All mathematical structure exist.

[vectorcube]

Appreciate the fact you are now taking the more careful epistemic approach of qualifying "if X is true". That is really helpful to serious discussion (and I do find the possibility worth discussing simply because it is the extremal position of a particular line of thought).

" is true" is metaphysics. It is not an epistemic notion.

 

[JoeDawg]

Example is fermat` s theorm. We now know that the theorm is true( because we have a proof), but conceivably, some alien from outer space would produce a proof that shows that theorm is true. The alien could never show that the theorm is falses even if they want to. Similarly, fermat` s theorm would be true even if fermat never formulated the conjuncture in the first place.

That confuses the theorem, with the evidence for the theorem. Evidence, facts about the world, are what is most often referred to as objective. The theorem describes something that may or may not be supported by evidence.

Proof is a mathematical concept, that generally has to do with logical consistency, not evidence.

Evidence is not 'proof' of anything. Evidence provides a basis for prediction.

 

[Redbelly98]

Not sure I agree.

Re: the elephant. How can there be an objective statement about something only you can see? It is, by definition, subjective. Same with Santa, it is only agreed upon by the general populace that he wears a red suit. That's not objective.

How would you go about falsifying such claims?

These philosophical arguments are tougher than I thought they would be.

 

[apeiron]

" is true" is metaphysics. It is not an epistemic notion.

And "if" - the actually relevant qualifying word here?

 

[vectorcube]

And "if" - the actually relevant qualifying word here?

Does that matter?

 

[vectorcube]

That confuses the theorem, with the evidence for the theorem. Evidence, facts about the world, are what is most often referred to as objective..

Did i say evidence?

The theorem describes something that may or may not be supported by evidence.

Empirical evidence? It is crazy to me why you would talk about evidence here. Math propositions are necessary true. As such, they cannot be falsified by evidence like any scientific theory.

roof is a mathematical concept, that generally has to do with logical consistency, not evidence.

Evidence is not the right word. What you want is deduction. Theorms are deduced from premises. What you say about logical consistency do little to explicated proof. I would say criterion are much more strict.

 

[JoeDawg]

Did i say evidence?

If it is objective, it exists independent of mind. Unless you base your mathematical axioms on some type of common existing 'evidence', then your axioms will be completely arbitrary, and so will your alien's axioms. In which case, you would come up with completely different theorems. Your theorems would untrue for your alien, and vice versa.

Empirical evidence? It is crazy to me why you would talk about evidence here. Math propositions are necessary true. As such, they cannot be falsified by evidence like any scientific theory.

Mathemathical axioms are definitions. Those definitions are based on human experience.

Evidence is not the right word. What you want is deduction. Theorms are deduced from premises. What you say about logical consistency do little to explicated proof. I would say criterion are much more strict.

Where do you get your premises and criterion?

 

[apeiron]

Does that matter?

If something is true does tend to have a different meaning that something is true. So yes, you could say it matters.

 

[vectorcube]

If it is objective, it exists independent of mind. Unless you base your mathematical axioms on some type of common existing 'evidence',

Again, there is no evidence. Math objects to not have any causal relation to physical matter.

then your axioms will be completely arbitrary, and so will your alien's axioms. In which case, you would come up with completely different theorems. Your theorems would untrue for your alien, and vice versa.

I get what you are saying, but the way you say it is wrong.

Platonism is the view that there are objective mathematical facts. Now, platonism is not without it` s problems. The most problematic( one would say the only problem) problem is the explication of how we come to know these mathematical facts. This is an epistemic problem. If mathematical facts exist, then they have no causal connection with the world, and thus, there is really no evidence.

Mathemathical axioms are definitions. Those definitions are based on human experience.

For platonist, math axioms are not definitions at all. The axioms are used to describe mathematical facts.

Where do you get your premises and criterion?

That is outside the issue. I rather we remain focus. At present, you seem to not know platonism, and i think you ought to read about it before you reply. I suggest you read about it, and ask me questions. That way, you can learn something.

 

[vectorcube]

If something is true does tend to have a different meaning that something is true. So yes, you could say it matters.

What does that matter for the topic at hand?

 

[qsa]

wiki quote

"It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the foundations of mathematics program."

I think there is big confusion here between "compelling inevitability" and the "source". I think people get the idea because the source can debated, the "compelling inevitability" is not so compelling. even the "fictional" interpertation does not doubt 2+2=4

no matter what source, these "compelling inevitability" is what existence is made of.Is there any other really "compelling" entities we can count on.

 

[vectorcube]

wiki articles usually suck. It is good to have a general overview, but for more meat, you ought to read the stanford philosophy site.

