What's Your Philosophy of Mathematics?

[apeiron]

Thus in the modern view, it is the statements of mathematics, not mathematical objects, that correspond to the properties of mathematical reality.

In what sense would these statements be physically real then? Where would they "bounce about"?

I know a lot of people say they are Platonists, but it would be nice if they could be more specific about what is entailed by their claims to actual existence (as opposed to just being emergent regularities, expressions of material potential).

 

[chiro]

In what sense would these statements be physically real then? Where would they "bounce about"?

I know a lot of people say they are Platonists, but it would be nice if they could be more specific about what is entailed by their claims to actual existence (as opposed to just being emergent regularities, expressions of material potential).

The one thing I see in terms of comparing and contrasting material properties as opposed to properties that are generic in a mathematic context has to do with the measurability aspect.

In terms of our material quantities, one thing that is striking is that the material components are highly quantified and constrained when it comes to measuring. When we measure something, we impose a kind of quantization in some form or another and thus we introduce a kind of implicit finite characteristic for the phenomena.

This is not only a property of an isolated part of a system but also of the system itself.

Materialistic things naturally carry a measurable quantity that is finite and thus can me measured and because of this, anything with non-measurable characteristics are hardly going to fit within a model of materialism.

Even with regards to potential, there is still this same kind of restriction whereas in a completely general context that includes non-materialism, then your representations and constraints include things that are simply out of the scope of materialism type frameworks.

I know this is a philosophical thread, but when people try and think about even beginning to reconcile this idea of mathematics pertaining to reality, or a subset of reality or whatever, then it is important to ascertain up front that materialism, as we know and practice it, will end up inducing finite characteristics in many ways and because of this, it will never correspond with the generic mathematical representations in their generic form given this subtle but very defining characteristic.

I would say it's kind of analogous to the situation with the Pythagoreans when one of them found that a RHS triangle with two sides of 1 gave a hypotenuse of SQRT(2) and claimed blasphemy (and ended up killing the guy who found it out: nice friends you have right there).

The thing was that in Pythagoras' time in that cult, it was unimaginable that anything non-rational could even exist. In the same vein, we have advanced a lot since then but again this kind of thinking is, IMO, one primary thing, be it direct or indirect, of materialism.

 

[John Creighto]

But what does category theory or Ramsey theory have to do with the physical world?

Category theory is quite similar to set theory and both I believe are isomorphic to logic. Well, set theory focuses on objects, category theory focuses on relations between objects.

From here I have an off topic questions:

A set is just another category (the category of sets) but how do we express a category in terms of sets? I know we can’t have a set of all sets but isn’t the category of sets suppose to be in some sense complete?

-------

For give my Ignorance here as my knowledge of category theory is nearly nonexistent. I do know that it is used in the Haskell programming language and sense programming langues are used to represent useful ideas, one might think that in some sense there is a tangible connection between category theory and the physically world.

http://en.wikibooks.org/wiki/Haskell/Category_theory
http://www.haskell.org/haskellwiki/Typeclassopedia

 

[apeiron]

Materialistic things naturally carry a measurable quantity that is finite and thus can me measured and because of this, anything with non-measurable characteristics are hardly going to fit within a model of materialism.

I agree this is one of the questionable assumptions that underpins a materialist ontology - that substances inherently possesses properties such as existence, rather than such properties being contextual and thus emergent.

Again, it is what happens when things are forced into the either/or style of thought. If you take substance to be real, form to be epiphenomenal, then any properties must be possessed by the material in question - as where else can these properties reside?

So you can appreciate how this ends up as the mirror view of Platonism. Except the idea of a material realm where the qualities of nature reside as ontic essences is not one that people seem to find troublesome.

Materialism gives substantial properties a place "to be". And Platonism wants to give formal properties an equivalent place to be.

I am taking the alternative view that everything in fact arises in the "place" inbetween these two ontic limits on potential being.

 

[jgens]

A set is just another category (the category of sets) but how do we express a category in terms of sets? I know we can’t have a set of all sets but isn’t the category of sets suppose to be in some sense complete?

Just use classes: http://en.wikipedia.org/wiki/Class_(set_theory )

 

[chiro]

I agree this is one of the questionable assumptions that underpins a materialist ontology - that substances inherently possesses properties such as existence, rather than such properties being contextual and thus emergent.

Again, it is what happens when things are forced into the either/or style of thought. If you take substance to be real, form to be epiphenomenal, then any properties must be possessed by the material in question - as where else can these properties reside?

So you can appreciate how this ends up as the mirror view of Platonism. Except the idea of a material realm where the qualities of nature reside as ontic essences is not one that people seem to find troublesome.

Materialism gives substantial properties a place "to be". And Platonism wants to give formal properties an equivalent place to be.

I am taking the alternative view that everything in fact arises in the "place" inbetween these two ontic limits on potential being.

I agree with your analysis.

The hardest thing to me has to actually do with the definition itself and that relates primarily to language.

To me our descriptive capability is such that our language classifies in a way to take big classes of things and make them smaller and this is more or less done in an inductive manner where complexity, understanding, and compressibility properties of representation end up translating in languages with more powerful descriptive power and hence better analytic use.

