is but note this passage:

If the lexicon is reduced to a single element, then Merge can yield arithmetic in various ways. Speculations about the origin of the mathematical capacity as an abstraction from linguistic operations are familiar, as are criticisms, including apparent dissociation with lesions and diversity of localization. The significance of such phenomena, however, is far from clear. They relate to use of the capacity, not its possession; to performance, not competence. For similar reasons, dissociations do not show that the capacity to read is not parasitic on the language faculty, as Luigi Rizzi points out.

Approaching UG from Below
http://www.punksinscience.org/kleanthes/courses/UCY10S/IBL/material/Chomsky_UG.pdf

With respect to different math abilities I always thought the concept of number versus concept of space may be potentially dissociated? Interestingly, this study argues that math ability, at least number ability is innate:

You Can Count On This: Math Ability Is Inborn, New Research Suggests
http://www.sciencedaily.com/releases/2011/08/110808152428.htm

 

[Storm89]

Being as new to physics and math as I am (despite having taken Math at B-level (C, B, A here in Denmark pre-uni), I think at the current point that it holds a bit of this and that and wouldn't know where to place my vote fully. I do believe in logic, but also that had we originally defined 1 as being 2, we would have just gone on from that as if nothing had happened. In which case 2 might have been 4 etc. So in some sense I feel that it's man-made as well.

I had this discussion with a friend not too long ago and I was thinking that the only time we can probably truly say if our math is universal is when we've met a couple of other civilisations and see whether or not they have come to similar conclusions.

So on that, math as most other things could be and probably is in constant development. Who's to say what it will be like in 20.000 years?

Again though, feel free to shoot the above down as I don't have the required math knowledge to really say anything.

 

[apeiron]

There's no "math bone" in there anywhere, and that is also why we cannot pick a single descriptor for what "math is" from the poll list.

But the brain is systematic in its organisation, not some arbitrary bundle of processing modules. It makes sense of the world via dichotomous or complementary analysis. For instance, you have the left/right hemisphere divide for focus~context, the ventral/dorsal divide for object~relationship, the frontal/posterior divide for motor~sensory, the prefrontal/striatal for attention~habit.

The same divisions are found within areas. The prefrontal is split into outwardly attending dorsolateral and inwardly attending orbital. And all the way down to neural integration level. Colour perception, for example, depends on opponent channel processing - red~green and yellow~blue.

So there is a deep principle, that also seems Platonic because it is hard to imagine any way that it could be done differently. Even an alien brain would have to dichotomise its world - analyse it in terms of complementary extremes. Differentiate so as to be able to integrate.

Coming back to the argument that maths is a language, and the brain handles it like a language, I take this to be true. The same networks light up, the same divisions - like syntactic handling vs semantic access - rule.

The same is the case for music as well. The brain treats it as a language - though with the expected differences in emphasis, such as a greater right brain activation for the prosodic aspects of what is being heard.

Stepping back again to the general questions posed by the OP, I repeat that there are three ways to view the possible answers.

You can try and make just one choice right - a Platonic uniqueness and perfection. You can go the other way and say it is a bit of everything - an arbitrary bundle with no deep structure. Or you can seek out the dichotomies that underpin systematic relationships, that can give you complex hierarchical variety as a result of deep process.

The dichotomies that the poll list touches on are primarily the necessary epistemic distinction between our models and the world. And then the general ontological distinction between material and formal cause. And then - which is where it gets tricky - the "epistemology as ontology" distinction, or semiotic distinction, between information and dynamics. The epistemic cut which is the deep structure of all "languages", genetic or otherwise, and allows for the construction of constraints, a formally modeled control over the material organisation of reality.

So the world just is a mixture of its materials and its forms, its constructions and its constraints.

And then we model that in a fashion that allows us to work out how to construct constraints - to be local actors taking globalised control over material flows through a "language".

This all seems Platonic because the interactions of the world are self-constraining and so cannot help but fall into regular, repeating, patterns. They have no choice.

And while in our heads, the realm of subjective modelling, we are free to choose, in fact that freedom is reduced to a choice about axioms. After that, syntax takes over and there is only deductive reasoning. Constraints in the form of allowable operations are rigidly imposed, and again everything falls into inevitable outcomes.

So maths could quite easily generate nonsense as much it generates truth. At least in terms of its ability to talk about reality.

But modelling itself is a constrained activity. It is constrained by the measurement of models against the world. And note the dichotomistic nature of the measurement process. You have both a generalised measurement in the formation of axioms (axioms are what seem reasonable as a result of empiricism or inductive experience). And then the particular measurements that are the checking for a match between the predictions of some actual model and how the world behaves (when it has been constrained in the fashion prescribed by the model).

To boil it down to a "philosophy of maths", maths is a modelling relationship with the world. And modelling in general involves an epistemic cut made possible by a machinery of language - a syntax for constructing constraints, an ability to stand back from the world so as to imagine controlling it. Maths is special in this regard because of its almost complete abstraction - it is the least materially constrained of all nature's languages and so has the most formal power.

It is a familiar trick now refined to the nth degree, and Platonic-feeling because it seems the end of the line in terms of how refined it is possible to be.

And yet. There is the achilles heel that the axioms are freely chosen but may not be as secure as people think. There could be some foundational failures built in now - such as an inability to deal correctly with issues of materiality, indeterminacy, causality and scale to name a few that spring to mind. Axioms force a choice, and that has a way of resulting in always telling just one half of the story.

 

[Ken G]

With respect to different math abilities I always thought the concept of number versus concept of space may be potentially dissociated? Interestingly, this study argues that math ability, at least number ability is innate:

You Can Count On This: Math Ability Is Inborn, New Research Suggests
http://www.sciencedaily.com/releases/2011/08/110808152428.htm

That study certainly finds interesting results, that "approximate number sense" in very young children is a predictor of future math ability. But it's easy to make questionable connections from that. For example, they wonder if maybe improving ANS might lead to better math skills later on, but to me that sounds like a classic case of "correlation is not causation." I think it's pretty obvious from experience that math ability is largely innate, and it's interesting that brains that are good in math are also good at developing ANS, but wondering if improving ANS might improve math ability sounds to me a bit like taking the fact that kids who are good at basketball (because they are tall or can jump) are also good at volleyball (because they are tall or can jump), and wondering if training them to play basketball will make them good at volleyball. I just think the brain is very complex, and it's not surprising that being good at one mathematical funciton, like ANS, is a predictor of being good at some other mathematical function, like proving theorems, but only because they both involving manipulations that we recognize as having some common elements.

 

[Ken G]

But the brain is systematic in its organisation, not some arbitrary bundle of processing modules. It makes sense of the world via dichotomous or complementary analysis. For instance, you have the left/right hemisphere divide for focus~context, the ventral/dorsal divide for object~relationship, the frontal/posterior divide for motor~sensory, the prefrontal/striatal for attention~habit.

OK, that's some interesting neurological information. I can accept the value in seeing the functioning in terms of dichotomies, but when you combine enough dichotomies, you have a very flexible and encompassing processor. I see it as a bit like a cooking recipe-- you don't just list the ingredients that are present vs. not present, you also mix in varying amounts of each, for a much wider range of results. Someone who is good in math may require strength on one side of more than one of those dichotomies, so math may require a mixture of different ingredients that the brain must get good at trying out. Maybe one brain "figures out the recipe" for math, while another "figures out the recipe" for foreign languages, or music, or whatever. It doesn't mean these different endeavors are themselves dichotomies, but can be successfully analyzed in terms of a rich enough set of dichotomies to choose from.

