On why mathematics appear to be non-cosmic

post by KyriakosCH · 2019-06-09T20:42:56.370Z · score: 5 (3 votes) · LW · GW · 19 comments

Preface

I do fear that perhaps this post of mine (my fourth here) may cause a few negative reactions. I do try to approach this from a philosophical viewpoint, as befits my studies. It goes without saying that I may be wrong, and would very much like to read your views and even more so any reasons that my own position may be identified as untenable. I can only assure you that to me it currently seems that mathematics are not cosmic but anthropic.


*

There are so many quotes about mathematics, from celebrated mathematicians, philosophers, even artists; some are witty yet too polemical to identify as useful in a treatise that aspires to discuss whether math is merely anthropic or cosmic, and others are perhaps too focused on the order itself and thus come across a bit like the expected fawning of an admirer to his or her muse.

Yet the question regarding math being only a human concept, or something which is actually cosmic, is an important one, and it does deserve honest examination. I will try to present a few of my own thoughts on this subject, hoping that they may be of use – even if their use is simply to allow for fruitful reflection and possible dismissal.

It is evident that mathematics have value. It is also evident that they allow for technological development. They do serve as a foundation for scientific orders that rest on experiment and thus are invaluable. However we should also consider what the primary difference between math as an order and scientific orders (physics, chemistry etc) easily let’s us know about math itself:

Primarily math differs from science in that it secures that its results are valid not from experiment, data and observation, but axiom-based proof. The use of proof in math is often attributed to the first Greek mathematicians, and specifically to either the first Philosopher, Thales of Miletus, or his students, Anaximander and Pythagoras. Euclid argued that the first Theorem that math presents is the one by Thales, which has to do with analogies between parts of 2D forms (eg triangles) inscribed in a circle. The idea of a proof proceeding from axioms, of a Theorem, is fundamental in mathematics – and it also is a crucial difference between math and orders such as physics. Fields of science that have to do with observing (and interacting with) the external world do significantly differ from a field (math) which only requires reflecting on axiomatic systems.

Given the above is true, it does follow that a human is far more connected to math than to any study of external objects: they are tied to math without even trying to be tied to it, given math exists as a mental creation and not one which requires the senses to intervene.  

But what does “being more connected” mean, in this context? Is math actually intertwined with human thought of all kinds? Obviously we do not innately know about basic “realities” of the external world, such as weight and impact; the risk of a free-fall is something that an infant has to first accept as a reality without grasping why it is so. On the contrary we do, by necessity, already have fundamental awareness of the (arguably) most basic notion in all of mathematics: the notion of the monad.

The monad is the idea of “one”. That anything distinct is a “one”, regardless of whether we mean to include it in a larger group or divide it to constituent parts: each of those larger groups are also “one”, and the same is true for any divisions. “Oneness”, therefore, as the pre-socratics already argued (and Plato examined in hundreds of pages) is arguably one of the most characteristic human notions, and a notion which is generally inescapable and ubiquitous. “One” is also the first digit and  the meter of the set of natural numbers (1,2,3,4…), and this is because the human mind fundamentally identifies differences as distinct, even when the difference may become (in advanced math) extremely complicated and of peculiar types. Yet the humble set of natural numbers also gives us an interesting sequence when altered a bit: the so-called Fibonacci sequence, which I think is a good example to use so as to show why I think that math are only human and not cosmic.

The Fibonacci sequence progresses in a very specific way: each part is formed by adding the two previous parts. The sequence begins with 1 (or 0 and 1), so the first parts of it are (0), 1, 1, 2, 3, 5, 8,13. The entire sequence diverges from both sides (alternating between the next part presenting a numerical difference just smaller or just larger) to the golden ratio, and forms a pretty spiral form (wiki image: https://en.wikipedia.org/wiki/Fibonacci_number#/media/File:FibonacciSpiral.svg). Yet for me it is of more interest that humans do happen to observe a good approximation of this specific, mathematical spiral, on some external objects; namely the shells of a few small animals.

It is pretty clear that the shell of some external being is not itself aware of mathematics. One could argue, of course, that “nature” itself is filled with mathematics, and thus in some way a few external forms happen to approximate a specific spiral, and the tie to the golden ratio etc is only to be expected given nature (and by extension, perhaps, the Cosmos itself) is mathematical. Certainly this can appear to provide an answer; or to be precise it would at least present a cause for this appearance of mathematics and of a specific spiral in the external world. Is it really a good answer, though? In other words, do we observe the Fibonacci or golden ratio spiral approximation on the external world because the external world itself is tied to math, or do we do so because we are tied to math in an even deeper way than we realize and could only project what we have inside of our mental world onto anything external?