But the problem I see is that while we can classify 'material' type things well, we can't really do this well for non-material things and this is emphasized mathematically if you ever study infinity in any shape or form.

This infinity concept baffles a lot of people because it is really hard for us to relate to it in any way and I would largely attribute this to our sensory limits that are more so physical (and hence material) than non-material.

It doesn't mean that we can't and never will make sense of it, but what I will say is that as long as we rely more so on our physical and hence material sensory apparatus and analytic methods, then we will not grasp the nature of the general representations which are in the level you are referring to.

The result of this is that we would need to understand infinity in the same kind of depth that we can make sense of 'finitey' of 'finite-ness' and do to this, it means relying on things like the mathematical representations and systems themselves that deal with these infinite characteristics rather than our physical intuition, and for a lot of people I imagine this is not an easy task.

It's very easy to see why people have trouble with this if you look at the questions of the people that were studying this like David Hilbert (with Hilbert-Space Theory) with the Hilbert Hotel example. But even things like the 0.99999999... = 1 is also another example that is simple enough to state by hard for some when they need to rely on some kind of materialistic reference point of some kind to make sense of it.

The question I think that you should first ask, is if the potential of the material world really is infinite or not and then construct a discussion to flesh out why or why not this may be the case.

You could use results from physics and mathematics to support such arguments, but ultimately what this will do is force people to either support or not support the idea that potential really is infinite or not infinite and doing this over time will clarify things in a way that make it easy and clear to state why or why not people think this is the case.

 

[dcpo]

I took a university course in the philosophy of mathematics in 2006 that went over some of this, and while I wouldn't place too much trust in my undergraduate mind to have really gotten to the heart of things (or my current mind for that matter), I remember finishing the course more confused than I was when I started, which was to be expected I suppose. I can't claim to be much clearer about things today, but I do have some thoughts.

First, is there a fundamental difference between the abstraction that takes place at the basic level in mathematics, and the abstraction necessary for everyday language? To me it seems the answer is 'not really'. In everyday language we constantly use abstract objects such as 'a dog' to convey information. For example, if someone says to me 'a dog is in the yard', what do I understand from that? If I am asked to imagine a dog, what do I see? Clearly there is no universal canonical dog, and without extra information I am unable to know in advance which actual dog is in the yard, but if I go into the yard and see a dog I feel I am not surprised (unless there is something else to surprise me). Similarly, if someone says ‘there is a triangle drawn on page 37’, I do not know in advance what exactly the triangle will be like, but if I go to that page and find a triangle I am not surprised.

Now, in everyday language it is very difficult or impossible to pin down our abstractions very exactly. I think Wittgenstein talks about this in the Philosophical Investigations. So it is not too difficult to imagine places where our natural conception of ‘dog’ runs into trouble. We could, for example, see something that is very similar to a dog, and be unsure if ‘dog’ was the proper word. Maybe we could disagree with someone else on the subject, and our communication could be compromised to an extent. Of course, advances in science allow us to get closer to unproblematic definitions of real world objects, and similarly axiomatic approaches in maths allow us to pin down ideas very precisely, but before science allowed us to look at DNA and the like, and without formal axioms, we knew a dog when we saw one, and a triangle too.

Of course, in practice mathematics is quite different from natural language, because we demand that the concepts we use stand up to a very high degree of scrutiny, so that when we use them we are very unlikely to run into trouble when trying to communicate with a fellow mathematician, but I’d argue that this a quantitative difference rather a qualitative one. We have merely reduced the uncertainty in our definitions to a point where it is very unlikely to cause a failure of communication. I’d argue that ultimately our concepts do rest on ideas that are not well understood and have a low degree of common acceptance, as arguments in foundations demonstrate.

Second, I see no reason why anyone of the theories mentioned should cover all of mathematics, or why a piece of mathematics could not be covered by several. Related to this, it is by no means clear to me what is meant by ‘mathematics’ or ‘all of mathematics’, though as with the dog and the triangle I know it when I see it. In my opinion, problems arise from the fact that ‘mathematics’ is a term in everyday language, at least when most mathematicians use it, and to me at least it seems unreasonable to suppose that a rather vague term arising from usage (as in, ‘mathematics is what mathematicians do’) should fit neatly into one particular characterization.

For example, using the general framework of universal algebra I can easily define a variety of algebraic structures far removed from anything that has been found interesting in mathematics so far. I can proceed to prove results about this class. I cannot see how what I would be doing would be anything other than formal symbol manipulation, and certainly it would be unlikely any person would find it interesting, maybe some would even say it wasn’t proper mathematics. Suppose then that by chance it turned out that these structures actually have an application, would what I had done retrospectively become more meaningful? Conversely, if I prove results about a class, believing myself to be following intuition, and subsequently prove that the class is empty what am I to retrospectively make of my actions? At the time I was convinced I was making meaningful statements, but then it turned out I was finding round about ways of formally manipulating contradictions.

I had some other thoughts but this is already rather long and confused so I’ll stop.

