What's Your Philosophy of Mathematics?

[ThomasT]

I think it starts with the ability to perceive the presence of a definite, bounded something and the absence of that something, and the ability to perceive congruence and incongruence.

It's an emergent phenomenon of an emergent phenomenon ...

I voted for physism.

 

[bohm2]

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.

 

[John Creighto]

But the epistemic trick of maths/logic is to jump to the end of the story - to presume that what emerges is simply existent. So stasis is taken for granted. Limits actually "are" - whereas in nature, limits are precisely what are not. Limits mark the edge of reality.

It is the old debate about infinity. A naturalistic perspective - one that insists on descriptions that are material - will argue that infinity is a limit that can only be approached. But maths - inventing its Platonic realm of forms - just drops the requirement of a material means and takes the limit as something that exists.

With regards to taking the limit, in section II of the introduction of Critique of Pure Reason (Translated by F. Max Muller) there is the following relevant quote.

"II.

We are in possession of certain Cognitions a priori,
and even the ordinary understanding is never without them.

All depends here on a criterion, by which we may safely distinguish between pure and empirical knowledge. Now experience teaches us, no doubt that something is so or so, but not that it cannot be different. First, then, if we have a proposition, which is thought, together with its necessity, we have a judgment a prior; and if, besides, it is not derived from any proposition, except such as is itself again considered as necessary, we have an absolutely a priori judgment. Secondly, experience never imparts to its judgments true or strict, but only assumed or relative universality (by means of induction), so that we ought always to say, so far as we have observed hitherto, there is no exception to this or that rule. If, therefore, a judgment is thought with strict universality, so that no exception is admitted as possible, it is not derived from experience, but valid absolutely a priori. Empirical universality, therefore, is only an arbitrary extension of a validity which applies to most cases, to one that applies to all:"

pg 25-26 of Basic writings of Kant, Edited and with an Introduction by Allen W. Wood, copyright 2001, ISBN: 0-375-75733-3

This quote is not found in the Gutenberg version which is available free online.
http://www.gutenberg.org/ebooks/4280

I warn anyone that Kant's writings are quite difficult to read and consequently I would not suggest him for an introduction to philosophy. If anyone wants to learn about the basic concepts of how we obtain knowledge about the world, I would suggest either Bertrand Russell’s, "The Problems of Philosophy" of Aristotle's "Metaphysics" as an Introduction. As an aside I here Hume is quite difficult as well.

 

[apeiron]

Empirical universality, therefore, is only an arbitrary extension of a validity which applies to most cases, to one that applies to all.

Yes, but here Kant is surely making the contrast with pure a priori universality? So the view he ends up taking is both related and subtly different.

My point here was that everyone recognises the underlying dichotomies at work - such as Kant's synthetic~analytic, or constitutive~regulative, distinctions. And people keep trying to force an either/or answer as to which is fundamental, instead of recognising how the answer is both/together.

This poll was set up as another prime example of that reductionist trope. Either maths has to be real or invented, rational or empirical, objective or subjective, etc.

Kant's answer on maths - that it is synthetic a priori - is in fact a powerful insight here.

The way I would describe it is that humans generalised their way to some ultimate abstractions such as the natural numbers and their fundamental operations. This was knowledge derived from experience of the world, and thus not a priori. But then there is that final step, that epistemic cut, which shifts us from an "imperfect" material world to the immaterial world of our imagination where we grant the unreal - the "in the limit" - a (Platonically) concrete reality. So now we are indeed dealing with analytic truth - what we deem to be just self-evident (having apparently "completely" eliminated the need for material foundations).

Then the genius bit. We start to synthesise with this "immaterial material" we have created. We can get going on constructing mathematical objects using numbers and their operations (or more broadly, structures and their morphisms). So truths become synthetic a prior - true by principles of constitutive judgement.

I say genius, but this semiotic trick was already discovered by nature. Genes and words are also symbolic means by which to construct states of regulative constraint. Logic, maths, computation, information theory, etc, are just taking this habit of nature to a higher level of abstraction and thus applicability.

So it is complicated. The material world creates material states of global regulative constraint via emergence. Then humans create immaterial descriptions of these global states. And from there, we use this mental material to construct immaterial worlds of our unlimited imagining. Then to complete the loop, we can measure our constructed worlds - our mathematical models - against the actual behaviour of the material world again.

So for instance, we give names to numbers, names to operations. A global concept like "many" is reduced to some particular actual Platonic thing, like 122,988.0879. These atomistic entities can then be combined by fixed rules such as "add" or "subtract". Then we can compare the behaviour of the model back to events in the world to show it is all "true" - that the trip into the realm of the rational, though the land of the analytic and synthetic a priori, maintained the empirical correspondence we ultimately must value (unless we are idealists or Platonists, I guess ).

