What's Your Philosophy of Mathematics?

[lugita15]

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

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

 

[chiro]

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

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

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

 

[lmoh]

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

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

 

[lugita15]

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

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

 

[chiro]

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

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

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

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

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

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

 

[lmoh]

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

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

 

[ThomasT]

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

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

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

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

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

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

 

[conradcook]

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

Conrad.

 

[lugita15]

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

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

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

 

[conradcook]

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

C.

 

[lugita15]

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

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

I can't believe those formalists would claim that!

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

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

 

[ThomasT]

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

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

 

[apeiron]

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

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

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

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

 

[ThomasT]

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

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

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

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

 

[bhobba]

Hi Guys

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

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

Thanks
Bill

 

[Paulibus]

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

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

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

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

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

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

 

[apeiron]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This is why the philosophy of maths actually matters.

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

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

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

 

[epenguin]

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

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

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

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

 

[apeiron]

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

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

 

[Paulibus]

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

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

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

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

Evolution is the Name of the Game.

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

 

[apeiron]

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

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

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

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

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

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

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

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

 

[Ken G]

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

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

 

[Paulibus]

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

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

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

Success breeds success is a very powerful motto.

 

[Pythagorean]

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

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

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

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

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

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

wheras:

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

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

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

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

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

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

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

 

[Hurkyl]

I might as well chime in with my opinions.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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


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

 

[apeiron]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[apeiron]

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

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

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

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

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

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

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

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

 

[Ken G]

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

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

wheras:

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

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

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

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

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

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

 

[Ken G]

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

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

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

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

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

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

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

 

[bohm2]

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

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

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

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

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

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

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

 

[Ken G]

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

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

 

[apeiron]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[NewtonianAlch]

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

 

[Paulibus]

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

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

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

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

 

[lugita15]

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

That would be formalism.

 

[Ken G]

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

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

 

[Paulibus]

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

 

[bohm2]

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

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

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

 

[Ken G]

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

 

[bohm2]

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

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

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

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

 

[Ken G]

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

 

[apeiron]

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

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

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

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

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

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

 

[Paulibus]

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

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

 

[John Creighto]

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

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

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

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

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

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

 

[Ken G]

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

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

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

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

 

[bohm2]

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

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

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

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

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

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

 

[Storm89]

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

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

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

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

 

[apeiron]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[Ken G]

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

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

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

 

[Ken G]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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