|View Poll Results: What is your preferred Philosophy of Mathematics?|
|Logicism - Mathematics is reducible to logic, and mathematical truths are just tautologies||29||35.80%|
|Formalism - Mathematics is just a meaningless symbolic game that happens to be useful||9||11.11%|
|Intuitionism/Constructivism - Mathematics is an arbitrary invention of the human mind/brain||12||14.81%|
|Platonism - Mathematical truths are truths ABOUT something objectively real, like "Platonic heaven"||13||16.05%|
|Physism - Mathematics is based on the patterns humans gleam from studying the physical world||23||28.40%|
|Fictionalism - Mathematics is just a made-up story that has its own internal logic||4||4.94%|
|Other - Please specify or elaborate||7||8.64%|
|Multiple Choice Poll. Voters: 81. You may not vote on this poll|
|Jun13-12, 09:55 AM||#86|
What's Your Philosophy of Mathematics?
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.
|Jun13-12, 10:18 AM||#87|
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!
|Jun13-12, 12:05 PM||#88|
While there are criticisms for this position, there are some linguistics/psychologists who believe that mathematics is derivative/parasitic from our language ability:
|Jun13-12, 01:23 PM||#89|
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.
|Jun13-12, 06:33 PM||#90|
|Jun14-12, 12:14 AM||#91|
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.
|Jun14-12, 12:35 AM||#92|
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.
|Jun14-12, 04:21 AM||#93|
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.
|Jun14-12, 01:37 PM||#94|
Blog Entries: 3
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.
|Jun14-12, 06:27 PM||#95|
|Jun14-12, 07:08 PM||#96|
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
|Jun15-12, 12:51 PM||#97|
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.
|Jun15-12, 06:26 PM||#98|
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 modelled 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.
|Jun16-12, 12:34 AM||#99|
|Jun16-12, 01:21 AM||#100|
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.
|Jun16-12, 08:07 PM||#101|
Again, you want to argue that models are just arbitrary ideas that we project onto the data. So if my chosen idea just happens to be "dichotomies" then I can go in and carve up some phenomenon in convincing fashion using as many dichotomies as it takes.
I agree that modelling does have an arbitrary, free, basis. We can try whatever works. But then it becomes interesting that only certain ideas seem to work really well, even universally. These ideas look to be the way nature actually works - although we can never "know" that, just observe it to be likely.
Reductionism (that metaphysical mix of atomism, determinism, monadism, mechanicism, local reality, effective causality, etc) is one general idea that works really well.
And then there is the complementary tradition of holism which is about dichotomies, hierarchies, top down causality, indeterminacy, etc. Which works better when it comes time to tell the whole story of course!
Would a dichotomies approach be stronger if all the brain's architectural divisions could be reduced to just a single description? Yes, it would certainly seem less arbitrary (a projection onto the data) and more like the deep structure of the data.
I would start out by saying we shouldn't expect a simple single answer because the brain is a product of both evolution and development. Development is free potential but evolution locks in past history. So the story on brain evolution is a complex interaction between accumulated design and the addition of new possibility (such as by creating new room at the top by expanding the cortex).
But if we step back to the purposes of brains, they are there to make decisions. To make choices. And how can you make a choice unless you have alternatives? And how can you make the most definite possible choices unless the alternatives are dichotomous - reduced to either/or, to a binary yes/no, like retreat/advance, attend/ignore, expected/surprising.
Again, you will probably say that intelligence is defined by having a variety of choices. But as I say, that describes the variety that emerges as a result of the deeper structure - the ability to break the world down by polarities.
My favourite example of the primitiveness of this is the flagella that drives a motile bacterium. Spin one way and the threads tangle, driving the cell forward. The bacterium can follow a chemical gradient, head towards a food source. But then reverse the spin and the threads untangle, the bacterium begins to tumble randomly. So if falling off the scent trail, the bacterium can switch to search or escape mode.
The asymmetry of choice - as determined/random - in a nutshell.
But on the other hand, reality must actually have some kind of deep causal structure. It does not seem like an arbitrary bundle of happenings does it? It does seem to have a developmental history, a systemic and patterned materiality. So it is not impossible that our models of its deep structure could be essentially correct.