 

[lugita15]

I think there may be a specific innate capacity for acquiring mathematical knowledge but unlike intuitionism/constructivism I don't believe it's an arbitrary invention of the human mind/brain. Unless I'm misunderstanding constructivism.

The fundamental question is, do you think that mathematics is "out there" to be found, an objective reality that is independent of what humans happen to think?

 

[ThomasT]

The fundamental question is, do you think that mathematics is "out there" to be found, an objective reality that is independent of what humans happen to think?

No. Math is a function of the human situation. An emergent phenomenon associated with our sensory capabilities.

 

[lugita15]

Category theory is quite similar to set theory and both I believe are isomorphic to logic.

I'm not sure what you mean by isomorphic to logic.

Well, set theory focuses on objects, category theory focuses on relations between objects.

You can define a relation as a SET of ordered pairs (or ordered n-tuples_ of objects that satisfy the relation. So I don't think the characterization of set theory as focusing on objects and category theory as focusing on relations is particularly useful.

A set is just another category (the category of sets)

There may be such a thing as the category of sets (usually denoted as SET), so sets are objects in that category, but that doesn't mean that sets are categories.

but how do we express a category in terms of sets? I know we can’t have a set of all sets but isn’t the category of sets suppose to be in some sense complete?

Here we need to invoke the notion of proper class. A class is a collection of objects. A set is a special type of class. The special property of sets is that they can be elements of classes. A class that is not a set is known as a proper class, and it is not allowed to be an element of a class. The canonical example of a proper class is the class of all sets. If this were a set, then it would be the set of all sets, from which you could construct the set of all sets that don't contain themselves, which due to Russell's paradox gives a contradiction. Thus we come to the conclusion that there cannot be a set of all sets that don't contain themselves, and thus there can't be a set of all sets, so the class of all sets is not a set and is thus a proper class. Note that there can't be a class of all classes, because if there were it would contain e.g. the class of all sets as an element, which is impossible because we have just shown that the class of all sets is proper and thus cannot be an element of a class.

How do we systematically decide which classes are sets and which are proper classes, without finding a paradox in each case? The standard procedure is to use the idea of "smallness". Let us call a class "big" if it can be put into a one-to-one correspondence with the class of all sets, and "small" if it cannot be put into such a one-to-one correspondence. Then we identify sets as small classes and proper classes as big classes. This isn't the only possible procedure, so you could say that opinions differ on which classes are sets, but it's the one that's used in things like ZFC and NBG.

Now a category is a class of objects together with a class of morphisms between the objects. The reason there can be a category of sets is that even though there may not be a set of all sets, there is still a class of all sets, which is good enough. Here is something that you may also like: Just like we classes small or big, we can also call categories small or big. A category is small if its class of objects is small, i.e. a set, and it's big if its class of objects is big, i.e. a proper class. (So clearly the category of sets is big.) Note if the class of objects of a category is a set, then its set of morphisms is also presumably a set, and so the category itself can be considered a set because obviously it's just made of its objects and morphisms. So small categories are the categories that are sets. And since they are sets, they can be elements of classes. So we can construct a class of all small categories. And thus we can make a category of small categories (known as CAT), with the morphisms being functors between the categories! (Obviously CAT is a big category.) No wonder category theory is called abstract nonsense.

For give my Ignorance here as my knowledge of category theory is nearly nonexistent. I do know that it is used in the Haskell programming language and sense programming langues are used to represent useful ideas, one might think that in some sense there is a tangible connection between category theory and the physically world.

I think that's far too tenuous a connection. If Hartry Field was capable of removing the notion of number itself from the class of things needed to understand the physical world, certainly shouldn't we be able to eliminate the need for computer programming, if indeed there is such a need as you claim? In any case, what is the use of computer programs other than speeding up tasks humans could in principle do without computers?

 

[lugita15]

No. Math is a function of the human situation. An emergent phenomenon associated with our sensory capabilities.

So then why is it that mathematics is so remarkably self-consistent, when so many of our other thoughts are not? Why is it that mathematics seems to give us so much more than we put into it? It seems like the most plausible answer to this question is that mathematics is not just a product of Man, but rather that it is ABOUT something real.

 

[John Creighto]

As an aside, here is one view of Platonism:
“God has the Big Book, the beautiful proofs of mathematical theorems are listed here.”
― Paul Erdos

http://www.goodreads.com/quotes/show/24501

 

[Maui]

So then why is it that mathematics is so remarkably self-consistent, when so many of our other thoughts are not?

Isn't it the very basic structure that the deterministic portion of reality is built on?

 

[apeiron]

I can't claim to be much clearer about things today, but I do have some thoughts.

In fact yours is the most lucid of posts. Plain commonsense.

I’d argue that this a quantitative difference rather a qualitative one. We have merely reduced the uncertainty in our definitions to a point where it is very unlikely to cause a failure of communication. I’d argue that ultimately our concepts do rest on ideas that are not well understood and have a low degree of common acceptance, as arguments in foundations demonstrate.

Yes, the problem with Platonism is that it reifies what it talks about. Creates the impression of a context independent existence for its objects. And language functions differently.