Kant was concerned with further issues, like where our judgements on time and space came from - whether to force them into basket of the empirical or rational. These were a problem at the time because they were clearly general ideas, but ones that seemed to arise right down at the level of basic perception rather than loftier a-perception.

We now know enough about the evolution of the brain to see how those concepts are the result of earlier pre-human rounds of semiosis. They are biologically evolved abstractions written into the brain's architecture, whereas maths is a subsequent culturally evolved abstraction that gets learnt.

Abstraction is indeed about "taking the limit" - crossing the line from material emergence to immaterial reification. And the evolutionary view can show how this has been happening in steps, with the biggest jump being enabled by the human invention of syntactic language.

 

[lugita15]

Formalism/constructivism.

How would you respond to the criticisms of each of those philosophies I give in the OP?

First of all, mathematics is not reducible to logic, something which should be obvious for anyone who know euclidean geometry.

What bearing does Euclidean geometry have on the question of whether mathematics is reducible to logic? Perhaps you mean that it is a matter of physical observation what geometry our universe conforms to. That may be true, but both Euclidean and non-Euclidean geometry constitute internally valid systems. We may of course choose to work with either system, but both can be modeled perfectly using the real number system, which is based on rational numbers, which are based on natural numbers, which are based on ... (see Frege's short book The Foundations of Arithmetic for the rest).

Platonism doesn't make any sense, what would it mean if it was so?

What do you mean what would it mean? In the Platonist view, humans discover, not invent mathematical truth. So when we find out that there are infinitely many prime numbers, we are finding something out about something objective real, just as when we find out how many moons Jupiter has. In the traditional form, Platonism says that there is an abstract realm called Platonic heaven in which all the mathematical objects reside, like the number 9 and the perfect circle, and that through reason, intuition, or both (depending on your flavor of Platonism), humans are capable of finding out truth about the properties of Platonic heaven and what exactly is there. A more modern version of the philosophy says that although mathematical truth concerns a reality every bit as real as physical reality, there are no actual squares and triangles bouncing around in Platonic heaven. Thus in the modern view, it is the statements of mathematics, not mathematical objects, that correspond to the properties of mathematical reality.

Physicsm is ambiguous, "based" in what way?

Physism is the belief that mathematics consists of the patterns the physical world possesses, at least those patterns which humans have noticed so far. To put it another way, in this view mathematics is just the set of recurring properties of the world of that underlies the laws of physics. For instance, if humans see that particle motion conforms to Newton's 2nd law of motion, which is a second-order differential equation, then at the very least humans conclude, or perhaps hypothesize, that calculus, and thus the real number system, is part of whatever makes the universe tick. According to physism, if we lived in a physical universe that behaved differently, then mathematics would be different in that universe. Consequently, if our knowledge of the physical universe changes, the philosophy requires us to re-examine our supposed knowledge of mathematics. (Unlike Platonism, where mathematical truth is independent of our sensory experience of the world.)

This view conveniently solves the indispensability problem: why is it that an abstract subject like mathematics is so often applicable to the study of the physical universe, and why is mathematics so necessary to the formulation of the laws of physics? The response of physism is that this is so definitionally, because mathematics is defined to be that which is necessary to account for the behavior of the physical world.

The main problem with this is that most of mathematics doesn't seem to be grounded in our knowledge of physics. Numbers seem fairly well-grounded: you can count how many stars there are, or how tall a tree is. But what does category theory or Ramsey theory have to do with the physical world?

I don't understand fictionalism, in what way does it contradict the others?

In the context of the works of Arthur Conan Doyle, consider the statements "Sherlock Holmes lived on Baker Street" and "Sherlock Holmes lived on Sesame Street". In reality both statements are wrong, because Sherlock Holmes is nonexistent. But can't we still say that within the fictional world dreamt up by Arthur Conan Doyle, the first statement is right and the second is wrong? In the same way, Hartry field said that mathematics is just a fiction, albeit one that is often convenient. In his philosophy known as fictionalism, there are no such things as prime numbers, but we can still say that within the fictional world of mathematics, there are infinitely many prime numbers. It is in most direct contradiction with physism. Whereas physism says that mathematics is that which is indispensable to physics, Field said that NO mathematics is indispensable to physics, and that it is just a useful fiction we can discard at any time. He wrote a book, Science Without Numbers, which attempts to show how all the known laws of physics can be formulated without any mathematics whatsoever, in particular without real numbers or even natural numbers!

 

[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.