So there is a deep principle, that also seems Platonic because it is hard to imagine any way that it could be done differently. Even an alien brain would have to dichotomise its world - analyse it in terms of complementary extremes. Differentiate so as to be able to integrate.

Yes, the power of the yin-yang symbolism again. I agree there is great merit in thinking along those lines. But is it Platonic in the sense that dichotomous juxtaposition is really what is happening, or is that just how we like to think about it? By analogy, any number has a binary digitization, but that digitization is not what the number "really is", it's just a way to think about that number, an arbitrary but successful labeling scheme.

Coming back to the argument that maths is a language, and the brain handles it like a language, I take this to be true. The same networks light up, the same divisions - like syntactic handling vs semantic access - rule.

The same is the case for music as well. The brain treats it as a language - though with the expected differences in emphasis, such as a greater right brain activation for the prosodic aspects of what is being heard.

It is those differences in emphasis I would stress, however. We can see enough parallels between math and language, and math and music, just from the nature of each, to expect some similar responses in brain processing. But which is more important for understanding that processing, the major similarities, or the minor differences? I would argue that "the devil is in the details", in much the same way that a human and a monkey have extremely similar DNA, but the differences lead to very different attributes (especially different brain functions).

I have in mind an effect akin to sensitivity to initial conditions in dynamics-- a seemingly small difference is leveraged into an extremely different outcome simply because we don't recognize the significance of the difference. Bedeviled by these small but crucial details, we have as much trouble saying what math is at its core, as we would have saying what music is at its core, because somewhere along the way of the complex brain activity, "a miracle happens", and we get a seminal math theorem or a great work of music, an accomplishment most human brains are incapable of even if they are superficially identical to that of the master.

So the world just is a mixture of its materials and its forms, its constructions and its constraints.

Yes, that's an interesting parallel. The actions of our minds are such a mixture of primarily epistemic functions (what we might call thought) and primarily ontological functions (which we might call neural dynamics). It is common to equate these aspects, but more for a lack of anything better to do that a real good reason. In the vein of "epistemology as ontology", I would instead hold that thought cannot emerge simply from neural dynamics, because it is thought that allows us to analyze neural dynamics in the first place. So neither is the cause of the other, they come together, they need each other to work-- again like yin-yang, a mixture of material and form as you said. That is indeed a theme that runs through the different choices in the poll, but again none of those choices make sense in isolation-- math can't be a Platonic truth any more than a map can be a territory, but similarly a map doesn't mean anything unless there is a territory to map in the first place.

This all seems Platonic because the interactions of the world are self-constraining and so cannot help but fall into regular, repeating, patterns. They have no choice.

But are the interactions of the world really self-constraining as you imagine, or is that just how you make sense of them? We must not beg the question by building the Platonism right in from the start. Instead, we should accept that all we will ever have is a description of what is real, and that description must necessarily be mathematical because that is the description we seek. So we may find value in using a language to help us understand the world, but that is still only going to be the "yin", we still need the "yang" that recognizes our language is an internal language, not an external one. Even the internal/external dichotomy is really a kind of unity, for what is internal to me is external to you, and you may analyze my mind as neural dynamics even as I perceive it as thought.

And while in our heads, the realm of subjective modelling, we are free to choose, in fact that freedom is reduced to a choice about axioms. After that, syntax takes over and there is only deductive reasoning. Constraints in the form of allowable operations are rigidly imposed, and again everything falls into inevitable outcomes.

Yes, another dichotomy that is actually a unity-- the axiom/theorem dichotomy, but axioms mean nothing until they are used to make theorems that allow us to judge the axioms, and theorems mean nothing independently of the axioms that lead to them. It's material/form once again-- the axioms are like the Platonic forms, and their theorems are like the material, the flesh on the axiom's bones. We can't claim that if the axioms are Platonic, then so are the theorems they inevitably lead to, because we can only judge the truth of the axioms by their theorems, since attributing meaning to an axiom is a type of theorem, or consequence, of that axiom. The structure falls apart unless it is anchored at both the form and material end, so we cannot say that math is accessing truth of forms that are independent of the materials, nor can we say that math is a study of the materials without having underlying forms to axiomatize those materials.

And note the dichotomistic nature of the measurement process. You have both a generalised measurement in the formation of axioms (axioms are what seem reasonable as a result of empiricism or inductive experience). And then the particular measurements that are the checking for a match between the predictions of some actual model and how the world behaves (when it has been constrained in the fashion prescribed by the model).

Yes, I think you are also referring to the principle of "anchoring at both ends", which I feel is the fundamental reason that math cannot be just one of the items in the poll, for math is not the sound of one hand clapping, if you will.

And yet. There is the achilles heel that the axioms are freely chosen but may not be as secure as people think. There could be some foundational failures built in now - such as an inability to deal correctly with issues of materiality, indeterminacy, causality and scale to name a few that spring to mind. Axioms force a choice, and that has a way of resulting in always telling just one half of the story.

Bingo, that's why I cannot feel the Platonic picture can provide the whole story.

 

[apeiron]

Maybe one brain "figures out the recipe" for math, while another "figures out the recipe" for foreign languages, or music, or whatever. It doesn't mean these different endeavors are themselves dichotomies, but can be successfully analyzed in terms of a rich enough set of dichotomies to choose from.

True, but this is talking about the divergent variety rather than the convergent deep structure. You do of course have both because what polarities make possible is the emergent spectrum that emerges inbetween (as various mixtures of what gets separated).

Again, you want to argue that models are just arbitrary ideas that we project onto the data. So if my chosen idea just happens to be "dichotomies" then I can go in and carve up some phenomenon in convincing fashion using as many dichotomies as it takes.

I agree that modelling does have an arbitrary, free, basis. We can try whatever works. But then it becomes interesting that only certain ideas seem to work really well, even universally. These ideas look to be the way nature actually works - although we can never "know" that, just observe it to be likely.

Reductionism (that metaphysical mix of atomism, determinism, monadism, mechanicism, local reality, effective causality, etc) is one general idea that works really well.

And then there is the complementary tradition of holism which is about dichotomies, hierarchies, top down causality, indeterminacy, etc. Which works better when it comes time to tell the whole story of course!

Would a dichotomies approach be stronger if all the brain's architectural divisions could be reduced to just a single description? Yes, it would certainly seem less arbitrary (a projection onto the data) and more like the deep structure of the data.

I would start out by saying we shouldn't expect a simple single answer because the brain is a product of both evolution and development. Development is free potential but evolution locks in past history. So the story on brain evolution is a complex interaction between accumulated design and the addition of new possibility (such as by creating new room at the top by expanding the cortex).

But if we step back to the purposes of brains, they are there to make decisions. To make choices. And how can you make a choice unless you have alternatives? And how can you make the most definite possible choices unless the alternatives are dichotomous - reduced to either/or, to a binary yes/no, like retreat/advance, attend/ignore, expected/surprising.

