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Nature is not inherently mathematical anymore than mushrooms are inherently psychedelic

(An ignorant rant on the nature of mathematics, and the mathematics of nature)

When we look at a plant (or fungus) and it happens to produce a chemical that happens to bind to a receptor in our brain that happens to produce a psychedelic effect, we don't conclude that there is an inherent psychedlicness to plants. We just view this as an coincidence of the fact that our brains, the brains of others, and the plant emerged in the same world.

We also realize that this relationship can be confusingly bidirectional. A chocolate cake is not inherently sweet. There is no inherent sweetness property. If we want to study sweetness we need to look at our brains, not at chocolate cake. Actually, we need to look at the interaction of our brain and thing 'like' the chocolate cake.

If we continue this thread, and ask: why do we like cake? Is it because the cake is sweet? Or is the cake sweet because we like it? An evolutionary biologists would need to operationalize these terms, and try to give a story of how high fat, high glucose foods made survival more likely and thus our ancestors that had a stronger preference for those foods over something less nutritious were able to reproduce with slightly higher probability than our less glucose predisposed great-aunts. Now the explanation seems very confusing: we need to look at the cake, our brain, our history, and the system in which we were embedded.

What of mathematics? Why do mathematical physicists so often conclude that mathematics is inherent in nature? Why is mathematics not an accident of our faculties of understanding having emerged in the same system as the things we wish to understand?

For me, mathematics is The Mind. It captures and studies that which is understandable. When I study mathematics, I am not exploring some Platonic ideal from which I am separate. As I study mathematics, I am exploring myself, the minds of others, and the limits of understanding.

When I am doing science, and I happen to write down an equation that describes some physical process. I am not amazed by the simplicity or beauty of nature. I am amazed by the power and breadth of our faculty of understanding in being able to describe another phenomena. Here's the bidirectional rub, though. If I couldn't understand it, would I study it? Could I study it? In this view, the reason mathematics is so unreasonably effective for describing the physical world is because we can't really look at the parts of the world that she can't describe. At least we can't understand that we are looking at those parts of the world.

Does this stance mean that something like the Large Hadron Collider is just a multi-billion dollar group psychoanalysis section?

/cc +Abel Molina +Sina Salek +Kyler Brown +Yunjun Yang
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11 comments
 
Say it ain't so! I been eating mushrooms right and left. Maybe I should try someplace besides the grocery store. :)
 
Interesting view ... I could have fun with this .... 
 
"Universe is mind." - Allen Ginsberg (via #math)

That guy did a lot of acid.
 
I believe the argument above is at least incoherent. 

I start with a point which is not particularly relevant, but worth noting. You mentioned "There is no inherent sweetness property." Although as a reductionist I sympathise with this view, but if we want to be accurate we can't take this as granted. I personally would like to think that the world consists of only particles and fields interacting with each other. But ultimately, I wouldn't base a second controversial argument on my hard reductionism. We don't even understand why worms, which we managed to map their entire nervous system, move in a certain way, let alone understanding of human perception by a reductionist view of the brain. 

The evolutionary basis for us liking sweetness (as opposed to the perception of sweetness itself) is certainly correct. I shall no dispute that paragraph. However, I believe it becomes incoherent when you draw conclusion about mathematical explanation of the world. In fact I have strong bias against it. Even if Maths, as Brouwer and his fellow intuitionsts say, is a languageless activity of the mind (which is by the way indirectly undermined by Chomsky's account of universal grammar), even then the analogy fails. Because it's not just a form of perception.

Your claim is that, to paraphrase you, mathematics is an accident of our faculties of understanding having emerged in the same system as the things we wish to understand. This would certainly make sense, but only to an extent. How do you explain the over-mentioned examples of theories which were first result of relaxation of an axiom, but then were found in foundational studies of the nature. Non-Euclidean geometry is among them. We are certainly not evolved to understand that our spacetime is curved. We can surely make analogies, say of curvature of light passing through some non-uniform lens, but this is just an analogy, not understanding the curvature of spacetime, if by understanding we mean the reduction to the common sense. Hence your fungus analogy requires interchangeability of common sense and understand and therefore fails.