My view is that humans are so bound to math (regardless of how knowledgeable one is in mathematics) that we cannot but view the world mathematically. Rockets are built, using math, and by them we can even leave the orbit of our planet – yet consider whether what allowed us to realize how to achieve so impressive a result was not math alone, but math as a kind of very anthropic cane or leg by which we slowly learned to move about:

In essence I do think that due to the human species being so obstructed from developing far more advanced mathematics (to put it another way: due to how difficult advancing math can be even for the best mathematicians) we tend to not identify that math itself is not the cause of development, not the cause of movement and progression, but a leg - the only leg - we have to familiarize ourselves with because we aspire to move on this plane. Imagine a dog which wanted to move from A to B, but couldn’t use its legs. At some point it manages to move one of them, and then enough so as to finally get to B. It is undoubtedly a major achievement for the dog. But the dog shouldn’t proceed to claim that the dirt between A and B is made of moving legs – let alone that it is the case for the entire Cosmos.

I only meant to briefly present my thoughts on this subject, and wish to specify (what very likely is already clear to more mathematically-oriented readers of this post) that my personal knowledge of mathematics is quite basic. I approach the subject from a philosophical and epistemological viewpoint, which is more fitting to my own University studies (Philosophy).

by Kyriakos Chalkopoulos (https://www.patreon.com/Kyriakos)

19 comments

Comments sorted by top scores.

comment by Viliam · 2019-06-11T22:25:56.482Z · score: 3 (2 votes) · LW · GW

If the hypothetical aliens live in the same universe, they will probably develop natural numbers, some version of calculus, probably complex numbers, etc. Because those are things that describe the universe.

They may not have things like Fibonacci numbers, or ZFC axioms, because those are things humans are paying attention to for random historical reasons. Analogically, they may have other concepts that never seemed important for us, such as other sequences, or other sets of axioms. Learning those things could be interesting, but it probably wouldn't feel like a dramatic improvement in math; more like another interesting puzzle to solve.

comment by KyriakosCH · 2019-06-11T22:44:53.019Z · score: 1 (1 votes) · LW · GW

" If the hypothetical aliens live in the same universe, they will probably develop natural numbers, some version of calculus, probably complex numbers, etc. Because those are things that describe the universe. "

I think that they might not. Of course I cannot be certain, but at least in the hypothetical I meant aliens which indeed do not have even the concept of a natural number or other similar concept. And in my view the notion of a sum (a oneness, something specific and easy to contrast to other objects or qualities) is quite possibly (tied to) the most crucial human mental characteristic. The basis of any thought is that it is sensed as distinct from any other thought, regardless of its baggage of unconscious elements.

I can imagine (as an idea) an intelligent alien species which does not have a notion of a sum or a oneness. To that hypothetical alien species there isn't really an external and an internal. This by itself does not have to prevent those aliens from having advanced science, but I personally doubt it would be mutually intelligible to our own.

In my view nothing describes the actual universe, but there are many possible (species-dependent) translations of the universe. Those are always tied to a phenomenon (what is picked up by sensory or mental organs) and not the actual thing.


*

That said, another interesting question might be (assuming math and science aren't cosmic) just why we identify quite a lot of significant patterns as relatively simple forms. For example the elliptic and parabolic trajectories of heavenly bodies, or the (near) spheroid form of some others. Again my suspicion is that has to do with human perception, but it is a good question why so specific a form would be picked up. Recall how even Kepler was originally regarding the ellipsis as way too easy and convenient a form to account for movement in space, and was considering complicated arrangements of the platonic solids :)

comment by Viliam · 2019-06-12T19:52:12.950Z · score: 3 (2 votes) · LW · GW
In my view nothing describes the actual universe, but there are many possible (species-dependent) translations of the universe.

Well, here is the point where we disagree. In my view, equations for e.g. gravity or quantum physics are given by nature. Different species may use different syntax to describe them, but the freedom to do so would be quite limited.