A word such as dog or triangle acts as a constraint on uncertainty. We go from thinking about potentially anything to some more definite state of thought. Further words add further constraints, so this is why it is a synthetic/constructive exercise. Dog [small, yappy, fluffy] narrows down your uncertainty still further to particular breeds. Like Yorkie, Pomeranian, Bichon Frise, etc.

So if maths is just a more sophisticated language for talking about reality, it must follow the same pattern. It may naively be taken to refer to real world objects, but in fact what it really is the application of constraints to the realm of thought.

To say "triangle" seems naively to be pointing at a Platonic object, but really it is referring to an action of triangle-making, the set of constraints needed to form the shape in question.

The question can then arise about the status of this set of constraints - is it objectively real, mind-independent, etc?

But now the question seems far less problematic. Triangles can arise in our minds and also in nature. In fact they seem pretty rare in nature. Whorls, branching and other patterns are more commonly observed. And where something like simple geometric shapes arise, they are closer to squares and hexagons (cracking patterns in mud, convection cells in heated fluids).

So right there is evidence that maths' idea of "the real" is in fact a little off-beam. The early fixation on polygons was not the most accurate reflection of the world as it is. As the world has a thermodynamic, dissipative materialism that is only much more recently becoming described by mathematical statements (fractals, scalefree networks, etc).

Of course, this discovery of regular polygons which are in fact unlikely objects in nature was a reason for Platonism. The polygons were real in that anyone of the right mind could scribble their (imperfect) outline in the sand. As a constraint on materiality, they could certainly be constructed.

But the actual reason why the triangle, as a psuedo-object, had fundamental importance is that it referred to something essential about spatial relationships. Triangles encode for the existence of flat Euclidean space. So it is not the material object that matters here, but the world it reveals. Platonism is thus in deep error for celebrating the wrong thing.

Digging deeper, you can appreciate that the power of maths is the way it in fact generalises away arbitrary material constraints so as to recover underlying symmetries of the world, the unlimited potentials from which it is derived. All sorts of rough shaped objects can fill space. A triangle becomes the most revealing object because it is in some fashion the simplest way to break the symmetry of a dimensional void. To construct an object that reveals a plane, you only need three sides. Well, a circle is simpler. But what a triangle can encode is the most essential aspect of spatial dimensionality - orthogonality. Directions which are different (a symmetry of action that is definitely broken).

The generalisation to arrive at higher states of symmetry, which can then be in turn broken in mind-controlled fashion - as a constructive choice - is part of regular language too. The concept of dog has higher symmetry than that of a Bichon Frise. The concept of animal, or lifeform, likewise are still more general. So this is just how language works.

The real world is at it is. It is a set of material potentials that exists in some constrained state. Then we imagine this given world as if those constraints have been successively removed. I see this Bichon Frise, but it could be any kind of dog. Behind the particular instance, there is this Platonic object - this state of more generalised constraint, of higher symmetry - that also is "real". And then these higher order terms can be combined to re-create states of more particular constraint. I say dog, small, hairy, yappy, and the space of possibility is again reduced back towards some particular species of pooch.

Langauge is more than just words of course. It also has grammatical rules. So there is both the labelling of reality's constraints and a set of agreed rules, a syntax, for combining them. And even there, the similarities between speech and logic/maths are easy to see.

So in summary, maths arose as an extension of the language game. But it was different because of the way it jumped to an extreme in generalisation. It turned attention away from the particular objects like dogs which are the everyday subject of conversation to the frame within everything must exist. It focused on the objects that most directly revealed the deepest symmetries of nature - objects that actually only were likely to exist in our minds, or in the diagrams minds might draw to communicate, but still, objects that did have a potential to exist, because unconstrained reality had the possibility to be locally broken in that fashion.

The support for Platonism comes from the feeling that the deep symmetries of nature are mind-independent truths. And I would agree they are. But the objects we chose to create to reveal these lurking truths are not themselves "the real". Triangles do not pre-exist the dimensionality they reveal. So there is no need for a Platonic heaven to give them a place to be outside of material reality. Material reality already implies them as possible states of constraint. Although, as said, triangles are not frequently found as actual forms of nature. And a perfect triangle is so hard for even a human to construct that it remains an "in the limit" mental ideal. It is in fact what is not possible in nature.

If you are thinking "triangle", you are referring to a mind-dependent operation - the concrete action that reveals something. But if you are thinking "a concrete act of symmetry breaking that reveals something", then you are now referring to a mind-independent reality - the potential that was there to be broken in such a way.

Platonism is just mis-placed concreteness. Confusing the symbol with its referent.

 

[apeiron]

So then why is it that mathematics is so remarkably self-consistent, when so many of our other thoughts are not? Why is it that mathematics seems to give us so much more than we put into it? It seems like the most plausible answer to this question is that mathematics is not just a product of Man, but rather that it is ABOUT something real.

No, this is just what you would expect from modelling. If you can extract the right principles from nature and encode them formally, then the operations of the model are going to correspond to the operations of the world.