Nonetheless, the paragraph the follows in your note suggests that by understanding you must mean something beyond the common sense; lets use Wittgenstein's definition: the ability to put the concept into use. To apply this definition to shed light on our problem I shall slightly change you conclusion namely that "we can't really look at the parts of the world that she can't describe". Because surely up to now we have encountered several cases of such observation, which may be limits of human understanding. For instance it might be the case that we can never describe human consciousness in terms of a mathematical model. However we don't conclude that the ontology of human consciousness does not posses a structure. And I deeply believe that any phenomena which have structure aught to have an ontological abstract mathematics, even if we cannot put that into use, i.e. they are beyond our cognitive capacities. So the way I would change your sentence is to subscribe again to Wittgenstein: "What can be said at all can be said clearly, and what we cannot talk about we must pass over in silence." One should not conclude this as a failure of mathematics, but the failure of cognitive capacities to study that bit of maths.  

I thought about this issue for a while. Given that I don't specialise in that area, I don't claim it to be credible.  Nevertheless, my conclusion was that the astonishing compatibility comes from the fact that in both cases of maths and nature we are dealing with self-consistent structures. The foundations of mathematics (you can reduce it to logic if you are mathematician and probability theory if you are physicist), in my view being our constructed language, was based on our intuitions gained initially from the nature. So if you will, you can consider a type of causality (or determinism in this case) in the way mathematics is constructed. Then because its boundary condition is fixed in the nature, the whole construction becomes an amazing language to describe nature.

These days I am working on a relaxed set of Kolmogorov axioms, and I see this even more than before.

Thanks for tagging me, it was a pleasure to read your opinion about the question I asked you back in 2010 :-)
 
Thank you for responding +Sina Salek! Sorry that I have been such an awful email correspondent. I can't wait to see you at another conference/summer school or mountain hike! My interaction with you in Vancouver has definitely been important in shaping most of my views on this. To add to the quantumness, I will also ping +Niel de Beaudrap since his philosophical thought is insightful.

> "There is no inherent sweetness property"

I was trying to follow Dennett, here. The key point is really that a property of sweetness is not independent of the mind that perceives it, or at least the sensory organs that sense it.

> Even if Maths, as Brouwer and his fellow intuitionsts say, is a languageless activity of the mind (which is by the way indirectly undermined by Chomsky's account of universal grammar), even then the analogy fails. Because it's not just a form of perception.

I don't understand how it fails, I think you are conflating sensation and perception. Under any reasonable definition, perception is an unbelievably active process. To a small extent it is shaped by your sensations, and to a large extent it is shaped by your background knowledge. I don't see how math (or Chomsky) conflict with this.

> How do you explain the over-mentioned examples of theories which were first result of relaxation of an axiom, but then were found in foundational studies of the nature. Non-Euclidean geometry is among them. We are certainly not evolved to understand that our spacetime is curved. We can surely make analogies, say of curvature of light passing through some non-uniform lens, but this is just an analogy, not understanding the curvature of spacetime, if by understanding we mean the reduction to the common sense.

I don't follow this point at all. When did I say anything about common sense? Common sense is an awkward beast and a very narrow subset of mathematics. I would be hard-pressed to say that even Euclidean geometry is common sense for most. Relaxations and manipulation of axioms is not somehow less natural to me than intuition. Just because mathematics is inherently tied to the mind(s), doesn't mean that there cannot be progressively exploration and growth of the field. Much as a Buddhist monk can discover oneself faster under the guidance of a Master.

> Because surely up to now we have encountered several cases of such observation, which may be limits of human understanding. For instance it might be the case that we can never describe human consciousness in terms of a mathematical model. 

I cannot think of an example for this "surely". There are definitely things like consciousness where we have poorly defined our terms and keep making category errors, but usually it is obvious when these errors are being made. (Tangent, for some mathematical treatments of consciousness, take a look at the answers here: http://cogsci.stackexchange.com/q/987/29)

What I am most afraid with my 'not-understandable' statement is that it is empty. I don't think that I can define a reasonable way to test if something is 'understandable' because every test seems to be inherently self defeating.