Recall how even Kepler was originally regarding the ellipsis as way too easy and convenient a form to account for movement in space, and was considering complicated arrangements of the platonic solids :)

The fact that Kepler tried to have it one way, but it turned out to be other way, is an evidence for "the universe having its own mind about the equations", isn't it?

Of course an alternative explanation is that the scientists -- mostly men, at least in history -- unconsciously prefer shapes that remind them of boobs.

comment by KyriakosCH · 2019-06-12T22:41:27.364Z · score: 1 (1 votes) · LW · GW

" Well, here is the point where we disagree. In my view, equations for e.g. gravity or quantum physics are given by nature. Different species may use different syntax to describe them, but the freedom to do so would be quite limited. "

Yet differences of syntax connote relative uniformity in the observers as well as presupposing science being cosmic (also math being cosmic, where it relates to scientific examination). In my view supposing that indeed the cosmos (something clearly external to our mind) is examined and accounted for in a way which allows some hypothesized own (cosmic) rules to be picked up albeit in a slightly or somewhat particular manner by each observer, is a little like assuming that current AI in computer games actually identifies a sprite as a horseman and merely picks up the horseman as what the code translates it as. When (at least in this example; ie the fault may lie in the analogy) the game AI is obviously entirely incapable of identifying any "horseman" or any other form or trait, and just runs a code which has "horseman" only arbitrarily and in-code be tied to anything the AI can pick up. Likewise, it seems to me, a human runs (as well as reacts to; cause contrary to a current game AI we also have the ability to self-reflect) a human code which inevitably turns anything external into something anthropic. In the end, much like that AI, us humans also only deal with our own code and nothing else, regardless of the fact that the code is applied to specific and distinct phenomena (move the horseman, check if it is good to use a low HP unit against a rebel, etc).


" The fact that Kepler tried to have it one way, but it turned out to be other way, is an evidence for "the universe having its own mind about the equations", isn't it? "

My point is that it is a bit suspect (granted, this is just intuitive) that so simple and distinct a 2d geometrical form as an ellipse, is actually for us humans front and center in phenomena including the movement of heavenly bodies. Sure, by itself it isn't against math being cosmic, but I really doubt humans are so important OR that the ellipse is not a human concept but something cosmic. I'd need to elaborate on this, but yes, it is impressive in my view.


" Of course an alternative explanation is that the scientists -- mostly men, at least in history -- unconsciously prefer shapes that remind them of boobs. "

I thought these forums were meant for discussing things which aren't perfectly clear :D


comment by Viliam · 2019-06-13T21:42:44.648Z · score: 2 (1 votes) · LW · GW
My point is that it is a bit suspect (granted, this is just intuitive) that so simple and distinct a 2d geometrical form as an ellipse, is actually for us humans front and center in phenomena including the movement of heavenly bodies.

Coincidentally, some complex mathematical things are also related to the movement of heavenly bodies. So I'd say humans are good both at noticing simplicity and noticing complexity.

comment by Pattern · 2019-06-10T19:33:50.754Z · score: 2 (2 votes) · LW · GW
In other words, do we observe the Fibonacci or golden ratio spiral approximation on the external world because the external world itself is tied to math, or do we do so because we are tied to math in an even deeper way than we realize and could only project what we have inside of our mental world onto anything external?

I wasn't clear on what this question meant, but the reason the Fibonacci sequence approximates the golden ratio becomes apparent upon seeing it's closed form solution (which contains the golden ratio).

comment by KyriakosCH · 2019-06-11T00:09:43.088Z · score: 2 (2 votes) · LW · GW

Hi, yes, I do not mean why the Fibonacci spiral approximates the Golden Spiral. I mean why we happen to see something very close to this pattern in some external objects (for example some shells of creatures) when it is a mathematical formation based on a specific sequence.

I referred to it to note that perhaps we project math onto the external world, including cases where we literally see a fully fledged math spiral.

There are other famous examples. Another is The Vitruvian Man (proportions of man by Vitruvius, as presented by DaVinci). One would be tempted to account for this by saying math is cosmic, yet it may just be it is anthropic and the result is a projection of patterns. That math is very important for us (both consciously and even more so unconsciously) seems certain; yet maybe it is not cosmic at all.

comment by Pattern · 2019-06-11T19:31:58.181Z · score: 2 (2 votes) · LW · GW

As for whether math "is cosmic" or not:

If we are projecting, then is this tendency one we developed (social) or one we inherited (evolution)? If it is evolutionary, then perhaps* if we ran into intelligent aliens (which evolved) they'd "have math" as well.