Again, you will probably say that intelligence is defined by having a variety of choices. But as I say, that describes the variety that emerges as a result of the deeper structure - the ability to break the world down by polarities.

My favourite example of the primitiveness of this is the flagella that drives a motile bacterium. Spin one way and the threads tangle, driving the cell forward. The bacterium can follow a chemical gradient, head towards a food source. But then reverse the spin and the threads untangle, the bacterium begins to tumble randomly. So if falling off the scent trail, the bacterium can switch to search or escape mode.

The asymmetry of choice - as determined/random - in a nutshell.

Yes, the power of the yin-yang symbolism again. I agree there is great merit in thinking along those lines. But is it Platonic in the sense that dichotomous juxtaposition is really what is happening, or is that just how we like to think about it?

I agree it is a legitimate question. And the default position will be "all models are the free creations of the human mind". We should be automatically suspicious of any jump from the epistemic to ontic.

But on the other hand, reality must actually have some kind of deep causal structure. It does not seem like an arbitrary bundle of happenings does it? It does seem to have a developmental history, a systemic and patterned materiality. So it is not impossible that our models of its deep structure could be essentially correct.

I have in mind an effect akin to sensitivity to initial conditions in dynamics-- a seemingly small difference is leveraged into an extremely different outcome simply because we don't recognize the significance of the difference.

The butterfly effect is not a good analogy for biological processes because that is dynamics unconstrained (the system is unpredictable even if deterministic because measurement error compounds exponentially).

The whole point of biology (and its use of languages to construct constraints) is that such dynamism is harnessed. Constraints are applied to channel what happens.

There would not be life/mind without this trick of being able to harness dissipation-driven dynamics. So this is why we can say what "math is". It is not some unpredictable consequence of blind evolutionary change, it is instead the very predictable development of the constraint machinery which in fact defines life/mind.

You want to argue that the brain could have evolved any old how. It's just one accident on top of the other. But this is old-style Darwinism (the "modern evolutionary synthesis" of the 1960s). Today you would talk about evo-devo, and this is based on the idea that there are in fact deep structural principles at work. Existence is based on the dissipation of gradients. Life/mind arise as informational structure that locally accelerates the entropification of the Universe.

So there is a deep general principle at work. But then also some happenstance about how things actually work out.

For example, life/mind arose on the back of one kind of language - genes to code for enzymes that could control dynamical chemical cycles. But then H.sapiens stumbled upon actual language - words to control the thoughts that determine our actions.

Was it Platonically inevitable that human grammatical language would arise? Would it have to happen on any planet where some kind of life/mind was happening in sufficient abundance - given enough variety, would some species have to luck into this structural attractor, this pre-existing, ready-waiting, niche?

Personally I would say there is a healthy dose of both - of both random luck and Platonic inevitability. The luck is down to the fact that brain evolution was not headed in that direction. The evolution of an articulate vocal tract - the imposition of a new kind of serial output constraint on vocalisation - looks a pretty chance direction for events to have taken. On the other hand, it was then a very short step for this exaptation to be exploited for symbolic/syntatic purposes. Once there was a species that could chop up a stream of sound into discrete syllables, the machinery for a new level of coding could be used for exactly that.

That is indeed a theme that runs through the different choices in the poll, but again none of those choices make sense in isolation-- math can't be a Platonic truth any more than a map can be a territory, but similarly a map doesn't mean anything unless there is a territory to map in the first place.

We seem to agree then. Because I am saying that maths is not monadically anyone kind of thing. Which is what the poll wants to make it.

And definitely this is all about modelling.

But then, modelling is dichotomous - not just in terms of the relationship between the map and the terrain, but even the map itself has the tension of an internal division.

Our mental mapping of the world divides into ideas and impressions, the theories or formal constructs that are a general inductive understanding, and then the measurements, or expectations, or predictions, that are the local deductive particulars.

Measurement is often claimed to be the objective part of the process of modelling, but of course it always remains some mind's particular impression (such as a reading on a dial, a number on a counter, etc). I know you favour the Copenhagen stance on these things!

So again, where does math stand in all this? It is caught up in the general business of modelling, so it is fictional, intuitive, constructive, etc, foundationally. But at the same time, it is trying to stand at one extreme pole of the modelling process. It is trying to go and stand over at the end of our most general possible ideas. It is trying to be a pure description of form. And then to the extent this division that emerges in our mapping is also true of reality, of the terrain, then maths is going to end up "Platonic".

As I say, this may yet be telling only half the story. But that can only be clear once the foundations of maths is actually clarified.

But are the interactions of the world really self-constraining as you imagine, or is that just how you make sense of them? We must not beg the question by building the Platonism right in from the start.

It should be clear by now that I would only argue for Platonism (the fact that reality has a deep structure which our modelling can hope to map) to the extent that observation appears to confirm it.

Yes, another dichotomy that is actually a unity-- the axiom/theorem dichotomy, but axioms mean nothing until they are used to make theorems that allow us to judge the axioms, and theorems mean nothing independently of the axioms that lead to them. It's material/form once again-- the axioms are like the Platonic forms, and their theorems are like the material, the flesh on the axiom's bones. We can't claim that if the axioms are Platonic, then so are the theorems they inevitably lead to, because we can only judge the truth of the axioms by their theorems, since attributing meaning to an axiom is a type of theorem, or consequence, of that axiom. The structure falls apart unless it is anchored at both the form and material end, so we cannot say that math is accessing truth of forms that are independent of the materials, nor can we say that math is a study of the materials without having underlying forms to axiomatize those materials.

Yes, I agree. You seem to have me now arguing for Platonic fundamentalism when I want to make it plain that Platonism can "exist" only as one of a pair of complementary bounds.

So maths is extreme because it goes as far towards Platonic rationalism as we can imagine going. Which is good because that then makes the other side of the equation, the need to measure the local material particulars of the world, a matchingly precise task.

The legitimacy of the maths is wholly dependent on empiricism as a result. If triangles in flat Euclidean space do not have angles that sum to pi, then the formal model is screwed.

for math is not the sound of one hand clapping, if you will.
Bingo, that's why I cannot feel the Platonic picture can provide the whole story.

Yes, maths goes to one extreme - tries to be the one hand clapping. And this works because it creates its own complementary extreme. It creates with equal decisiveness the idea of a local, particular, material measurement. The other hand needed to make some noise.

The maths comes to seem like it is "all subjective". It is a realm of ideal forms discovered rationally. And the measurements likewise come to seem "all objective". They are the brute material facts that exist out in the world.

Yet really, both formalised models and material measurements are only ever in our heads as part of the dichotomy of mapping.

This is just a restatement of Copehagenism (which followed from Bohr's shocked need to deal with a world that actually appears foundationally dichotomous - always at root complementary in nature).

The problem with the Copenhagen interpretation is then that once the simple mechanical view of causality had been shown to fail (at the extremes of its range), the choice was to reject then any chance of a "true" model of causality. The observation were whatever they were within whatever the framework of observation happened to be. It was all taken to be quite arbitrary, with no possibility of systematisation.

Yet in my view, a constraints-based approach to causality fits QM like a glove. Asking questions of reality can reduce its inherent uncertainty to the point it seems very certain - but cannot in principle eliminate all uncertainty.

