What's Your Philosophy of Mathematics?

apeiron

But the rules of logic are just as axiomatic at base. They depend on certain critical assumptions, like the law of the excluded middle, which by definition may be true within the realm of our modelling, but not necessarily true of the world.

Fictionalism just seems to be making the modelling point to me. Whether the concepts we employ seem more concrete, or more abstract, they are still in the end all just concepts - general ideas derived by inference from experience.

And I don't really take physism to be refering to physics - especially something so particular as Newtonian mechanics - but rather a codification of metaphysical concepts. So maths/logic is based on our fundamental categorisation of nature - general sharp ontological distinctions such as discrete~continuous, chance~necessity, substance~form, stasis~flux, etc.

My point there was that maths has an intrinsic freedom which means it can be used to talk about real things, but also to talk about imaginary things. Just as language is free to talk about horses and unicorns.

So the same rules of syntax can carry the semantics from the realm of the real to the realm of the imaginary.

Mathematicians set up a machinery to generate patterns. Then they get busy discovering every pattern that can exist as a result of this machinery. Wolfram's exhaustive cataloguing of cellular automata is a good illustration here. Then some of these patterns are discovered to model reality in a useful way. And we feel tempted to believe this is because reality works in this way - although we can never in truth leap that epistemic divide.


The constraints of nature would be self-organised limits on nature's inherent dynamism. So whatever is stable in nature is emergent. The necessity of reality lies in its developmental history.

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.

What mathematicians call logical necessity, the universe would call historical inevitability.

The difference is that mathematicians presume they are unlimited in their pattern spinning - any possible pattern is also (within Platonia) an actual pattern. Whereas reality (as a mix of material and formal cause) erases the possible in developing into something actual.

The tricky thing at the centre of all this is that maths works so well in describing the patterns of reality because it does chop away the material limits of reality - Plato's chora. So the world is modelled in terms of forms, and the material aspects of the world are left unformalised as the separate business of making the measurements which might animate the models.

apeiron

I agree that every -ism seems to apply to some degree. And I would argue that this is because each attempts to mark some definite philosophical boundary on our modelling of the world. We want to be either "completely this" or "completely that", when actually being so extreme is not possible. We must always remain within the boundaries that we can define.

So the actual task would be to narrow down the -ism spinning to its simplest division.

Platonism (our experiencing of form, reason, computation) certainly appears to be one of these limiting extremes. And then our particular material experience of the world seems to be the other.

Maths tries to divorce itself as much as possible from the material and the particular (so as to maximise its abstract generality). But then in doing so, it is defining itself just as much by what it is moving away from as what it is moving towards.

And thus all these -isms, all these attempts to say that maths is founded monadically on "one thing", seem to carry some truth. But look closer, and it is always going to involve this kind of epistemic manouevre. Thesis and antithesis. To become one thing, you have to also become not the other thing. And so every self must include its other.

CS Peirce tried to fix this by arguing that abduction paves the way for induction and so, in turn, deduction.

So all that is actually needed to start the ball rolling is some kind of creative fluctuation, some random or spontaneous leap. A guess is good enough.

Although Peirce also pointed out that humans seem to make unreasonably good guesses. And so our actual starting point for reasoned thinking looks already highly evolved. Brains are natural generalising engines, and the formalised machinery of induction and deduction could emerge quite easily once humans developed the necessary syntactical machinery of speech.

disregardthat

Formalism/constructivism.

The other alternatives are ridiculous! First of all, mathematics is not reducible to logic, something which should be obvious for anyone who know euclidean geometry. Platonism doesn't make any sense, what would it mean if it was so? Physicsm is ambiguous, "based" in what way? I don't understand fictionalism, in what way does it contradict the others?

apeiron

So what are your grounds for believing maths is objective rather than subjective?

As I say, I take the modelling approach where all knowledge is subjectively derived even if rationally structured. Reality may be "mathematical" and so our impressions of it will come to match if we observe closely enough, but there is always going to be an epistemic gap that means our knowledge is never actually objective.

And this stance in turn seems more consistent with our actual beliefs about triangles and other Platonic forms.

The essential attribute of a Platonic form is that it is perfect, absolute, eternal. And Platonists agree that the material world we are modelling is always imperfect. An actual triangle can never exactly match the ideal.

A Platonist responds by saying, well, if my ideal does not exist out there in the real world, then it must exist in some other dualistic realm - that happens to be objectively accessible to the human mind (in some way that doesn't get explained, although divine souls are historically invoked).

So that answer lacks commonsense.

On the other hand, the modelling approach would say a Platonic form is our idea of a material limit. A triangle is a model of perfection which stands as an absolute boundary on what can actually be.

And hey what do you know, out in the real world, material reality is giving the same answer. A triangle is a limit on what it can achieve. It is - as I mentioned - the very place that reality cannot arrive at. Perfection is exactly what lies that infinitesimal step outside what can exist.

So this view now seems like commonsense.

We model the world in terms of its limits - what cannot actually objectively be the case. And the world indeed does not have perfect triangles or anything else in the Platonic bestiary of ideal mathematical objects.

Platonic forms "don't exist" in our heads (even though we can treat limits as conceptual objects and give them names). And they don't exist out there either. So no metaphysical difficulties are raised.

chiro

I don't know if mathematics is a 'random' invention of the human mind, but what I do think is that it is going to most probable being realized given our linguistic abilities.

To me mathematics is like any language although it's focus and application is different.

Lots of people think that language and analysis are two separate things but they are not. When you define something very clearly, you have taken the necessary steps to analyze something and hopefully you are representing something in an optimal way.