 

[qsa]

wiki articles usually suck. It is good to have a general overview, but for more meat, you ought to read the stanford philosophy site.

I agree, but I was trying to give conclusion type statement to arguments. It would take too
much time to argue every little concept (philosophy does a good job in not closing issues and openning new ones). My interest is finding how reality works in the physics sense, but I use just enough pertinent philosophy(tammed in Einstien's word) to excute my goal. I learned that when I did my Master's Degree in UK; the stress is on research.

I wonder if you have any thought on the articles in fqxi site.

 

[JoeDawg]

Again, there is no evidence. Math objects to not have any causal relation to physical matter.

Useful mathematical statements like 1+1=2, are abstract representations of the physical observable world. If you are going to claim they exist independently, you need some sort of evidence to show that this is so. The axioms of modern mathematics are not random, they have a solid foundation in the physical, which is why they can describe the physical so well.

Platonism is the view that there are objective mathematical facts.

Plato was wrong. There is no higher reality of forms. Its not necessary, nor is there any evidence for such a thing. The ancient greeks were overly impressed with abstract thinking because their understanding of the physical world was so rudimentary. It was thought that the physical world was chaotic, ruled by the whim of the gods. They could use mathematics and geometry, which was logical and predictable, as a foundation.

But the reason mathematics was logical and predictable is because it is abstract and constructed. It was like the difference between living in a cave, and building a house. The latter was preferable to the greeks because they could design it to fit what they needed. Mathematics was designed and constructed to address certain needs, which is why it appears more solid than say the english language, which is more chaotic.

If mathematical facts exist, then they have no causal connection with the world, and thus, there is really no evidence.

Then they have no relation to this world, and are pure fantasy.

That way, you can learn something.

LOL. whatever.

 

[JoeDawg]

wiki articles usually suck. It is good to have a general overview, but for more meat, you ought to read the stanford philosophy site.

If you really want to learn something, don't rely on secondary sources.

 

[vectorcube]

I agree, but I was trying to give conclusion type statement to arguments. It would take too
much time to argue every little concept (philosophy does a good job in not closing issues and openning new ones). My interest is finding how reality works in the physics sense, but I use just enough pertinent philosophy(tammed in Einstien's word) to excute my goal. I learned that when I did my Master's Degree in UK; the stress is on research.

I wonder if you have any thought on the articles in fqxi site.

I don`t have comment, because i did not have time to look over it.

Here is a site that sort of like arxiv for philosophy of science:http://philsci-archive.pitt.edu/

 

[vectorcube]

Useful mathematical statements like 1+1=2, are abstract representations of the physical observable world. If you are going to claim they exist independently, you need some sort of evidence to show that this is so. The axioms of modern mathematics are not random, they have a solid foundation in the physical, which is why they can describe the physical so well.

Again, mathematical objects do not have any causal relation to physical matter. In philosophy, not many people agree on anything, but this is something everyone agrees if there are mathematical facts.

I see you want to look for a reason that the axioms "are not random". The platonist line of thought is to assume humen have special intuition to know mathematical facts. People can be inspired by nature, but mathematical facts are real.

Plato was wrong. There is no higher reality of forms. Its not necessary, nor is there any evidence for such a thing. The ancient greeks were overly impressed with abstract thinking because their understanding of the physical world was so rudimentary. It was thought that the physical world was chaotic, ruled by the whim of the gods. They could use mathematics and geometry, which was logical and predictable, as a foundation.

But the reason mathematics was logical and predictable is because it is abstract and constructed. It was like the difference between living in a cave, and building a house. The latter was preferable to the greeks because they could design it to fit what they needed. Mathematics was designed and constructed to address certain needs, which is why it appears more solid than say the english language, which is more chaotic.

Well, it is fine if you want to think that way, but platonism( as understood by modern philosophiers) is a coherent view. As with any coherent view in philosophy, there is always problems. I guess you have to make up your own mind.

Then they have no relation to this world, and are pure fantasy

It can `t be pure fantasy. If it is fantasy, the it is pretty easy to deny situations from obtaining, but it is hard to deny "4=2+2". There seems to be an objectivity to "2+2=4" that is independent of sense experience.

 

[vectorcube]

If you really want to learn something, don't rely on secondary sources.

why can ` t you learn something from secondary sources? When people study relativity, i highly doubt they learn it from reading einstein` s paper. In fact, i don ` t see people learning from primary sources quite that often if they are not experts. For their jobs, they need to know details, and publish papers. I think it is quite acceptable to know the argument as expressed in the from of concise propositions, and conclusions.

 

[magpies]

Wouldn't it be interesting if one day we found out electrons come in the shape of the number 1.