You can see how these themes keep repeating. We spend so much time trying to disentangle epistemology from ontology - to form that crisp foundational dichotomy between map and terrain. And then we find that the two seem in fact deeply entangled.

In the realm of our minds, the maps are dichotomised into "subjective" rational forms and "objective" material measurements.

Then the bigger shock (perhaps). Out in the world, the terrain is also ontically dichotomised into its "subjective" forms and "objective" materials. Or rather, the self-constructing causality of global constraints in dynamic interaction with local degrees of freedom. A Universe that decoheres itself into structured being via some kind of semiotic or "self-observation".

So this would be where we differ.

I think we can develop a legitimate model of reality in which the ontology involves an epistemic aspect - the necessary decohering observer is made part of the entire system (in the guise of top-down constraint, the contextual information, a generalised environment). We can hope to make a map of the entire process.

But you would defend the more agnostic Copenhagen position where there is a map, and there is a world, and we can never say much more except that epistemology and ontology are fundamentally divided in this fashion. So the default philosophy is that modelling-associated activities like maths are arbitrary at the foundational level, even if useful in a pragmatic fashion.

As world views, we thus have naive reductionist realism, agnostic Copenhagenism, and constraints-based systems thinking.

I agree Copenhagenism is the correct default position - the place you have to retreat back to under pressure. But naive realism is a highly pragmatic choice. It works in the middle ground where humans mostly live. And systems thinking holds out the hope of getting "closer to the ultimate truth", to seeing the whole of reality within the one model.

 

[Ken G]

Very interesting post, it stimulates a lot of reactions on my part.

But if we step back to the purposes of brains, they are there to make decisions. To make choices. And how can you make a choice unless you have alternatives? And how can you make the most definite possible choices unless the alternatives dichotomous - reduced to either/or, to a binary yes/no, like retreat/advance, attend/ignore, expected/surprising.

That is a valid way to slice the choices our brains make, yet I would still argue it is how our brains think constructively about what brains do. The brain making sense of itself will model itself, but the model will, on purpose, take a projection and throw away what doesn't fit. It's a kind of template, the dichotomous analysis. The irony is, we can apply the same template to that analysis-- we can dichotomize, or unify, so we even have complementary choices around the issue of complementarity itself.

I think what happens is, each of our choices, taken to an extreme, tends to come "full circle" back to the seemingly opposite choice. Complete unity is too bland to convey meaning, while contrast is "crisp", as you might say. But crispness is a kind of intentional illusion, inventing distinctions out of the unity that underlies those distinctions-- nothing is ever actually crisp, crispness is not the way of the world. I recall a famous military general, I forget, who joked that he never retreats-- but sometimes he advances in an opposite direction. It was intended to get a laugh, but there is also a truth to it-- the attack/retreat dichotomy is invented from the unity of strategic military maneuvers, just as a cornered animal might lash out aggressively in what is actually a desperate attempt at escape, or a retreating army might actually be luring their pursuer into a trap.

A classic example of this "coming full circle" effect in philosophy is the rationalist/empiricist dichotomy. We all know that we combine mental analysis with sensory perception to make sense of our environment, but the rationalist emphasizes the mental analysis as the "truth" of the matter, while the empiricist emphasizes the sensory perception as the deeper arbiter of what is real. But if we take the empiricist approach to its logical extreme, we say that a sensory perception is not the light entering the eye, for light can enter the eye of a dead person-- the perception is the signal in our brain that is made when light strikes our retina. And it is not just the neuron that fires, for a neuron can fire even if we are distracted and fail to register the perception, it is a complex process going on in our brains that registers the perception. But complex processes going on in our brains are just what we normally call thought, and that's the seat of rationistic truth! So extreme empiricism is actually a form of rationalism, the crisp dichotomy disintegrates under the microscope.

Similarly, if we take rationalism to its logical extreme, we say that the mind is able to connect with truth, but the way we connect with truth is we perceive our own thoughts. Since our brains are also natural systems, presumably, then perceiving our own thoughts is also a form of empiricist truth, something that has a place at the table of reality simply because we perceive it to be there. These dichotomies we make as a useful tool are not actually true in any deeper sense.

Again, you will probably say that intelligence is defined by having a variety of choices. But as I say, that describes the variety that emerges as a result of the deeper structure - the ability to break the world down by polarities.

We agree that the variety is what is crucial, I'm just saying the way we break down that variety is itself a kind of simplified replacement. We write the digits of a number as a replacement for the number, and we can manipulate those digits in ways that mirror how numbers are manipulated, but the manipulation of a number is not entirely syntactic the way the manipulation of digits is-- the digitization does not replace the semantic meaning of the number, it is merely a placekeeper for it. I see dichotomies similarly, as placekeepers for the varieties, a useful labeling tool, but which does not actually capture the underlying variety-- that variety is irreducible, any syntactic construction of that variety is just a shell, like a robot programmed to mimic the actions of a human being.

My favourite example of the primitiveness of this is the flagella that drives a motile bacterium. Spin one way and the threads tangle, driving the cell forward. The bacterium can follow a chemical gradient, head towards a food source. But then reverse the spin and the threads untangle, the bacterium begins to tumble randomly. So if falling off the scent trail, the bacterium can switch to search or escape mode.

The asymmetry of choice - as determined/random - in a nutshell.

Yes, I like that metaphor a lot. I think it underscores the fallacy of "choosing sides" in any debate centered on a dichotomy (like "is life deterministic or random", when we find that life sometimes follows a deterministic scheme and sometimes a random one). I'm just taking that a step farther, and saying that even the dichotomy itself is not something we should commit to, for one such dichotomy is "embrace dichotomies vs. reject dichotomies." The moment we assert a dichotomy is true we find that taking it to an extreme causes it to be a circle rather than an axis, but if we use that logic to assert there is not that dichotomy we lose the analytical power of invoking it. I think there must be some truth to the idea that all analysis is judicious lying.

I agree it is a legitimate question. And the default position will be "all models are the free creations of the human mind". We should be automatically suspicious of any jump from the epistemic to ontic.

Indeed. We should also be suspicious there is any true distinction between the two, or that either of them even exists. Yet we must not completely reject Platonic thinking, for then we lose its analytic power. A map is a particular kind of lie, but a good map is a judicious lie that leads us where we want to go.

But on the other hand, reality must actually have some kind of deep causal structure. It does not seem like an arbitrary bundle of happenings does it? It does seem to have a developmental history, a systemic and patterned materiality. So it is not impossible that our models of its deep structure could be essentially correct.

I think that is indeed impossible. The problem is that if an atom cannot know itself, then neither can a huge and complex array of atoms. The only difference is the atom has not the required structure to invent a judicious lie, but the array of atoms has. That's the key shortcoming of pure reductionism, I agree with the systems perspective there.

There would not be life/mind without this trick of being able to harness dissipation-driven dynamics. So this is why we can say what "math is". It is not some unpredictable consequence of blind evolutionary change, it is instead the very predictable development of the constraint machinery which in fact defines life/mind.

I would agree, yet I would still call that a judicious lie. It's like when I tell a class that planetary orbits are ellipses, I know that I am lying, judiciously. Feynman said that science is a way to avoid fooling ourselves; I would add it is a way to avoid fooling ourselves that works by lying to ourselves judiciously.