The whole point of biology (and its use of languages to construct constraints) is that such dynamism is harnessed. Constraints are applied to channel what happens.
There would not be life/mind without this trick of being able to harness dissipation-driven dynamics. So this is why we can say what "math is". It is not some unpredictable consequence of blind evolutionary change, it is instead the very predictable development of the constraint machinery which in fact defines life/mind.
You want to argue that the brain could have evolved any old how. It's just one accident on top of the other. But this is old-style Darwinism (the "modern evolutionary synthesis" of the 1960s). Today you would talk about evo-devo, and this is based on the idea that there are in fact deep structural principles at work. Existence is based on the dissipation of gradients. Life/mind arise as informational structure that locally accelerates the entropification of the Universe.
So there is a deep general principle at work. But then also some happenstance about how things actually work out.
For example, life/mind arose on the back of one kind of language - genes to code for enzymes that could control dynamical chemical cycles. But then H.sapiens stumbled upon actual language - words to control the thoughts that determine our actions.
Was it Platonically inevitable that human grammatical language would arise? Would it have to happen on any planet where some kind of life/mind was happening in sufficient abundance - given enough variety, would some species have to luck into this structural attractor, this pre-existing, ready-waiting, niche?
Personally I would say there is a healthy dose of both - of both random luck and Platonic inevitability. The luck is down to the fact that brain evolution was not headed in that direction. The evolution of an articulate vocal tract - the imposition of a new kind of serial output constraint on vocalisation - looks a pretty chance direction for events to have taken. On the other hand, it was then a very short step for this exaptation to be exploited for symbolic/syntatic purposes. Once there was a species that could chop up a stream of sound into discrete syllables, the machinery for a new level of coding could be used for exactly that.
And definitely this is all about modelling.
But then, modelling is dichotomous - not just in terms of the relationship between the map and the terrain, but even the map itself has the tension of an internal division.
Our mental mapping of the world divides into ideas and impressions, the theories or formal constructs that are a general inductive understanding, and then the measurements, or expectations, or predictions, that are the local deductive particulars.
Measurement is often claimed to be the objective part of the process of modelling, but of course it always remains some mind's particular impression (such as a reading on a dial, a number on a counter, etc). I know you favour the Copenhagen stance on these things!
So again, where does math stand in all this? It is caught up in the general business of modelling, so it is fictional, intuitive, constructive, etc, foundationally. But at the same time, it is trying to stand at one extreme pole of the modelling process. It is trying to go and stand over at the end of our most general possible ideas. It is trying to be a pure description of form. And then to the extent this division that emerges in our mapping is also true of reality, of the terrain, then maths is going to end up "Platonic".
As I say, this may yet be telling only half the story. But that can only be clear once the foundations of maths is actually clarified.
So maths is extreme because it goes as far towards Platonic rationalism as we can imagine going. Which is good because that then makes the other side of the equation, the need to measure the local material particulars of the world, a matchingly precise task.
The legitimacy of the maths is wholly dependent on empiricism as a result. If triangles in flat Euclidean space do not have angles that sum to pi, then the formal model is screwed.
The maths comes to seem like it is "all subjective". It is a realm of ideal forms discovered rationally. And the measurements likewise come to seem "all objective". They are the brute material facts that exist out in the world.
Yet really, both formalised models and material measurements are only ever in our heads as part of the dichotomy of mapping.
This is just a restatement of Copehagenism (which followed from Bohr's shocked need to deal with a world that actually appears foundationally dichotomous - always at root complementary in nature).
The problem with the Copenhagen interpretation is then that once the simple mechanical view of causality had been shown to fail (at the extremes of its range), the choice was to reject then any chance of a "true" model of causality. The observation were whatever they were within whatever the framework of observation happened to be. It was all taken to be quite arbitrary, with no possibility of systematisation.
Yet in my view, a constraints-based approach to causality fits QM like a glove. Asking questions of reality can reduce its inherent uncertainty to the point it seems very certain - but cannot in principle eliminate all uncertainty.
You can see how these themes keep repeating. We spend so much time trying to disentangle epistemology from ontology - to form that crisp foundational dichotomy between map and terrain. And then we find that the two seem in fact deeply entangled.