 

[vectorcube]

Wouldn't it be interesting if one day we found out electrons come in the shape of the number 1.

i will be surprise if 1 is in the shape of 1.

 

[magpies]

Hello surprise nice to meet you.

 

[JoeDawg]

Again, mathematical objects do not have any causal relation to physical matter.

That is your claim. You have yet to show why you think this is so. The fact Plato believed it is not a proof.

In philosophy, not many people agree on anything, but this is something everyone agrees if there are mathematical facts.

Sure their are mathematical facts, there are all kinds of facts, but your claim was that they were objective. Objective facts are independent of mind. Mathematics is something we learn when we are very young, so its not really surprising that we take it for granted.

humen have special intuition to know mathematical facts. People can be inspired by nature, but mathematical facts are real.

Special intuition? Human intuition is about pattern recognition. Its an evolved capacity, that manifests through our accumulated knowledge. When we are babies we learn to distinguish between objects, when we are older we learn to count, then to add...etc... We are taught about numbers and mathematics. We learn all this through examples, through experience. Once we have the basics down, we can creatively mix and adjust these patterns to deal with new situations. There is nothing magical about it.

Plato didn't have our modern understanding of things. You don't need a magical soul, with a priori truths embedded in it.

Well, it is fine if you want to think that way, but platonism( as understood by modern philosophiers) is a coherent view.

If math has no causal relation to experience, then it would be useless to us, because it wouldn't reflect what we experience. One could certainly develop a completely alien form a mathematics, based on random axioms, but that's not the math we use. We have created a mathematics that reflects how our world works.

but it is hard to deny "4=2+2".

Its hard to do so, because our experience shows us how addition works.

There seems to be an objectivity to "2+2=4" that is independent of sense experience.

That's because 2+2=4 is not an axiom. Its a formulation that relies on axioms that were abstracted from exprience, and taught to you when you were young.

All kinds of things seem objectively true... because we are used to them, because we grew up with them and have developed an intuition about them. Intuition is not objective, it is, by definition, subjective.

 

[JoeDawg]

why can ` t you learn something from secondary sources?

You can learn lots from encyclopedias and wikis. You can learn more from primary sources.

 

[vectorcube]

That is your claim. You have yet to show why you think this is so. The fact Plato believed it is not a proof.

Actually this is not my claim. This is the standard claim in any philosophy textbook.
No, mathematical facts do not interact in any physical process, or physics interaction.
Are you sure you really want me to prove this?

Sure their are mathematical facts, there are all kinds of facts, but your claim was that they were objective. Objective facts are independent of mind. Mathematics is something we learn when we are very young, so its not really surprising that we take it for granted.

This is for technical reasons. To say that it is a fact implies that it is objective, and mind independent. The word "fact" also means "state of affair" in philosophy in case your are interested.

Special intuition? Human intuition is about pattern recognition. Its an evolved capacity, that manifests through our accumulated knowledge. When we are babies we learn to distinguish between objects, when we are older we learn to count, then to add...etc... We are taught about numbers and mathematics. We learn all this through examples, through experience. Once we have the basics down, we can creatively mix and adjust these patterns to deal with new situations. There is nothing magical about it.

Plato didn't have our modern understanding of things. You don't need a magical soul, with a priori truths embedded in it.

Focus on the topic!

I am not at all trying to argue for platonism. I am tell you that platonism has a lot of modern following, and it is a consistent view as any philosophical views can be.

If math has no causal relation to experience, then it would be useless to us, because it wouldn't reflect what we experience. One could certainly develop a completely alien form a mathematics, based on random axioms, but that's not the math we use. We have created a mathematics that reflects how our world works.

Stay focus, please.

I hope you know that to say non causal is really mean to have a priori knowledge.
A priori knowledge is not at all useless. math is a priori necessary. mathematical proposititons are true in all possible worlds, so they obviously work in the actual world.

Its hard to do so, because our experience shows us how addition works.

Perhaps, but platonism is a metaphysical thesis. Perhaps, people learn how math work from experience, but the math is objective.

That's because 2+2=4 is not an axiom. Its a formulation that relies on axioms that were abstracted from exprience, and taught to you when you were young.

All kinds of things seem objectively true... because we are used to them, because we grew up with them and have developed an intuition about them. Intuition is not objective, it is, by definition, subjective.

As in the previous reply. Platonism is a metaphysical thesis. You can come up with as much stories as you want of how people come to know math, but the math is objective( according to platonist).

You can learn lots from encyclopedias and wikis. You can learn more from primary sources.

Textbooks are usually secondary source. They tend to summerize arguments in digestable bits from many primary works in the form of research papers, and books.