Life/mind arise as informational structure that locally accelerates the entropification of the Universe.

And yet entropy, as an ontology, is a classic example of a judicious lie. The universe is in one state-- so always has zero entropy, formally speaking. But the concept emerges when we, as analysts, decide that we don't know that state, we know only a class of states that satisfy what we care about. We know not the territory/state, we know the map/class of states. So the concept of entropy is born-- the natural log of the number of states in the class. The map has entropy, the territory does not. But entropy is a mapmaker's key, one of the most judicious lies of all time that some feel underpins at the deepest level all of our understanding of nature.

For example, life/mind arose on the back of one kind of language - genes to code for enzymes that could control dynamical chemical cycles. But then H.sapiens stumbled upon actual language - words to control the thoughts that determine our actions.

An interesting point, the way one form of language gave rise to another. Yet I would say that DNA is not really a language, it is we who understand language who also understand DNA that way. A gene has no need for even the concept of language, so certainly has no need to participate in one. It is we who need to see the gene's action in that light, the judicious lie comes from us-- in fact, we invented judicious lying when we invented language.

And of course, even that last statement is a judicious lie about judicious lying, it can't really be true because humans cannot be separated well enough from language to say we invented it, for as you put it, as soon as we say we invented language, we find that DNA satisfies our meaning, but then DNA invented us, so we end up with one language inventing another, which tells us nothing about what language is or where it comes from.

Logic is no better off than language. If we say that logic is based on the true/false dichotomy, then I say it is based on a lie, albeit a very judicious one. So what do we do with logic when we see it as a lie that works? You could claim that a lie that works is not a lie, but I don't mean it is lie in the sense that something else would be true, I mean it is a lie in the sense that truth is something we just invented, and logic is its syntax. If we invented truth, then Platonic truth is a lie, but it is a judicious lie that allows us to invent the concept of truth in the first place. Truth requires a lie to even be possible, and the true/false dichotomy comes full circle.

Once there was a species that could chop up a stream of sound into discrete syllables, the machinery for a new level of coding could be used for exactly that.

Which raises another interesting question: what is the meaning of syntax? Syntax is supposed to be distinct from meaning, yet it requires a meaning or else we don't know how to use it in a sentence. We can't connect vocal patterns with DNA patterns unless we understand that the patterns represent something deeper. Another dichotomy comes full circle. And can whatever is the meaning of syntax be a Platonic truth about DNA and language, when we cannot even enforce a Platonic separation between syntax and semantics? The escape hatch is to recognize they are all judicious lies, all of them: syntax, semantics, language, DNA, the works.

We seem to agree then. Because I am saying that maths is not monadically anyone kind of thing. Which is what the poll wants to make it.

Yes, getting back on topic, we agree there. The poll is trying to get us to commit to a lie about mathematics that is not judicious because any of the choices either sell math short, or are grandiose and unsubstantiated wishful thinking. A more judicious lie about math is that it combines all those elements in a complex way, but of course if that were really true, then it would have to be true in some Platonic sense, which would make math Platonic, so the argument would come full circle.

As I say, this may yet be telling only half the story. But that can only be clear once the foundations of maths is actually clarified.

And I would say the very idea that math has foundations at all is another judicious lie. Math has attributes that let us recognize it, that's all we can really say because that's how we defined it ourselves. Everything on that list is like a hobo jumping a train simply because it is going in the same direction that they want to go.

It should be clear by now that I would only argue for Platonism (the fact that reality has a deep structure which our modelling can hope to map) to the extent that observation appears to confirm it.

But this isn't really any kind of confirmation, because it is our nature to frame our analysis of observations in those terms. We are looking into the mirror, not at something that transcends us. What observation could come out X if our way of understanding that observation is also something Platonic, or Y if our way of understanding it comes from us, when it is the outcome itself that we are trying to understand?

Yes, I agree. You seem to have me now arguing for Platonic fundamentalism when I want to make it plain that Platonism can "exist" only as one of a pair of complementary bounds.

That form of existence is probably pretty close to what I mean by a judicious lie, so perhaps we are not so far apart on this. I'm just adding that the complementarity is also part of the lie, as is seen by how it tends to come full circle if you take it to its extremes. The opposite poles are just directions, they don't exist as destinations because the destinations come full circle.

So maths is extreme because it goes as far towards Platonic rationalism as we can imagine going. Which is good because that then makes the other side of the equation, the need to measure the local material particulars of the world, a matchingly precise task.

Here I believe you echo a similar sentiment.

As world views, we thus have naive reductionist realism, agnostic Copenhagenism, and constraints-based systems thinking.

Yes, and I see each as a hat we put on when it serves us. Three different maps that each lie about the terrain in different ways, like a bus schedule, a road map, and a topographic map-- lies when regarded as the full story that become judicious enough to help us achieve our goals when not so framed. Like the poll itself.

I agree Copenhagenism is the correct default position - the place you have to retreat back to under pressure. But naive realism is a highly pragmatic choice. It works in the middle ground where humans mostly live. And systems thinking holds out the hope of getting "closer to the ultimate truth", to seeing the whole of reality within the one model.

Yes, there are different times when each flavor of falsification becomes the closest thing we get to a truth that does not, in fact, exist in the absence of falsification.

 

[apeiron]

The irony is, we can apply the same template to that analysis-- we can dichotomize, or unify, so we even have complementary choices around the issue of complementarity itself.

Is it an irony or instead a logical truth that for once is successfully self-referential? The set that now includes itself without contradiction?

But crispness is a kind of intentional illusion, inventing distinctions out of the unity that underlies those distinctions-- nothing is ever actually crisp, crispness is not the way of the world.

All of this is about limits and dynamics, so yes, actual crispness is never attained. It is just the bounding limit of a process.

If we come up with some standard metaphysical dichotomy like discrete~continuous, then the claim would be that these are the two extremes of what might be the case, and material reality would lie within these extremes. But it could never definitely attain either of them (because then it would become just perfectly one of these things).

I recall a famous military general, I forget, who joked that he never retreats-- but sometimes he advances in an opposite direction. It was intended to get a laugh, but there is also a truth to it-- the attack/retreat dichotomy is invented from the unity of strategic military maneuvers, just as a cornered animal might lash out aggressively in what is actually a desperate attempt at escape, or a retreating army might actually be luring their pursuer into a trap.

Proper metaphysical strength dichotomies have the quality of being asymmetric or orthogonal. So attack/retreat is a simple and thus unstable, because easily reversed, anti-symmetry. To go forward is just much the same as going backwards - an inverse operation.

But real dichotomies are reciprocal operations. It is a symmetry breaking across scale that leaves the two extremes as unlike as possible.

So again take a full-strength dichotomy like discrete~continuous, as might be illustrated by a dot marking a line. The dot is so infinitely small, it no longer even has dimensionality, whereas the line is infinitely large as a dimension. The relationship is an asymmetry, a really, really broken symmetry.

But complex processes going on in our brains are just what we normally call thought, and that's the seat of rationistic truth! So extreme empiricism is actually a form of rationalism, the crisp dichotomy disintegrates under the microscope.