> However we don't conclude that the ontology of human consciousness does not posses a structure. And I deeply believe that any phenomena which have structure aught to have an ontological abstract mathematics, even if we cannot put that into use, i.e. they are beyond our cognitive capacities. 

That would still be understanding, no? In the sense that we are aware of it possessing a structure? Once again, I think I have pushed myself into a corner.

> So the way I would change your sentence is to subscribe again to Wittgenstein: "What can be said at all can be said clearly, and what we cannot talk about we must pass over in silence." 

I like this reformulation

> One should not conclude this as a failure of mathematics, but the failure of cognitive capacities to study that bit of maths.  

If there is not a cognitive process that can study it, then for me it is not mathematics. That is basically a restatement of my previous stance that mathematics is the Mind. That being said, my individual failure to be able to understand some part of mathematics has no bearing on all cognitive processes. Of course, I've once again corner myself by not being able to define cognitive process adequately, without taking an unreasonable stance on mathematics (for instance I could define cognitive processes as Turing Machines, but then I think I am throwing away certain parts of maths that are considered kosher). I am also being circular.

> Nevertheless, my conclusion was that the astonishing compatibility comes from the fact that in both cases of maths and nature we are dealing with self-consistent structures.

Isn't a self-consistent structure a mathematical concept?

> The foundations of mathematics (you can reduce it to logic if you are mathematician and probability theory if you are physicist)

Do people still look for foundations of mathematics? I was under the impression that such a quest has been resigned as futile (maybe an example of unknowable?). We can look for foundations of particular branches of math, or theories, but I think we can no longer look for foundations for all of mathematics. We just hope that the part we're standing on continues to not crumble.
 
+Artem Kaznatcheev This is self-referential of course and I mean no personal criticism. ;-)

The range of human performance is veiled by the limitations of the observers.
 
"When I study mathematics, I am not exploring some Platonic ideal from which I am separate. As I study mathematics, I am exploring myself, the minds of others, and the limits of understanding."

The other day I had a similar conversation with my officemate on why I am slightly uncomfortable with the Platonic view of mathematics. A realm of purely mathematical ideas that exist independent of minds that can comprehend them is something that introduces an unnecessary split in my worldview -- you have a natural world and then you have mathematics that is independent of that natural world. Since I am predisposed to a (broadly speaking) "everything that is, is material" view of the world, it seems to me that the possibility of math must come from the fact that the natural world allows us to infer things about its behaviour, and from our inferences about what we learn, we invent new mathematics to describe it. Once we sufficiently abstract away from the original motivation that made us invent the math, we can do the math without any reference to the physical world. How much we can abstract from specific instances to the generic principles is indicative of the limits of our ability to deal with abstraction, and perhaps our ability to invent new math to suit new purposes. The key thing we need to understand is the capacity of our minds for abstraction.

In the beginning was the world. We came along and saw the world wasn't all haphazard, it worked according to principles we could infer, obeyed some patterns that we could understand. We figured out those patterns, abstracted from them the language of mathematics. We went from experience to abstraction, and let loose our unbridled mathematical imagination. Mathematics that needs no analog in the physical world is like poetry that isn't always useful, but often beautiful. And beauty, of course, is a relationship we share with what we find beautiful. I like to think the world is about interrelationships between its constituents rather than any absolute properties of those constituents. Mathematics is a way to study of some of these interrelationships, either in the physical world, or in the conceptual world (the basis of which, I should stress, is physical -- our brains  provide the substrate for this conceptual world to even exist). While all of mathematics is physical in this sense, not everything physical may be amenable to mathematical treatment (until we get smarter and figure out how!).

In this sense (risking some circularity) mathematics is a relationship we share with things we find mathematical. It's not the thing in itself that is mathematical, it's our relationship with it that is. And it's a meaningful relationship insofar as it lets us better comprehend and deal with that which we find mathematical.
 
>>I don't understand how it fails [...]

Trying to be concise, I the distinction will be more clear if you agree that learning maths uses the same cognitive capacities as learning language. This capacity (the hard-wire of the universal grammar) does not exist in non-human animals. Whereas perception of what has been sensed is believed to be universal in a larger family of animals in the kingdom!