To me, you really cannot say primary source are better, because it depends on your purpose. If you are a kant scholar( say), then you get pay to read the original work, and it is your responsibility to read the original work. If you want an overview of the field, i suggest you read textbooks because they focus on what are the essentials for the most recent debate, and what are the arguments for\againist a thesis. I personal think it is better, because it draws on the perspective of many people all at onces, and what separate them are the strength of their arguments.

 

[Pagan Harpoon]

Old, convetnional mathematics is designed such that it reflects very closely the nature of our universe as we see it. As we progress in science and maths, it gets closer and closer to perfectly describing everything in it. I don't doubt that there exists a system of mathematics that can perfectly describe our universe at every level and I think that most people would agree with me there. I view the fact that there are perfectly consistent and mathematical systems that don't relate to the physical existence of our universe or our experience to be evidence that mathematics is not subordinate to our physical existence. Again, whether or not mathematics is a human thing that we invented to describe the universe that we see or a preexisting code that underlies the operation of everything that we just discover and examine cannot be demonstrated or argued either way, either accept it or refuse it, but you are mistaken if you believe that your view of the matter is necessarily correct.

I would like to make an addition to Vectorcube's mention of mathematical facts. That 2+2=4is not necessarily an objective mathematical fact. However, I think that it is reasonable to suppose that if objectivity exists at all, then that 2+2=4 follows from the definitions and axioms that are usually understood is as objective a fact as there ever can be. Whether or not those definitions and axioms are good ones to take, on the other hand, is subjective.

 

[apeiron]

That's because 2+2=4 is not an axiom. Its a formulation that relies on axioms that were abstracted from exprience, and taught to you when you were young.

This is an excellent point. All axioms would seem to be statements that could be true, could be false. They don't seem to have an absolute necessity about them.

The consequences that flow from an axiom would be necessary, but not the axioms themselves - the usual Godellian modelling point.

Perhaps if vectorcube really wants to focus, he can suggest some axiom, like axiom of choice, that he believes has necessary truth (no choice but to exist). And prove to us how it is a necessary truth.

 

[vectorcube]

One more time,

The axioms of math according to the platonists are used for descriptive purposes.

 

[apeiron]

I am not at all trying to argue for platonism. I am tell you that platonism has a lot of modern following, and it is a consistent view as any philosophical views can be.

It is impossible to ever know what you are arguing for as you say this kind of thing every time you find you really have to explain something. Very weak.

But platonism is only consistent in the way ghosts, gods and other things are impossible to disprove because they are treated as being not part of our world. In fact platonism is even worse according to you if the forms have no causal connection with the world.

Plato did try to account for how forms brought organisation to the chora with his flickering shadows on the cave wall allegory. So he did think there had to be a causal connection of some kind.

I suspect the reason quite a few philosophers and mathematicians like platonism is that it gives their disciplines greater weight - science and its observations can again play second fiddle to those employing pure reason.

But I've actually found true platonism to be quite rare even there. Notable exceptions would be people like Roger Penrose and David Chalmers.

I remember asking one philosopher at the first Tuscon consciousness conference why Chalmers was suddenly the big star when he talked such nonsense. Oh, he is bringing back dualism again, said this guy exactly like a naughty schoolboy.

Anyway, once philosophy starts putting its explanations in realms where their being true or false makes no observable difference - well I think we can say internal consistency becomes irrelevant. It is no longer modelling as there are no measurements.

It is the same as religion, psychic powers or any other just-so stories. At any point where an idea ceases to have consequences for observables, then it is no longer part of what I would consider to be the knowledge developing process.

 

[Pagan Harpoon]

I am inclined to think of a model that makes predictions about the world that can be tested so as to prove them true or false as science rather than philosophy.

Both mathematics and physical existence are there, let's not bother calling that into question. I don't think it's an unreasonable position to suppose that physical existence came before mathematics and gave rise to it, though that view does not appeal to me. It is no less reasonable to propose that mathematics comes before physical existence and is on a higher level than it. This is not equivalent to gods, as that involves creating a new entity that has never been observed and can't ever be.

 

[apeiron]

I don't think it's an unreasonable position to suppose that physical existence came before mathematics and gave rise to it, though that view does not appeal to me. It is no less reasonable to propose that mathematics comes before physical existence and is on a higher level than it.

It seems pretty easy to make a distinction between ideas with measurable consequences and ideas without. Whether we call it science, metaphysics, philosophy, maths, or whatever, is beside the point. Within all these disciplines (witness the string landscape debacle) it seems a fair dividing line. Those doing modelling without possible consequences are simply doing something else.