Yes, as I say, mental processes are dichotomous in this way. Impressions build ideas and ideas form our impressions. Or in other words, empiricism is the basis for rationalism and rationalism the basis for empiricism.

This is the basis of CS Peirce's triadic metaphysics. Or models of the brain such as Stephen Grossberg's ART neural networks.

The mind starts off in the unity of ignorance - vague unformed potential - and then becomes organised by discovering structure in experience. In a newborn brain, visual pathway neurons will fire off to just about anything. Then quickly they learn to narrow their responses as higher level ideas (top-down constraints) develop.

These dichotomies we make as a useful tool are not actually true in any deeper sense.

Well they seem true both by observation and reason surely? Not all dichotomies would have to be true of course. But the ones that seem philosophically deep - such as discrete~continuous - do seem to be both what is fundamental about the world, and also self-referentially true in the logical sense.

From the rational standpoint, if things are not continuous then they must be broken up into the discrete. And if things aren't broken, then they must be continuous. All other possibilities are exhausted by these mutually exclusive universal concepts. The only third possibility is the unifying one - things are too vague for us to tell.

And then by observation, things are generally either discrete or continuous. When we actually stop to make particular material measurements, that is what we find.

I see dichotomies similarly, as placekeepers for the varieties, a useful labeling tool, but which does not actually capture the underlying variety-- that variety is irreducible, any syntactic construction of that variety is just a shell, like a robot programmed to mimic the actions of a human being.

Yes, there may be many varieties of dichotomies. But then that very statement is saying there is only the one underlying process - dichotomisation.

And again, there would be two levels of claim here. One that dichotomisation is the universal operation underlying epistemology, and then the stronger claim that it underlies ontology as well.

If the second is true, the "irreducible variety" of reality will have been generated by dichotomies (ie: symmetry-breakings).

I'm just taking that a step farther, and saying that even the dichotomy itself is not something we should commit to, for one such dichotomy is "embrace dichotomies vs. reject dichotomies."

As said, there is a consistent position here because holism and reductionism are taken to be mutually contradictory views of causality. So holism "rejects" reductionism - and in doing so, proves that it is a complementary truth.

Reductionism of course sees itself as the sole truth. Which is why it can only be ever half the truth.

You are quite right to keep pointing to the fact that dichotomies demand to be understood in a dichotomistic light. But that is a strength rather than a weakness. It is a logic that applies to itself with consistency.

A map is a particular kind of lie, but a good map is a judicious lie that leads us where we want to go.

Or another way of putting it is that a map tells the least amount of truth needed to do its job. The terrain has the complete information. A useful map represents what is meaningful by using the least possible information.

Maps don't actually lie. They are judiciously selective about the truth they represent. And less information results in more meaning.

Tor Norretranders wrote a good pop sci book based on this principle - The User Illusion.

Truth requires a lie to even be possible, and the true/false dichotomy comes full circle.

I know what you mean by lie, but it is misleading. As I say, the point of modelling, or maps, is to reduce the truth to a bare minimum - to discard as much information as possible so as to focus all attention only on what matters.

And my claim is that actual Reductionism is dichotomistic. But it wants to reduce too far by saying things are either/or. But even atoms require the matching "truth" of a void. You end up needing both, even though the claim is that only one "actually exists".

So the least truth you can get away with boils down to the "both" of a dichotomy. The simplest map has to be both the line that sketches the path, and the blank paper that symbolises all the rest that has been actively discarded. The information deliberately left out because it is just "noise".

But this isn't really any kind of confirmation, because it is our nature to frame our analysis of observations in those terms. We are looking into the mirror, not at something that transcends us. What observation could come out X if our way of understanding that observation is also something Platonic, or Y if our way of understanding it comes from us, when it is the outcome itself that we are trying to understand?

Well, if you frame your hypothesis in dichotomous fashion, you can reject one of the choices. So here, the choice is between the Platonic unity that would follow from the existence of deep structure, and the alternative of an unstructured variety - a mereological bundle.

If on observation the sum of the angles of triangles always measured differently, then you might suspect the hypothesis that a triangle was a Platonic truth.

I'm just adding that the complementarity is also part of the lie, as is seen by how it tends to come full circle if you take it to its extremes.

Coming the full circle is again a strength rather than a weakness here. Again, a true dichotomy is a difference that is broken across scale - it is the canonical local~global division of systems science or hierarchy theory. So what you call a circular relationship is in fact an interaction across scale. It is the fact that there is the bottom-up in interaction with the top down.

Just as I said about the easily reversibility of an inverse relationship, unless you have a scale difference, any symmetry breaking is only weak, unstable. But break across scale and things are far enough apart for their interactions to become interesting because they now look very different in kind. They are no longer going around in a tight circle that leads nowhere. The interactions are coming from opposing limits of scale.

 

[Ken G]

Coming the full circle is again a strength rather than a weakness here. Again, a true dichotomy is a difference that is broken across scale - it is the canonical local~global division of systems science or hierarchy theory. So what you call a circular relationship is in fact an interaction across scale. It is the fact that there is the bottom-up in interaction with the top down.

Just as I said about the easily reversibility of an inverse relationship, unless you have a scale difference, any symmetry breaking is only weak, unstable. But break across scale and things are far enough apart for their interactions to become interesting because they now look very different in kind. They are no longer going around in a tight circle that leads nowhere. The interactions are coming from opposing limits of scale.

I think we are largely in agreement about the importance of dichotomization in analysis, and the way seemingly opposite options tend to come full circle when pressed to their limits. We agree that a dichotomy is not "truth" in an either/or sense that one or the other extreme should be regarded as correct. This is relevant to the math poll-- if we juxtapose math as Platonic truth vs. math as arbitrary human construct, we might tend to expect math to have to be one or the other, for how could it be two opposite things at once, yet on further thought we see no difficulty at all in being two opposite things at once, since they are not actually "opposite choices" but more like "elements in opposition".

What I'm adding to this already somewhat controverisal view is an objection to the following stance. It might be said that the opposite poles of the dichotomy should be viewed as opposite destinations that could each have their own "Platonic truth", as it were, such that the real truth is some kind of mixture, like splitting time between a Winter and Summer home. But I'm framing those opposite poles as "judicious lies" (where "lie" is chosen to be somewhat melodramatic for effect more so than precision) because neither one actually exists at all-- their lack of an independent existence is revealey by pushing on each until they turn into the other. They break a symmetry, yet introduce a new one, an interchange symmetry, like in this Escher print: http://4.bp.blogspot.com/-btPpdTyzL3k/TdDwi8al6wI/AAAAAAAAABQ/qOeXVRlgN28/s1600/escher.gif The print gives the illusion of crispness, but a moment's reflection reveals the paradox behind that crispness.

Your point about rationalism and empiricism being responsible for each other is very much along the same vein-- I'm merely saying that this means neither of those concepts can really be a potential truth, for if they could, they could exist independently of each other. It's like, if you go too far to the left, you end up on the right (as even happened in the book Animal Farm), so there is "leftness" and "rightness" as relative orientations, but there is no such destination as "the left" or "the right." This means "the left" is a judicious lie to give us a concept of "leftward", the latter being a relationship with rightward. More to the point of this thread, I would say that every choice on the math poll is a judicious lie-- none of those options really exist, they only have meaning in relation to each other, as pointers to certain relative directions or angles of perspective.

