What's Your Philosophy of Mathematics?

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:
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

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:
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

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, 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

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

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

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:
Approaching UG from Below
http://www.punksinscience.org/kleant...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...0808152428.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

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 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.

Ken G

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

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.

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

apeiron

True, but this is talking about the divergent variety rather than the convergent deep structure. You do of course have both because what polarities make possible is the emergent spectrum that emerges inbetween (as various mixtures of what gets separated).

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.

I agree it is a legitimate question. And the default position will be "all models are the free creations of the human mind". We should be automatically suspicious of any jump from the epistemic to ontic.

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 butterfly effect is not a good analogy for biological processes because that is dynamics unconstrained (the system is unpredictable even if deterministic because measurement error compounds exponentially).

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.

We seem to agree then. Because I am saying that maths is not monadically any one kind of thing. Which is what the poll wants to make it.

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.

It should be clear by now that I would only argue for Platonism (the fact that reality has a deep structure which our modelling can hope to map) to the extent that observation appears to confirm it.

Yes, I agree. You seem to have me now arguing for Platonic fundamentalism when I want to make it plain that Platonism can "exist" only as one of a pair of complementary bounds.

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.

Yes, maths goes to one extreme - tries to be the one hand clapping. And this works because it creates its own complementary extreme. It creates with equal decisiveness the idea of a local, particular, material measurement. The other hand needed to make some noise.

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.

Ken G

Very interesting post, it stimulates a lot of reactions on my part.That is a valid way to slice the choices our brains make, yet I would still argue it is how our brains think constructively about what brains do. The brain making sense of itself will model itself, but the model will, on purpose, take a projection and throw away what doesn't fit. It's a kind of template, the dichotomous analysis. The irony is, we can apply the same template to that analysis-- we can dichotomize, or unify, so we even have complementary choices around the issue of complementarity itself.

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.
We agree that the variety is what is crucial, I'm just saying the way we break down that variety is itself a kind of simplified replacement. We write the digits of a number as a replacement for the number, and we can manipulate those digits in ways that mirror how numbers are manipulated, but the manipulation of a number is not entirely syntactic the way the manipulation of digits is-- the digitization does not replace the semantic meaning of the number, it is merely a placekeeper for it. I see dichotomies similarly, as placekeepers for the varieties, a useful labeling tool, but which does not actually capture the underlying variety-- that variety is irreducible, any syntactic construction of that variety is just a shell, like a robot programmed to mimic the actions of a human being.
Yes, I like that metaphor a lot. I think it underscores the fallacy of "choosing sides" in any debate centered on a dichotomy (like "is life deterministic or random", when we find that life sometimes follows a deterministic scheme and sometimes a random one). I'm just taking that a step farther, and saying that even the dichotomy itself is not something we should commit to, for one such dichotomy is "embrace dichotomies vs. reject dichotomies." The moment we assert a dichotomy is true we find that taking it to an extreme causes it to be a circle rather than an axis, but if we use that logic to assert there is not that dichotomy we lose the analytical power of invoking it. I think there must be some truth to the idea that all analysis is judicious lying.
Indeed. We should also be suspicious there is any true distinction between the two, or that either of them even exists. Yet we must not completely reject Platonic thinking, for then we lose its analytic power. A map is a particular kind of lie, but a good map is a judicious lie that leads us where we want to go.
I think that is indeed impossible. The problem is that if an atom cannot know itself, then neither can a huge and complex array of atoms. The only difference is the atom has not the required structure to invent a judicious lie, but the array of atoms has. That's the key shortcoming of pure reductionism, I agree with the systems perspective there.

I would agree, yet I would still call that a judicious lie. It's like when I tell a class that planetary orbits are ellipses, I know that I am lying, judiciously. Feynman said that science is a way to avoid fooling ourselves; I would add it is a way to avoid fooling ourselves that works by lying to ourselves judiciously.
And yet entropy, as an ontology, is a classic example of a judicious lie. The universe is in one state-- so always has zero entropy, formally speaking. But the concept emerges when we, as analysts, decide that we don't know that state, we know only a class of states that satisfy what we care about. We know not the territory/state, we know the map/class of states. So the concept of entropy is born-- the natural log of the number of states in the class. The map has entropy, the territory does not. But entropy is a mapmaker's key, one of the most judicious lies of all time that some feel underpins at the deepest level all of our understanding of nature.
An interesting point, the way one form of language gave rise to another. Yet I would say that DNA is not really a language, it is we who understand language who also understand DNA that way. A gene has no need for even the concept of language, so certainly has no need to participate in one. It is we who need to see the gene's action in that light, the judicious lie comes from us-- in fact, we invented judicious lying when we invented language.

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.
Which raises another interesting question: what is the meaning of syntax? Syntax is supposed to be distinct from meaning, yet it requires a meaning or else we don't know how to use it in a sentence. We can't connect vocal patterns with DNA patterns unless we understand that the patterns represent something deeper. Another dichotomy comes full circle. And can whatever is the meaning of syntax be a Platonic truth about DNA and language, when we cannot even enforce a Platonic separation between syntax and semantics? The escape hatch is to recognize they are all judicious lies, all of them: syntax, semantics, language, DNA, the works.Yes, getting back on topic, we agree there. The poll is trying to get us to commit to a lie about mathematics that is not judicious because any of the choices either sell math short, or are grandiose and unsubstantiated wishful thinking. A more judicious lie about math is that it combines all those elements in a complex way, but of course if that were really true, then it would have to be true in some Platonic sense, which would make math Platonic, so the argument would come full circle.
And I would say the very idea that math has foundations at all is another judicious lie. Math has attributes that let us recognize it, that's all we can really say because that's how we defined it ourselves. Everything on that list is like a hobo jumping a train simply because it is going in the same direction that they want to go.
But this isn't really any kind of confirmation, because it is our nature to frame our analysis of observations in those terms. We are looking into the mirror, not at something that transcends us. What observation could come out X if our way of understanding that observation is also something Platonic, or Y if our way of understanding it comes from us, when it is the outcome itself that we are trying to understand?
That form of existence is probably pretty close to what I mean by a judicious lie, so perhaps we are not so far apart on this. I'm just adding that the complementarity is also part of the lie, as is seen by how it tends to come full circle if you take it to its extremes. The opposite poles are just directions, they don't exist as destinations because the destinations come full circle.
Here I believe you echo a similar sentiment.

Yes, and I see each as a hat we put on when it serves us. Three different maps that each lie about the terrain in different ways, like a bus schedule, a road map, and a topographic map-- lies when regarded as the full story that become judicious enough to help us achieve our goals when not so framed. Like the poll itself.Yes, there are different times when each flavor of falsification becomes the closest thing we get to a truth that does not, in fact, exist in the absence of falsification.