The difference - what makes maths and logic seem so fecund - is that modelling principles derived from the actual world can be used to generate an unlimited number of other imaginary worlds.

The actual world has a materiality that constrains its operations. But with a mental model, we can just choose to ignore that and plug in any kind of materiality we like. So we can pretend to count to infinity, draw perfect triangles, ignore holographic bounds on information, add and subtract dimensions or constants of nature to our content.

So maths abstracts away the materiality of reality (as the actual values are variables which can be plugged back in as the occasion requires). This then makes it no longer about "the real" at all. It becomes a game of the formally possible.

Of course, what then makes maths unreasonably effective - a game that often seems to run ahead of physics and science - is that nature is also exploring the space of the formally possible.

Humans are deconstructing the reality they most directly perceive - one in a highly developed state of broken symmetry - to recover the most primitive possible symmetries.

Nature is headed the other way, developing into a highly particular state by breaking the pre-existing symmetries, the unbroken potentials.

Humans are imagining more and more possible worlds. Nature is reducing possibility to less and less to make an actual world.

So there is a correspondence in what is happening in our minds and out in the world. There is a shared logic. But Platonism reads this correspondence the wrong way round. We are working our way back to what was possible. Nature is working its way forward to what is actual.

 

[lugita15]

Isn't it the very basic structure that the deterministic portion of reality is built on?

Your belief, that mathematics is grounded in the underpinnings of the physical world, is known as physism. The question of why mathematics is so self-consistent is a criticism best leveled at constructivism. As far as physism goes, perhaps the most significant objection is that there doesn't seem to be that much mathematics that is absolutely necessary for the functioning of the physical world. Especially abstract branches of mathematics like Ramsey theory seem to not be grounded in our knowledge of the physical world. And Hartry Field's work in fictionalism is an attempt to formulate all the known laws of physics without any reference to the notion of numbers at all! You can read his book Science without Numbers. So then the question becomes, if most or all of mathematics is not based on the patterns and structure of the physical world, what is it based on?

 

[chiro]

The fundamental question is, do you think that mathematics is "out there" to be found, an objective reality that is independent of what humans happen to think?

If we can 'physically discover' what corresponds to a kind of uncountability in phenomena, then the answer to this question would be yes.

The philosophers IMO, should be working on this question and generating some discussion to give points for and against the premise, but understanding this will help understand some of the why issues for this question.

 

[lmoh]

As far as physism goes, perhaps the most significant objection is that there doesn't seem to be that much mathematics that is absolutely necessary for the functioning of the physical world. Especially abstract branches of mathematics like Ramsey theory seem to not be grounded in our knowledge of the physical world. And Hartry Field's work in fictionalism is an attempt to formulate all the known laws of physics without any reference to the notion of numbers at all! You can read his book Science without Numbers. So then the question becomes, if most or all of mathematics is not based on the patterns and structure of the physical world, what is it based on?

Well, some of these other branches, as I see it, may just be extrapolated from the basic mathematics that we already know, by logic and reasoning, so the connection is still there. As for those that aren't, well I am not sure how much importance they play in the field at all.

 

[lugita15]

Well, some of these other branches, as I see it, may just be extrapolated from the basic mathematics that we already know, by logic and reasoning, so the connection is still there. As for those that aren't, well I am not sure how much importance they play in the field at all.

But as I said, the work of Hartry Field tries to show that real numbers and even natural numbers aren't necessary to the formulation of the laws of physics, so how much mathematics is really grounded in human understanding of the physical world?

 

[chiro]

But as I said, the work of Hartry Field tries to show that real numbers and even natural numbers aren't necessary to the formulation of the laws of physics, so how much mathematics is really grounded in human understanding of the physical world?

One question in response to yours would be not whether something is necessary per se, but rather: Which representation and analysis is 'better' in any respect than another?

I don't disagree that you don't need mathematics per se to really formulate behaviour for anything, but in terms of its use or utility, it makes sense to use mathematics because of its advantages in some respects over other descriptive and analytic systems.

It should be pointed out that we have lots and lots of different languages that are used for many different purposes and each language is often designed in a way that for it's particular use in a particular context, it is optimal. However for other uses it becomes highly non-optimal for that particular context and subsequent use.

We have written languages for writing, spoken languages for speaking, languages for writing code in procedural and non-procedural contexts, mathematics of every kind of form, design languages like flow-charts and other similar constructs, languages for writing music, data structures for representing lots and lots of different things, graphical languages for describing things, and so on. We have braille, sign language, basically anything you can think of, we have some kind of language for it.

The utility of each language is different for different things, and analyzing the utility of mathematics for describing the world against other forms of representation and analysis will answer the questions you are asking.

 

[lmoh]

But as I said, the work of Hartry Field tries to show that real numbers and even natural numbers aren't necessary to the formulation of the laws of physics, so how much mathematics is really grounded in human understanding of the physical world?

But mathematics is still derived from observations in the physical world, regardless of whether or not we can make sense of the physcial world without reference to numbers (but I am not sure what kind of model Field is proposing). It is just a different method of understanding, but that does not mean that it is any less relevant than the Field's approach.