I think the issue of broken symmetries across scale is crucial to this point. In string theory, there is a concept of "duality", which seems like it is trying to become the next really profound insight in physics. Though I am no expert, the basic idea of duality is that two seemingly different theories or descriptions of nature can be mapped into each other if one simply inverts the large and the small-- so if each theory has very different behavior at large and small scales, then one acts like the other at the opposite scale (say, in quantum mechanics we have indeterminacy at small scales yet macroscopic determinacy by the correspondence principle, so there should be some theory which hasn't been found yet that is indeterminate at large, say cosmological, scales, but determinate at atomic scales). So I agree that the symmetry breaking across scale is what has significance there, but not the scales themselves-- there is no large, no small, no left, and no right. Simply changing our perspective swaps these, all that is retained is their relationship to each other.

So if one takes this perspective to its logical conclusion, and applies it to all dichotomies, one can also apply it to the dichotomy "do dichotomize, don't dichotomize". When we do that, we find that what matters is we have the choice to dichotomize, but it doesn't matter which choice we make, because whatever choice we make, in some other perspective we will have made the opposite choice, especially if our choice is taken to its extreme limit. If you choose to treat reality as discrete, I will take your choice to break reality up, and break it into fewer and fewer pieces until it is just one piece-- at which point it is back to a continuous description. There cannot be a law that says "reality is fundamentally about dichotomies", for then the dichotomous perspective is "reality is fundamentally about unities." So presenting a dichotomy, on grounds that the dichotomy is the truth of the matter, is choosing a judicious lie.

The reason I choose "lie" and not "reduce the information to its minimum necessary truth content", as you framed it, is that I feel the danger of imagining that a dichotomy represents destinations applies to the most fundamental of all dichotomies, "true/false." Referring to the Escher print above and imagining this is truth and falseness in that picture, we see that even the invention of the concept of "what is true" is a judicious lie. We cannot hold that "Platonic truth" underlies any of these other dichotomies, or maps, because the only way we can even give meaning to the concept of "truth" is by giving meaning to the concept of "false." Since neither exist without the other, neither is a destination in itself. What's more, I suspect a duality there as well, where a change of perspective interchanges everything we regard as true and false, with no effect on the overall structure because the "deep structure" only required the opposition between truth and falseness, the structure does not identify one or the other independently any more than it does left and right. Truth is one hand clapping, so there is "no such thing" as truth, so we invented it-- it is itself a judicious lie (of course, so are lies).

I think the reason that we do all of this can be summed up in this other Escher concept:
http://thefalloutgirl.files.wordpress.com/2011/10/escher.gif
Our goal is to understand an environment that has us in it, so all we will ever be able to do is look over our own shoulders. This forces every concept, every word, every meaning we glean, to be a kind of judicious lie, because a brain is a device for doing that.

 

[apeiron]

What I'm adding to this already somewhat controverisal view is an objection to the following stance. It might be said that the opposite poles of the dichotomy should be viewed as opposite destinations that could each have their own "Platonic truth", as it were, such that the real truth is some kind of mixture, like splitting time between a Winter and Summer home.

Yes, the further step in this ontological story is that what is simply divided then gets complexly mixed. So this was what yin-yang/I Ching said about dichotomies, and also it was the metaphysics of Anaximander.

It is also the QM/classical story I would suggest. All material entities are complex mixtures of position and momentum for example. At least that is how they look over the middle range of scale between the complementary poles of the Planck energy and the Planck distance. But then eventually that breaks down asymptotically as you approach the Planck limits. Try to make one pole of being "the truth" - to obtain an exact value for position or momentum - and the other pole becomes radically indeterminate.

So we seem to in fact have strong empirical support for this very ancient, if somewhat controversial, metaphysics. The world does actually seem like an isotropic, homogenous, scalefree mixture of polar properties over its middle range of scale. And then it breaks down if you attempt to push all the way towards one or either of the limits.

So the basic principle of differentiate~integrate holds I believe. It makes sense that to have anything definite, you have to have some fundamental division into complementary "truths". And then what gets separated must be also freely mixed. If things get separated by splitting across scale, then across scale is where they will be evenly mixed. This type of logic is well modeled mathematically these days by scale-free networks and nested hierarchies, for example.

But I'm framing those opposite poles as "judicious lies" (where "lie" is chosen to be somewhat melodramatic for effect more so than precision) because neither one actually exists at all-- their lack of an independent existence is revealey by pushing on each until they turn into the other. They break a symmetry, yet introduce a new one, an interchange symmetry, like in this Escher print:

Again, the Escher print is a misleading image because it portrays a symmetric symmetry breaking rather than the asymmetric or hierarchical one that I am talking about. (Anyone who has read Gödel, Escher, Bach: An Eternal Golden Braid will know that this was the same mistake Hofstadter made in his attempts to make sense of "strange loop" causality).

An asymmetric dichotomy is divided across scale and so pushing in one of its two direction cannot eventually lead back to where you started. Again, think of HUP uncertainty. The more precise your measurement of one direction, the increasingly indeterminate becomes the other. You don't eventually go from measuring distance and breaking through the Planck scale to find yourself measuring energy.

Your point about rationalism and empiricism being responsible for each other is very much along the same vein-- I'm merely saying that this means neither of those concepts can really be a potential truth, for if they could, they could exist independently of each other.

I agree that they cannot exist independently. They exist only in opposition.

And then the further assertion - if they really are a legitimate dichotomy and follow the logic of asymmetric (or hierarchical!) dichotomies - the practice of maths would have to mix the two over all its scales of action. So maths would have some standard balance of empiricism and rationalism over all its scales of operation. As argued, maths is in fact quite distorted in this regard because it attempts to reduce its empirical content to a minimum so as to maximise its rational content. It is trying to be "in the limit" Platonic, deductive, etc. And it does this by constraining the empirical, the inductive truth, to the business of forming axioms. And perhaps also the subjective checks necessary at each stage of a deductive proof to "know" that each step is watertight.

This means "the left" is a judicious lie to give us a concept of "leftward", the latter being a relationship with rightward.

Again, this is symmetric symmetry breaking not asymmetric or hierarchical symmetry breaking. In the asymmetric case, the polar directions would seem orthogonal. On a sphere, going westward, you might indeed end up going eastward. But you could keep going forever and never find yourself going north- or south-wards. So you are working with a different mental intuition here.

More to the point of this thread, I would say that every choice on the math poll is a judicious lie-- none of those options really exist, they only have meaning in relation to each other, as pointers to certain relative directions or angles of perspective.

I agree, with the difference that I am arguing a foundational view would be based on a single dichotomy ideally. Or maybe a pair.

So one obvious dichotomy they all try to orientate themselves by is the modelling relation - the map~terrain. Either maths is just a map, or it is actually a terrain. But this is a general foundational issue for epistemology, not just maths. It is true of all knowledge. And the modern resolution would be that we only just map, and then that it is the map which is dichotomised into models and measurements, or general ideas and particular impressions. This is what actually fits what we know about how brains work.

So again, the foundational dichotomy for maths would seem to be that maths is a language, a machinery, for constructing constraints. It describes the forms that bound materiality. And describes them in an atomistic fashion so the forms can actually be built additively, step-by-step, as a series of effective causes.