If it is a property of living things in the external world (which seems to be the case), then it may be the way they are (as opposed to a projection). And that may also be the result of evolution*. So we may be seeing such things as they are (readily) because we have a tendency to see patterns of certain forms, with the downside of occasionally seeing patterns where there are none as a consequence of this fitting.

*While evolution might "work the same way" in other places, what is specific to Earth isn't super clear, and how much things generalize remains to be seen.

comment by KyriakosCH · 2019-06-11T20:32:50.426Z · score: 2 (2 votes) · LW · GW

I will try to offer my reflection on the two matters you mentioned.

1)First of all whether this development may have been social. It would - to a degree - but if so then it would be a peculiar and prehistoric event:

If I was to guess, at some point (in deep prehistory) our ancestors could not yet be able to communicate using anything resembling a language, or even words. Prior to using words (or anything similar to words) prehistoric humans would only tenuously tie their inner world (thinking & feeling) to formulated or isolated notions. It is highly likely that logical thinking (by which I mean the basis of later formalization of logic, starting -at the latest- with Aristotle) wasn't yet so prominent a part in the human mentality. It is not at all impossible, or even (in my view) that improbable that some degree of proto- rationalization had to occur so that prehistoric humans would manage to think and sense less of something less organized, and move towards becoming able to establish stable notions and consequently words and a language.

2) Secondly, this would be also inherited. I do suppose that ultimately math (by which I mean more complicated math than the one we currently are aware of) serves somewhat as a dna-to-consciousness interface. But even if this is so, due to point 1 it wouldn't really connote mathematical parameters as being more important than other parameters in the human mind or overall organism.


But there is another point, regarding your post. I think that a non-mental object (for example any external object) cannot be identified as it actually is by the observer/the one who senses it. In philosophy there is a famous term, the so-called "thing-in-itself". That term (used since ancient times) generally means that any object is picked up as having qualities depending on the observer's own ability and means to identify qualities, and not because the actual object has to have those qualities or anything like them. The actual object is just there, but is not in singularity with the observer; the observer translates it through his/her own means (senses and thought). Your point about the object possibly having math inherently is interesting (I do understand you mean that its form is shaped due to actual, real properties, and those are just picked up by us), yet it should be supplemented with the note that even if the object (for example one of those shells) had properties itself which create that spiral and then we notice it, it would have to follow that either we noticed the spiral without distorting the thing-in-itself as an observer of it, or that we picked up some property which didn't actually have any mathematical value but was (in some strange way) isomorphic to the spiral when translated for a human observer's sensory organs. If the latter somehow was true then the external object had no mathematical property, and we picked up some math property because we seem to project math even in more ways than one. If the former was true then we are in singularity with the observed object and nothing is actually distinct in the cosmos (certainly anyone senses their own self as distinct from something external). And in both cases it would not connote that math are cosmic, given the case where math were part of the observed object would present a case where we are so full of illusions that we think (incorrectly in that case) ourselves distinct from a shell, when in "reality" we would not have been.


I do realize this may seem way too "philosophical" (and in a bad way). Philosophy has had problems since ancient times (this itself is already examined by Plato himself: how philosophy may seem very alienating and problematic). Yet the gist of the matter is that (in virtually all serious philosophers' view) there is no reason to think that we as observers pick up any actual non-anthropic reality. We do pick up a translation of something, and this translation is enough to allow us to advance in various ways, including being able to build space-traveling rockets. This is so because we always stay within the translation, and to us the cosmos is witnessed in translation. But a translation of something is not in tautology with the thing itself. My own suspicion is that different intelligent species will not have compatible translations (because they would likely even lack fundamental notions we have; for example they may not sense space or movement or other parameters, and sense ones we cannot imagine. Intuitively I suspect even so alien a species could develop tech and science of a very high level).



comment by Slider · 2019-06-10T12:55:56.717Z · score: 2 (2 votes) · LW · GW

I had a misunderstanding regarding a very similar topic. I was thinking of arithmetic being very core to mathematics when other people included all of formal deductive reasoning (ie stuff other than that has to do with numbers). Humans do have subconcious thoughts which are not well captured by mathematics.