 

[ThomasT]

So then why is it that mathematics is so remarkably self-consistent, when so many of our other thoughts are not?

Because one of the requirements of making coherent mathematical statements is playing by the rules. Expressions in ordinary language (and its variants) are not so proscribed.

Why is it that mathematics seems to give us so much more than we put into it?

I don't think it does that. It's just sets of rules wrt the manipulation of symbols. You can't get any more out of it than the rules allow. Inferring that a mathematical statement is applicable to or in accordance with a certain physical phenomenon isn't the mathematics itself, but rather the philosophy of the mathematics.

It seems like the most plausible answer to this question is that mathematics is not just a product of Man, but rather that it is ABOUT something real.

Everything that we humans do can be said to be ABOUT something real. I suggested in an earlier post that the root of complex math is our ability to discern differences in perceivably bounded structures/objects/groupings. This is a function of our, apparently limited, sensory capabilities. But from that, and with data from experiments using instruments which augment our senses, we're able to make certain reasonable inferences about an underlying reality which isn't amenable to our senses. There's nothing particularly mysterious about that exercise per se, or why mathematics is able to communicate it less ambiguously than ordinary language.

 

[conradcook]

"Rulesism" -- Mathematics is a set of rules, in theory and application.

Conrad.

 

[lugita15]

"Rulesism" -- Mathematics is a set of rules, in theory and application.

Sounds like formalism to me. Here is what I said about formalism in my OP:

Formalism is yet another philosophy; it was all the rage a century ago, but now it's fallen out of favor. Formalists like David Hilbert believed that math is just a formal game we play using strict axioms and rules. But Godel's Incompleteness Theorems cast doubt on this: it turns out that mathematics is too expansive and bountiful (the technical term is "indefinitely extensible") to be captured by a single formal system. Also, it's hard to be absolutely sure that the system we're dealing with doesn't have some inconsistency lurking within. Finally, it seems too much of a coincidence that the universe behaves exactly according to the rules of a formal system we came up with millennia ago. (Unless you believe in computationalism, in which the universe really is just a big computer).

 

[conradcook]

No, no, I didn't mean that anyone formal system is to be the only correct mathematics. I can't believe those formalists would claim that!

C.

 

[lugita15]

No, no, I didn't mean that anyone formal system is to be the only correct mathematics. I can't believe those formalists would claim that!

All right, so are all formal systems part of mathematics in your view, or only some of them? If the former, then there are plenty of formal systems that are not self-consistent, and plenty more that are not consistent with each other. If the latter, what determines what formal systems are part of mathematics and which ones aren't? Whatever criteria you think determines this, is it possible for a computer program to test which formal systems satisfy the criteria and which do not? If so, then it is possible to make one formal system that contains all the others.

I can't believe those formalists would claim that!

That is precisely what formalists believe, although different formalists have various opinions as to what formal system is the right one. David Hilbert, the most famous formalist, believed that Primitive Recursive Arithmetic (PRA), a very weak formal system concerned with the natural numbers, constituted all of mathematics. He argued that we can encode any other formal system using natural numbers (akin to Godel numbering), so that we can reason about all formal systems within PRA. And he had a grand project he was working on, of using PRA to determine which formal systems are consistent and which weren't. But then Hilbert's program was thwarted in 1931 by Godel's theorem, which states that for any "sufficiently strong" formal system F (a criterion that PRA satisfies), F cannot prove the consistency of any system that is "stronger" than F.

Nowadays there are some formalists who think that ZF, ZFC, or ZFC with large cardinal axioms, is the right system. And on the other extreme, there's Edward Nelson, who believes that the correct formal system is a Predicative Arithmetic, a system of natural numbers do weak that you're not even allowed to do exponentiation. Nelson is trying to encode as much mathematics as he can within his system, with the hope of proving that exponentiation is not total, meaning that there are natural numbers x and y such that x^y does not exist! If you're interested I can give you more information about Nelson.

 

[ThomasT]

Nature is reducing possibility to less and less to make an actual world.

I don't quite understand this statement. Could you elaborate a bit?

 

[apeiron]

I don't quite understand this statement. Could you elaborate a bit?

It is just saying that nature is dissipative. For example, take the weathering of the landscape. When rain hits a flat hillside, it can take many paths. There is a state of high symmetry because so many paths are possible and none are preferred. But after a while, grooves and channels start to form. The symmetry becomes broken. Outcomes are now definitely constrained. The drainage patterns become something actual and particular, a unique history. Paths that were once possible are now completely lost.

So humans look at the world around them and extrapolate back towards the earlier unbroken possibilities. We can look all the way back to the unformed potential of the Big Bang.

But the Universe itself has already run down that entropic gradient to become what it is. And it will continue to spread and scatter into the future.

 

[ThomasT]

It is just saying that nature is dissipative. For example, take the weathering of the landscape. When rain hits a flat hillside, it can take many paths. There is a state of high symmetry because so many paths are possible and none are preferred. But after a while, grooves and channels start to form. The symmetry becomes broken. Outcomes are now definitely constrained. The drainage patterns become something actual and particular, a unique history. Paths that were once possible are now completely lost.