>> When did I say anything about common sense?

I didn't claim you said so. In fact in the paragraph the follows is I made it clear that I understand what you say is beyond that. However, the fungus analogy holds only for a small class of common sense concepts. The claim is to refute the analogy.

>> I cannot think of an example for this "surely". There are definitely things like consciousness where  [...]

If consciousness argument rises more questions than answers, then replace it with any reductionist programme. We don't have any (and maybe can't have any) mathematical model for the experience of the colour red. Or even any phenomena in biology. Given the simplest problems, we don't know how birds migrate, the problem which has been a subject of intense study in quantum biology. We don't know the nature of most of the phenomena even in chemistry. And the list goes on.

>> That would still be understanding, no?

Knowing that something has structure is equivalent of understanding it?! Or maybe I don't get what you say.

>> If there is not a cognitive process that can study it, then for me it is not mathematics

Of course it the main subject of philosophy of mathematics. The Nominalist, the Platonist and the Constructivist will give completely different answers to that. However, I'm certainly not sympathetic with your position. Because I find it hard to believe nature has structure, only up to the point that I understand it. Nature cares the same amount for me as for a rat. It just happens that I have higher cognitive capacities than the rat. It shouldn't have any effects of the structure of the world.

>> Isn't a self-consistent structure a mathematical concept?

 I don't know what you mean by that. To me self-consistency is related to compatibility of the results from a truth-maker. If the truth-maker belongs to a mathematical structure then it's a mathematical concept. If it's natural, then it's a natural concept.

>> Do people still look for foundations of mathematics?

Well, it's less fashionable than the time when Quine was doing it. But why you consider it as dead?

By the way, there is a SEP article you might be interested in http://plato.stanford.edu/entries/fitch-paradox/
 
I'm a little late to the conversation, but here are my two cents.

I think that the aesthetic component of mathematics as an activity, while recognized in practise, is deeply underappreciated in terms of its theoretical implications; particularly in the extent to which we unconsciously bias our activity towards arriving at "simple" (or failing that, highly compressed) models of phenomena and answers to questions.

Human cognition is an expensive activity; and when considering cognition which does not concern the particular aims and eccentricities of social primates and other medium-sized mammals, it is all the harder for being a somewhat different activity than what our brains are likely developed to do. But just as we have apparently in-born intuitions about anger, hunger, envy, etc. and readily develop intuitions regarding sharing, detente, and trust, we are primed to readily grasp certain concepts of space and of quantity which we can bootstrap to richer and useful concepts of the real line. And just as we have moral aesthetics based on what seems appropriate, due to cultivated tastes which are founded on primal emotional principles, we have mathematical aesthetics based on what seem to be the important and profitable higher-order properties of concepts which we found on primal notions of number and line.

The LHC and the Hubble telescope are not experiments in mass-psychology; theoretical physics is a project in mass psychology, to formulate ways of concieving what we see from the LHC and its like, which are effectively at the forefront of flint-stone technology and astrological (sic) lore.

So-called "pure" mathematics is an activity of exploration, but what is the space that it actually explores? As a formalist and a fictionalist, I certainly don't consider pure mathematics to be "true" in any sense except for the conventional one of pure mathematics; no more than To Kill A Mockingbird is true, whatever plausible elements it contains, however plausible the historical setting, or whatever the teachable moments it contains. The converse remark, however, is also true; pure mathematics does communicate plausible structures in a sense, instructive examples, and those principles which are meant to represent earnestly realistic models of the world. And I do believe that there is technical skill involved in the deriving of theorems and an aethetic sense of what to prove, and that furthermore that very aesthetic sense has been instiled in me, by nature and by nurture, to bias towards results which at least rhyme with the ways in which the world works.