The use of the spoken and written words help us do exactly the same things that we often do using mathematics, but the exception is the nature of the language as mathematics is not only broader in its scope, but also extremely precise and these two seemingly contradictory properties create something that is extroadinarily powerful.

To me mathematics as a whole field endeavors to do a few things: it tries to generalize the representation and thus extend the language, it tries to gauge some level of internal consistency within the language to bring clarity to its descriptive capacity, and it tries to create a way of looking at transformations in a general way so that one can build multiple perspectives on an otherwise single thing.

In addition to this, it ends up with these goals to create a language that is remarkable in terms of what can be encoded in only a few set of symbols: again this relates the idea of the breadth of what can described as well as the low information content that needs to describe something that is so broad in its definition.

Like any language, it has its exploratoy aspects where people explore mathematics and create problems and new language within the language just like poets and writers create poems, stories, and plays that create whole new themes, conflicts, and ways of expression and thought. I think it was Hardy that said that he was like a poet and in many ways I agree with him if that was the case.

Language is used to analyze, to create, to abstract, to solve problems, to express oneself and to communicate amongst other things and I think it does all of these better than most other languages.

alt

Is there an alternative missing ? Ie, 'Mathematics is a language'

I have often seen it stated here, even by long term, well respected mentors, that

'Mathematics is a language - like the French language for example'

I didn't vote because of this missing alternative which I think is quite a compelling one, but if I had to, would go for Physism.

lugita15

That is just formalism. It is the view that mathematics is just a symbolic language made up by humans, and that it has no underlying significance or truth to it, except perhaps truth concerning the properties of the language itself. In my write-up in the OP, I discuss some problems with this philosophy; perhaps the most significant issue is Godel's theorem.

Whovian

And, similarly, one might come up with a completely different way to express mathematics, which would be considered yet another language. I think the question here is about the concepts in mathematics, not how we express them.

Pythagorean

Well firstly, I agree that mathematics is a language (but a very accurate one that allows for things natural languages don't!)

but I agree with Whovian; we could ask the same kind of questions about natural language and the fact that "they're a language" isn't really an answer. The question is really whether our conceptual perspective of nature is "realistic".

In the case of mathematics though, a special case, I think it's a question of whether the axioms would still be true if it weren't for humans.

Our natural languages aren't explicitly axiomatic (but then again, I'm not sure if mathematics as a field is really explicitly axiomatic or if that is just an ideal or only applied to particular subfields, or what)

But... I think logical truth statements in philosophy are the comparable natural language version of axioms, though I'd think the subjectivity of natural language contaminates the axioms a bit.

fuzzyfelt

Nice thoughts in this thread.

Paulibus

As I write, this thread has 27 posts. I guess others will follow. This poll has the makings of a long story that, so far, illustrates very well a point I’ve mentioned in the recent “Ultimate Question...” thread: the discussions of philosophers don’t seem take much account of the progress of biology and palaeontology over the last hundred and fifty years or so.

To recapitulate briefly: It is now accepted knowledge, especially from evidence gathered over the last few decades, that we are one of several species of great apes that evolved in Africa over the last few million years. We are the species which, driven by the forcing hand of evolution, somehow acquired the ability and compulsion to invent and communicate with rapidly evolving languages. In the latest few evolutionary instants our drive to talk has led first to the invention of numbers and from this beginning to the evolution of mathematics. And by stimulating technology our ability to count and quantify has helped to elevate our numbers to our six or seven billion chattering individuals that now infest this planet. Think Facebook.

Acknowledging this success should play a part in selecting among the seven ‘isms’ offered in this poll. We now understand better than earlier disadvantaged folk what we are, and should take this into account when discussing what we do and why.

lugita15

Frege wrote his famous book, The Foundations of Arithmetic, to refute Kant's view that arithmetic is synthetic. I find his arguments quite persuasive.But you can define the different systems of geometry, Euclidean and non-Euclidean, axiomatically, without reference to the physical world. You can say that human use of logic, like human use of physics has a psychological origin. But do you really think the basic laws of logic are contingent upon human psychology or the laws of physics? For instance, what about the law that says a statement always implies itself? For instance, "If logic has a psychological origin, then logic has a psychological origin." Do you really think that this law could possibly be wrong, regardless of the universe in which we live or the nature of our psyche? Because if you're skeptical of that, you're questioning the very basis of argument itself.
This is not an accurate characterization of Godel. Roughly speaking Godel's 1st theorem says that any sufficiently strong consistent formal system must be incomplete, and his 2nd theorem says that any sufficiently strong consistent formal system cannot prove its own consistency. I have no idea what you're talking about here.

apeiron

I'd say John Creighto is correct that logic arises out of cognitive evolution. But then, as I say, the modern mind does its little trick of "taking the limit".

So the animal mind is quite happy to induce a general idea, such as a bell means the expectation of food, but the idea would not have the absolutism that we demand of logic.

So an animal would be thinking the equivalent of "the bell is as reliable a signal as possible". While the logician would be thinking either the bell is true or false.

The animal's expectation remains semantically bounded - so it is realistic. The logician switches the game to talking in terms of those bounds, so any subsequent utterances are now applies unreal labels to the world.

If you want to talk about the foundations of maths, this epistemic cut that maths/logic/semiotics makes is critical. It is the trick that moves you across the line into a world that is formally unlimited (where thought, logic, induction, whatever, is no longer materially bounded in the same way).

The grounds of maths/logic is untruth . It pretends the world is full of definite things. And that proves to be a very useful new cognitive trick.

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

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

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.