So humans look at the world around them and extrapolate back towards the earlier unbroken possibilities. We can look all the way back to the unformed potential of the Big Bang.

But the Universe itself has already run down that entropic gradient to become what it is. And it will continue to spread and scatter into the future.

Ok. Thanks. I now understand the statement in question.

 

[bhobba]

Hi Guys

Thomas T posted in another thread he didn't understand my Platonic view of Math and Physics. It's one of the voting options and it seems abut 13% are on my side. It's not something I really am interested in debating but just in the interest of getting it out there here is a link:
http://www.scienceandreligiontoday.com/2010/04/01/is-mathematics-invented-or-discovered/

I do not fully agree with Penrose's position eg his belief the brain does wavefunction collapse but do believe in the literal existence of a realm where mathematical truth exist and it is that realm that really determines how the physical and mental realm behave. But I will have to leave it to you guys to pursue - its not something that moves me to debate.

Thanks
Bill

 

[Paulibus]

They say "a little knowledge is a dangerous thing". I therefore have some trepidation in
posting in a forum where most participants probably know lots about Physics and/or about Philosophy; both formidable disciplines. Of course that doesn't stop me banging on about facets of these subjects that seem to me to be neglected. Here's another one relevant to this thread. The operations of mathematics, like addition, multiplication etc. Do folk here think they are invented or discovered?

Consider an aspect of change, namely the operation 'to Increase': make more or bigger. In mathematics an operation that effects such change is addition, say of real numbers. An example is the sentence: One plus One makes Two. In physics this quantification of change is used to add like quantities --- physical things like mass, and distance (once appropriate units have been defined). But such algebraically scalar stuff is not all that physics deals with.

Distances often go hand in hand with one or more directions. Even in the simplest case of adding straight-line distances that lie in different directions one needs to talk of mathematical objects called Vectors. These may be scalarised, as it were, analysed into components of a coordinate system and then added. Or addition can be thought of holistically and geometrically; vectors can be added by linking them tail to head in a segmented chain. Their sum stretches from the chain’s start to its end.

When addition involves distances directed along the points of the compass and measured
along the surface of the Earth, the geometrical addition of ‘curved vectors’ (so to speak) on a sphere can lead to the concept of an abstract mathematical object called a Spinor (beautifully illustrated in Roger Penrose’s The Road to Reality, Fig. 11.4). Unlike familiar real physical objects, a spinor has to be rotated about an axis twice, by 2pi, rather than once, to complete a symmetry operation.

Spinors have been a sophisticated feature of elementary addition for more than 80 years now. Is the operation of constrained and non-planar addition on which spinors are based a discovered and eternal Platonic truth? Or is it an evolved description of change in special circumstances that we have invented to rationalise for human purposes our probably specist perception of reality?

I’d vote for a specist take, but it’s not on the list here.

 

[apeiron]

Spinors have been a sophisticated feature of elementary addition for more than 80 years now. Is the operation of constrained and non-planar addition on which spinors are based a discovered and eternal Platonic truth? Or is it an evolved description of change in special circumstances that we have invented to rationalise for human purposes our probably specist perception of reality?

Again, this illustrates the point I was trying to make. Maths is a formal machinery for the construction of constraints.

So out in the real world, a triangle, a path, a channel, an object, a whatever, comes to exist as a matter of top-down constraints. There is some source of potential, some unbound, undetermined, degrees of freedom. And things happen to constrain those freedoms to have a particular form.

Then in our heads, maths is a way of modelling states of constraint via bottom-up construction. We can define a triangle, a path, etc, in terms of step-by-step operations. So we can describe what is out there in the real world using a language - the construction of meaningful statements using words and rules.

Out in the world, a triangle would just happen as an emergent feature of reality. But humans create a recipe for making such things happen.

Crucially, there is nothing special about this kind of construction of constraints via a "language". It is the secret of life. That's what genes do too. Out in the world, a complex protein might form by accident because - like a triangle - you just happen to have an unlikely combination of contraints impinge on a locale. But genes are a mechanism for constructing the set of environmental contraints that will produce such a molecule with a high degree of inevitability.

Actual language - words and grammar - do the same thing at a idealistic level. Left to itself, a large brained animal might happen to form some kind of idea. The constraints that happen to impinge on a mind at some point might create a certain firm impression (such as I see a cat). But language can be used to construct such states of mental constraint with a high degree of inevitability.

Then maths is just a further development of this general epistemic trick. The kind of objects~operations that maths talks about are so generalised, so abstract, that they can be used to construct constraints in the most universal possible fashion.

Genes talk about very concrete stuff - the constraints that regulate metabolic processes. Langauge mostly talks about concrete stuff too - this cat, that dog. The material and the formal aspects of "what exists out there" are still entangled. Though language of course can progress to high abstraction, as in philosophy (so paving the way for science and maths). The particular, local, material aspects of "what is" can be generalised away to leave only the Cheshire cat's grin of the notion of the formal limits that might bound that materiality. So language can come up with pure ideas such as the good, the one, the discrete, the infinite.