In the realm of our minds, the maps are dichotomised into "subjective" rational forms and "objective" material measurements.
Then the bigger shock (perhaps). Out in the world, the terrain is also ontically dichotomised into its "subjective" forms and "objective" materials. Or rather, the self-constructing causality of global constraints in dynamic interaction with local degrees of freedom. A Universe that decoheres itself into structured being via some kind of semiotic or "self-observation".
So this would be where we differ.
I think we can develop a legitimate model of reality in which the ontology involves an epistemic aspect - the necessary decohering observer is made part of the entire system (in the guise of top-down constraint, the contextual information, a generalised environment). We can hope to make a map of the entire process.
But you would defend the more agnostic Copenhagen position where there is a map, and there is a world, and we can never say much more except that epistemology and ontology are fundamentally divided in this fashion. So the default philosophy is that modelling-associated activities like maths are arbitrary at the foundational level, even if useful in a pragmatic fashion.
As world views, we thus have naive reductionist realism, agnostic Copenhagenism, and constraints-based systems thinking.
I agree Copenhagenism is the correct default position - the place you have to retreat back to under pressure. But naive realism is a highly pragmatic choice. It works in the middle ground where humans mostly live. And systems thinking holds out the hope of getting "closer to the ultimate truth", to seeing the whole of reality within the one model.
|Jun18-12, 02:31 AM||#102|
Very interesting post, it stimulates a lot of reactions on my part.
I think what happens is, each of our choices, taken to an extreme, tends to come "full circle" back to the seemingly opposite choice. Complete unity is too bland to convey meaning, while contrast is "crisp", as you might say. But crispness is a kind of intentional illusion, inventing distinctions out of the unity that underlies those distinctions-- nothing is ever actually crisp, crispness is not the way of the world. I recall a famous military general, I forget, who joked that he never retreats-- but sometimes he advances in an opposite direction. It was intended to get a laugh, but there is also a truth to it-- the attack/retreat dichotomy is invented from the unity of strategic military maneuvers, just as a cornered animal might lash out agressively in what is actually a desperate attempt at escape, or a retreating army might actually be luring their pursuer into a trap.
A classic example of this "coming full circle" effect in philosophy is the rationalist/empiricist dichotomy. We all know that we combine mental analysis with sensory perception to make sense of our environment, but the rationalist emphasizes the mental analysis as the "truth" of the matter, while the empiricist emphasizes the sensory perception as the deeper arbiter of what is real. But if we take the empiricist approach to its logical extreme, we say that a sensory perception is not the light entering the eye, for light can enter the eye of a dead person-- the perception is the signal in our brain that is made when light strikes our retina. And it is not just the neuron that fires, for a neuron can fire even if we are distracted and fail to register the perception, it is a complex process going on in our brains that registers the perception. But complex processes going on in our brains are just what we normally call thought, and that's the seat of rationistic truth! So extreme empiricism is actually a form of rationalism, the crisp dichotomy disintegrates under the microscope.
Similarly, if we take rationalism to its logical extreme, we say that the mind is able to connect with truth, but the way we connect with truth is we perceive our own thoughts. Since our brains are also natural systems, presumably, then perceiving our own thoughts is also a form of empiricist truth, something that has a place at the table of reality simply because we perceive it to be there. These dichotomies we make as a useful tool are not actually true in any deeper sense.
And of course, even that last statement is a judicious lie about judicious lying, it can't really be true because humans cannot be separated well enough from language to say we invented it, for as you put it, as soon as we say we invented language, we find that DNA satisfies our meaning, but then DNA invented us, so we end up with one language inventing another, which tells us nothing about what language is or where it comes from.
Logic is no better off than language. If we say that logic is based on the true/false dichotomy, then I say it is based on a lie, albeit a very judicious one. So what do we do with logic when we see it as a lie that works? You could claim that a lie that works is not a lie, but I don't mean it is lie in the sense that something else would be true, I mean it is a lie in the sense that truth is something we just invented, and logic is its syntax. If we invented truth, then Platonic truth is a lie, but it is a judicious lie that allows us to invent the concept of truth in the first place. Truth requires a lie to even be possible, and the true/false dichotomy comes full circle.
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