The world is mathematical in the sense that mathematics is the set of tools with which we explore the world and attempt to find its structure; the world is mathematical in that we find that our structural narratives are useful. But perhaps this is just a result of the fact that we are an intelligent tool-using species; or more to the point, perhaps we are an intelligent tool-using species to the extent that our structural imagination proves useful for describing the world. "Mathematics" is just an iteration of our tool-usage technology to extend our structural imagination by linguistic means: hypothetical and fictional in the same way that sagas and stories are hypothetical and fictional, but nevertheless codifying and presenting examples to learn from of what structures arise in the world, and are reasonable to consider, even if they involve implausible entities of infinite extent or power or severe deviations of the usual rules of space, time, and action. If we find the world to be mostly mathematical, it is because we find that these particular tools are useful for solving many problems. But there are many problems which mathematics has yet to grasp, or provide effective tools for treating. What your opinion is about how easily, or whether it is possible, to push back that boundary is a philosophical position on the general effectiveness of mathematics. But perhaps the technology of mathematics itself is beside the point; perhaps there are methods which are not in any meaningful way mathematical, perhaps not even linguistic, which would prove to be the appropriate set of tools.

To our benefit, mechanics is very nearly linear in quantities which are readily concievable; and furthermore, many systems which are not linear can be modelled as being "linear enough" over short periods of time that calculus proves to be a useful tool. But now we struggle with deeply non-linear systems, such as fluid mechanics and General Relativity; or with systems which seem linear but only if you involve a tremendous explosion of parameters, such as quantum theory (never mind the fact that we get that linearity at the price of nondeterminism). We can make lucky guesses involving symmetries to chance upon plausible theories of subatomic particles, so we may feel that nature is very mathematical; but as this is once again a return to very simple physical systems (just as slow-moving mostly-rigid geometrically-regular weakly-interacting bodies in vacuum are simple systems) so that we should not be deeply impressed with the fact that the answers prove simple.

Probability theory is notable in that it can only arise if one surrenders to the fact that there are things (such as dice rolls) which perhaps could have been considered deterministic, but are at least temporarily too uncontrollable to seriously attempt to predict. A probabilistic explanation for a phenomenon is preferable to none at all, but to be satisfied with one is to suppose that, either in practise or in theory, no better answer is forthcoming -- that the world has too little structure, or that structure too subtle and recondite, for us ever to be able to figure it out; or that the relationships involved are between events so delicate and precise that we can never find out the premisses to sufficient precision to arrive at a definite conclusion. A goal achieved by probabilistic methods is one achieved by moving the goalposts; it is a compromise rather than an unconditional triumph over the problem at hand. The fact that probability theory nevertheless works at giving us confidence in averages and bounded probabilities of significant deviations from averages is comforting. But it is not clear what it says about the universe that this should be true -- how seriously deranged or complicated would a world be in which a sequence of events could not be described as a probability distribution? If a form of intelligent life arose in such an unpredictable world -- where we consider the notion of intelligence to be that they can plan and adapt to that world -- would they perhaps have to be sufficiently clever to arrive at a theory of prediction which is to their world, what probability theory already is to our own relatively tame one? And would it be any more or less of a confession of unpredictability, if their theory failed to give definite answers to the questions they asked of the world?

I'm obviously enamoured of mathematics, but in the end it is just a relatively well-defined toolset with which we can tackle a broad range of problems, which is heavily steeped in linguistic modes of reasoning (or linguistic coding of spatial modes of reasoning). We live in a world which is consistent enough that just by using our imaginations, we can arrive at mathematical structures which describe fundamental particles just by starting with theories of how to count days between the full moon. But most cultures on Earth still have cultural hangoverse from centuries of various semi-mysterious religious tradition (and I count Europe in this as much as any). We are still learning to stop giving our scientific theories a pass when they fail to grasp complicated situations; and even when we're not, we're still a bit drunk on the confidence we have gotten from Hooke, Newton, and Fourier. We should not forget the fact that there is a lot which mathematics yet fails to do, and that there are some things which we strongly suspect it will never be able to do.
 
Ha I had much the same line of thinking in this post but took a different line. https://plus.google.com/100656786406473859284/posts/jXpUN8mSbKm

The basic idea is that it is tautological that mathematics works. A universe you can learn about and understand is one whose description is compressible. It turns out that math is one of those ways to describe it. Programs are another (constructive only). If the universe was not compressible we could not, by definition learn anything about it and math would not work.
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