And from a hierarchy theory point of view, when you are talking about maths and physical existence, this loses all its angst if we just accept them both as accounts of the world at different levels of generality or abstraction. They are not different in kind, just different in degree.

So we could say we have a naive direct experience of reality, an experience of reality mediated at the level of physical modelling, an experience of reality mediated at the even more rarified level of mathematical modelling.

At no stage do we ever know the world directly. It is always impressions mediated through ideas. But we ascend from a very subjective level of modelling (what it is like for me looking out through my eyes), to increasingly objective - what it is like for me to look with my eyes through these instruments and controlled laboratory situations, then through my eyes employing these philosophical and mathematical concepts. By the time I get to the top level, it seems like I am seeing objective reality.

 

[vectorcube]

apeiron,

It is impossible to ever know what you are arguing for as you say this kind of thing every time you find you really have to explain something. Very weak.

Explain what? What is difficult? The majority of the mathematics and philosophy community are platonist. Every one knows the pros, and cons of this view( until you don` t know it, and that is pretty weak). The problem is the epistemic problem inherent in the platonist view, and it is the only major problem with this view. If you know anything at all of the other views in the philosophy of math, you know the others are in a much worst situation.

But platonism is only consistent in the way ghosts, gods and other things are impossible to disprove because they are treated as being not part of our world. In fact platonism is even worse according to you if the forms have no causal connection with the world.

You just know this? come on? Do you think i am making this up? Go read about it, and stop claiming that i invented the whole idea. How do you even know i am a platonist anyway?

Plato did try to account for how forms brought organisation to the chora with his flickering shadows on the cave wall allegory. So he did think there had to be a causal connection of some kind.

I suspect the reason quite a few philosophers and mathematicians like platonism is that it gives their disciplines greater weight - science and its observations can again play second fiddle to those employing pure reason

out of topic. Not related to motivation.

Anyway, once philosophy starts putting its explanations in realms where their being true or false makes no observable difference - well I think we can say internal consistency becomes irrelevant. It is no longer modelling as there are no measurements.

This don` t even make sense. Something can be logically consistency, but unfalsifiable.

It is the same as religion, psychic powers or any other just-so stories. At any point where an idea ceases to have consequences for observables, then it is no longer part of what I would consider to be the knowledge developing process

.

This is no argument for or againist. Opinion.

 

[vectorcube]

I am inclined to think of a model that makes predictions about the world that can be tested so as to prove them true or false as science rather than philosophy.

Both mathematics and physical existence are there, let's not bother calling that into question. I don't think it's an unreasonable position to suppose that physical existence came before mathematics and gave rise to it, though that view does not appeal to me. It is no less reasonable to propose that mathematics comes before physical existence and is on a higher level than it. This is not equivalent to gods, as that involves creating a new entity that has never been observed and can't ever be.

You can think whatever you want. I think it is good character to actually know the view you are talking about. You comment about the axioms being subjective is not the platonist line of thought. You ought to know it.

 

[Pagan Harpoon]

And from a hierarchy theory point of view, when you are talking about maths and physical existence, this loses all its angst if we just accept them both as accounts of the world at different levels of generality or abstraction. They are not different in kind, just different in degree.

To not accept both views as consistent and unfalsifiable is idiotic, regardless of whatever effect it has on the "angst" of the issue. That is the way it is.

To challenge the view that mathematics is above our physical universe in this way challenges not just that particular viewpoint, but the whole basis for this discussion. That criticism can be applied equally to the opposing side, there is no reason to claim that it is derived from physical experience either, that just seems to be the popular take.

If you feel that the issue lacks angst, then you shouldn't have wasted your time talking about it.

You can think whatever you want. I think it is good character to actually know the view you are talking about. You comment about the axioms being subjective is not the platonist line of thought. You ought to know it.

I have never mentioned the words Plato, Platonism or anything similar. I am not necessarily talking about any particular labeled philosophy, just what seems to make sense. To deny that someone else in a different universe might decide that a different set of axioms are better than the ones used most by us is to deny the whole idea of the ultimate ensemble.

 

[vectorcube]

At no stage do we ever know the world directly. It is always impressions mediated through ideas.

in case you are interest. Your view here has an anti-realist favor, and it will benefit you to read more of this view.

 

[vectorcube]

I have never mentioned the words Plato, Platonism or anything similar. I am not necessarily talking about any particular labeled philosophy, just what seems to make sense.

Great.

To deny that someone else in a different universe might decide that a different set of axioms are better than the ones used most by us is to deny the whole idea of the ultimate ensemble.

What is this suppose to mean? Someone else in a different universe can do whatever they want.