So maths is formal syntax that can construct states of materially constrained semantics. The world just is. Maths is a tool that can rearrange it within its limits. And those limits actually "exist". The material world is ontically bounded. The Universe is constrained (somehow) to be Euclidean flat (over the middle range of our observation at least) and so triangles add up to 180 degrees as a "Platonic truth".

I think the issue of broken symmetries across scale is crucial to this point. In string theory, there is a concept of "duality", which seems like it is trying to become the next really profound insight in physics. Though I am no expert, the basic idea of duality is that two seemingly different theories or descriptions of nature can be mapped into each other if one simply inverts the large and the small-- so if each theory has very different behavior at large and small scales, then one acts like the other at the opposite scale

Yep, these are reciprocal dualities - scale based. There are three of them (which suggests that they might reduce to a single duality in some fashion). And there are arguments that you can go through the Planck scale and come out the other side. With topological or T-duality, a wound string would become an unwound vibrating string by this manouvre and going east would be suddenly going north. Though this seems more a mathematical passing through the eye of a needle than a physically realistic story.

With S-duality, or "soliton" like duality, you could pass from what looks like a solid particle with weak interactions to a clump of excitation bound by its strong interactions.

And then there is the AdS/CFT correspondence that people are so excited about. Note how it depends on a "conformal world" - the kind of scaled realm where mathematically you represent both a space and its bounding extremes of scale. You take a limit and can then place another complementary "world" on the other side of that boundary, making one the simple description of the complex mixture.

(say, in quantum mechanics we have indeterminacy at small scales yet macroscopic determinacy by the correspondence principle, so there should be some theory which hasn't been found yet that is indeterminate at large, say cosmological, scales, but determinate at atomic scales). So I agree that the symmetry breaking across scale is what has significance there, but not the scales themselves-- there is no large, no small, no left, and no right. Simply changing our perspective swaps these, all that is retained is their relationship to each other.

I think it is an error here to equate the Planck scale with just a single limit. It is in fact a dualised description of an inflection point, a yo-yo point, the vertex of a parabolic relationship.

So the Planck scale describes two dichotomous (orthogonal) extremes - the greatest heat and the smallest distance. It is how small can you shrink in spatiotemporal terms, how large you can grow in material density or other measures of energy. So again, a classic form vs substance dichotomy - the least degree of one and the maximal presence of the other.

This of course then requires a further "dimension" to our ontology. Back at the Big Bang, the Universe seems to have been both very small and very hot. The two aspects of reality were united - and therefore in the logic of dichotomies, we would say that existence was radically vague, or indeterminate. It is only once the Universe expanded/cooled that you could actually make crisp measurements that might say some event was definitely small (in relation to the general largeness of the Universe), or hot (in relation to its generalised coolness).

So through the construction of constraints - setting up the kind of experiments that can measure the quantum boundaries of our existence - we can recover this naked quantum indeterminacy either by making things really hot (as in a collider) or very small (as in observing single buckyballs going through twin slits at near absolute zero).

There cannot be a law that says "reality is fundamentally about dichotomies", for then the dichotomous perspective is "reality is fundamentally about unities." So presenting a dichotomy, on grounds that the dichotomy is the truth of the matter, is choosing a judicious lie.

But the full ontological position is that it is not just about dichotomies. Dichotomies refer only to the middle bit, the process by which a vagueness (Anaximander' apeiron, Peirce's firstness, QM's indeterminacy) becomes transformed into a hierachical, complex, realm (Peirce's thirdness, Classical crispness, etc).

So there is monadic unity in the initial vagueness. And also a kind of unity in the stability of the final triadic outcome, a hiearchical order where you have the three things of bottom-up atomistic construction, top-down global constraint, and then the ambient equilibrium balance of those two complementary actions that creates a conformal spectrum inbetween.

If this is all a judicious lie, it is a lie with an intricate causal structure.

The reason I choose "lie" and not "reduce the information to its minimum necessary truth content", as you framed it, is that I feel the danger of imagining that a dichotomy represents destinations applies to the most fundamental of all dichotomies, "true/false."

I don't follow that. True/false would be a false dichotomy, in the technical sense of being forced to chose one pole over its alternative. It is the step too far that reductionism makes.

A dichotomy is A and not-A. A division of reality (or of our ignorance) into a something and everything that it is not. And both sides of this partitioning would still by definition be "the real". At most, all we are saying is that there is one side of the partitioning of reality that we are chosing to ignore, or to generalise away.

All I am talking about then is returning to the fact that this is what logic is about - the process of excluding middles so as to set up dichotomous alternatives. And instead of rejecting one pole (labelling it "the false"), or ignoring one pole (labelling it entropy, or noise, or void, etc), actually giving it an equally definite name (making it also "the true" because it is the yin to the yang, or the continuous to the discrete, or the position to the momentum, etc).

The logic of dichotomies is not about the simplicities of true/false. It is about the complexity of crisp/vague. Things only become definitely anything if they are dichotomous. So in terms of our maps of the terrain, we are talking about knowledge/ignorance. If we can definitely frame a view of reality based on an A~notA distinction, then the law of the excluded middle applies and we are lifted out of vague, indeterminate uncertainty about what may be the case.

Truth is one hand clapping, so there is "no such thing" as truth, so we invented it-- it is itself a judicious lie (of course, so are lies).

Truth can only be true if lies are also true? Where have we heard these modal paradoxes before? Yes, and the logic of dichotomies would demand that complementaries have equal claim to "truth". Except as I say, the actual dichotomy by which the antimonies would be measured would be vague~crisp, or indeterminate~certain. So for a truth to be certain, so does the not-A that is the lie. Truths cannot just simply exist, they have to be formed within an equally definite context that allows the process of a judgement.

Truths thus imply always a knower of the truth - an observer. Modal logic generates paradoxes because it fuzzes over this issue. It wants to take the limit and presume that truths can have brute existence, independent of the business of any measurement process.

So the reductionist view of "truth" based on truth/lie is a really big judicious lie (both very useful and extremely untrue), and the dichotomistic view of truth as a process of discrimination is still a lie, but not so much of one, perhaps.

I think the reason that we do all of this can be summed up in this other Escher concept:
http://thefalloutgirl.files.wordpress.com/2011/10/escher.gif
Our goal is to understand an environment that has us in it, so all we will ever be able to do is look over our own shoulders. This forces every concept, every word, every meaning we glean, to be a kind of judicious lie, because a brain is a device for doing that.

That is the AdS/CFT correspondence again. Conformal symmetry can represent the full story of a world and its limits. And by imagining a world like that, you can step outside it to see it all.

It is no accident that every direction you turn, people are backing into the same story. Worlds are the product of limits - of global constraints on local degrees of freedom. And to model this fully, you have to be able to model the development of those global constraints as well.

Do I have to mention the whole renormalisation schtick? Or holography? As our vision of reality has expanded far enough to now observe its bounding limits of scale, we are now groping towards the models that can include those limits as the final facts. And slippery dualities are what must be tamed. You have to have ways (in the language of maths and logic!) to hold both ends of the beast pinned down. Reductionists are stuck in the game of wondering which end they need to pin down (and so always getting whacked on the back of head by the other end flailing about).