Maths then deals only in purified formal notions. It wants to leave materiality completely behind (to the point where mathematicians can despise intuitive mental imagery or illustrations cluttering up textbooks). If materiality is needed, it can be put back in by measurement. One what? Well, one apple, or one cat, or whatever. But leave the messing around with measurement to science.

So discovered or invented? Again, this question is being posed as a forced choice, a case of either/or, when really maths has aspects of both.

What maths is discovering/inventing is the formal half of reality - the fact that reality is the product of constraints on material potential, and so how to (re)construct those constraints.

So the Platonic forms are "out there" in that the potential to materially construct them really exists.

But they are also not "out there" because in our heads they are idealised descriptions. We imagine a realm of perfect triangles and true infinities that are beyond material actualisation (because they are the limit description on acts of material construction).

On the whole, maths still seems more invented than discovered because it does not relate so obviously to the world we directly experience. If we are looking for naturally-occurring patterns about us, we are far more likely to see vortexes and fractals than triangles and infinities. This is because the world is dissipatively material. The constraints that form its patterns arise in way that maths only recently began to model.

But as I say, the early maths - the initial geometric breakthrough - was so striking because it found a way to objectify the symmetry-breakings that must have occurred right at the start of the universe. A triangle is a pretty unnatural pattern to come across as a product of material dissipative structure. But it does reveal the existence of flat Euclidean spatial dimensionality.

We now know thanks to physics and cosmology just how particular and material that "deep geometry" actually is. It is not Platonically existent as Newton assumed to simplify his modelling. Some event - like inflation possibly - had to create a material flatness. And some even more remote event perhaps constrain spatial dimensionality to just three directions.

So physics knows that it has to push backwards from the highly constrained material state of the current universe to a description of the least constrained possible states from which the universe might have arisen.

And maths too has been following the same sort of path by relaxing the constraint on its Platonically existent objects (the impossibly perfect versions of possible material constructions).

As you say, for example, maths has gone from scalars, to vectors, to spinors. It has gone from confinement to a location, to confinement to a straight path, to confinement to a curved path.

This would be why there are the striking parallels between mathematical invention and scientific discovery. Exploring the Platonic realm of increasingly unconstrained form is retracing the steps by which a reality formed by constraints could have developed.

However there is then the question of whether that mathematical expedition has really focused on the meat of things. As I say, natural patterns, natural states of constraint, are the result of material dissipation. So working your way backwards from vortexes and scalefree networks rather than points or triangles might be ultimately the more fruitful path.

This is why the philosophy of maths actually matters.

If you believe maths is invented and arbitrary, you won't care much about the relationship of the formal world to the material world. Only the material world really exists for you.

If you believe maths is Platonic, again the relationship doesn't matter because the formal world exists in its own independent right.

But if you believe that form and material are in interaction to create reality, then this should be your philosophy of maths too. It would guide the way you developed math further, focus your attention on core issues like the way nature constructs its own constraints via material dissipation. And then how "language" can come in over the top of that to take control over natural processes.

 

[epenguin]

I (nonmathematician) put down constructivism though open to argument. Except though it is an invention, that does not make it arbitrary, nor does that deny it is objective.

E.g. arithmetic. We set down (postulate, define) some rules for use in e.g. economic life. We by and large obey and adhere to these conventions and their consequences; sometimes I would like to pull off an operation equivalent to making 0 = 1,000,000 but this violates the set conventions of the arithmetic and if I am found out I am sanctioned. (Of course you will recognise there is some idealisation in my description, which is typical of mathematics.)

I rule out physism. "Mathematics is based on the patterns humans gleam from studying the physical world". Patterns are mathematics. You cannot 'observe' that a year has 365 days until you have the concept 365, or the concepts that contain 365.

Earth and comets travel round sun in paths that are conic sections, or nearly. Objectively true. Earth, sun and comets are not mathematical concepts. Conic sections are. They are our concepts, though true and objective. But they are not 'out there'. The sun which is at one of their foci is out there, and you can see it. But you can't see anything at the other focus. Likewise the cones, of which the paths are sections, are not to be seen in any physical material way though we know exactly where they are. And F = GmM/r² is not to be seen written anywhere in the universe, except in books which we wrote.

 

[apeiron]

I rule out physism. "Mathematics is based on the patterns humans gleam from studying the physical world". Patterns are mathematics. You cannot 'observe' that a year has 365 days until you have the concept 365, or the concepts that contain 365.

Some define maths as the science of patterns. It abstracts the formal description away from the material description so, as you say, there is the mathematical concept (such as "number") and then the pragmatic quantification (observing that number to "exist" materially).

 

[Paulibus]

Thanks for a long and interesting reply, Apeiron. I agree with pretty much all that you say. And my conclusions are similar to yours. Your leading statement that:

...Maths is a formal machinery for the construction of
constraints.

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.

We adapt such machinery to suit discovered circumstances, for example by inventing spherical trigonometry when it was discovered that we need to navigate on a round rather than a flat Earth. Or by postulating spinors when genius is inspired (I'm thinking of Dirac here). 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.