The case for math being beyond human extends far bigger than cosmic. If you had a fictional story that fullfilled some axioms then math based on those axioms would be in full effect.

There are some activities that rely heavily on math yes, but I think we do have observations we don't arrange into neat systems. For example we can't do 3-body problems but we have general gravity kind of locked down (a bit math adjaccent but still an example how we can do without a mathematical theory despite knowing we would like one).

The concept of "one" can be made problematic. And there are systems were the concept is not an elementary one but emerges from deeper principles. For example one has to somehow argue why sexual reproduction doesn't make an example of "1+1=3". In some systems it could be argued that 0 is actually the first digit and more fundamental.

comment by KyriakosCH · 2019-06-10T17:44:00.492Z · score: 2 (2 votes) · LW · GW

Thank you. Intuitively I would hazard the guess that even non-obvious systems (such as your example of the story which rests on axioms) may be in the future presented in a mathematical way. There is a very considerable added hurdle there, however:

When we communicate about math (let's use a simple and famous example: the Pythagorean theorem in Euclidean space) we never focus on parameters that go outside the system. Not only parameters which are outside the set axioms which define the system mathematically (in this case Euclidean space) but more importantly the many more which define the terms we use: I do not communicate to you how I sense the terms for A, B, squared, equality or any other, regardless of the likelihood of myself sensing them in my mind in a very different way to you. It's the same relative communication which is used in every-day matters: if one says "I am happy" you do not sense what is very specifically/fully meant, although the term is a fossil of specific connotations, so some communication is possible, and often no more is needed. Likewise no more is needed to present a math system like that, but certainly far more will be needed to present a story or the subconscious in math terms (and within a given level; outside of that set the terms will remain less defined).

comment by wizardcheetah · 2019-06-10T04:27:06.097Z · score: 2 (2 votes) · LW · GW

Why does this matter? It felt like it was missing.

Is your question well formed? I'm not sure what a proof would demonstrate.

Have you considered the original purposes of math? My understanding is that it was for accounting, rather than elaborating on axioms.

Newton was the first to try to model the universe mathematically. Others had taken quantitative observations and even noted that specific things in nature could be modelled with math. But Newton was the one that sought and found universal laws that could be expressed in math, like his law of universal gravitation.


comment by KyriakosCH · 2019-06-10T09:14:53.816Z · score: 2 (2 votes) · LW · GW

Thanks for the reply. I think that it does matter, because if math is indeed anthropic then it should follow that humans are in effect bringing to light parts of our own mental world. It isn't a discovery of principles of the cosmos, but of how any principles (to the degree they exist in parts of the cosmos) are translated by our mentality. I do find it a little poetic, in that if true it is a bit like using parts of yourself so as to "move" about, and special kind of "movement" requires special knowledge of something still only human.

To use another common metaphor: people who are born blind have no sense of how the world looks. They do come up with theories. To a degree those theories, coupled with sensory routines (counting steps to known routes, hearing and noticing smells) provide a personal model of some environment, translated in their own way. Yet the actual phenomenon, the visible world, is not available. Likewise, it seems that math is not part of anything external, and is an own, human tool, composed of particularly human ingredients and enough to model something of the world that it may allow quite complicated movement through it (including space travel).

comment by Slider · 2019-06-10T12:58:49.849Z · score: 2 (2 votes) · LW · GW

I don't understand how the heard world is less real than the seen world.

comment by KyriakosCH · 2019-06-10T17:38:39.004Z · score: 1 (1 votes) · LW · GW

If you mean my example using people who were born blind, I meant that much like they develop their own system and theory to identify what they cannot sense (visible space), so do all humans in regards to identifying how the external world/the cosmos functions. It isn't about which one is "less real", unless we claim that there is one being (or one group of beings etc) which witness an actual reality. That itself is highly debatable (eg Descartes, and some other thinkers, usually reversed such a role for a deity).

comment by wizardcheetah · 2019-06-10T12:06:51.232Z · score: 2 (2 votes) · LW · GW

So you are comparing math to a pursuit that is clearly an exploration of the human mind like graphic design or other arts. But I am still fuzzy on the 'why'? I can share that wild crows, too, can count up to 4. But because I am not clear on your why I'm not sure how this observation will affect you. It shows that there are at least some part of math that are useful to non-humans. But perhaps you are referring to more sophisticated math systems like ZFC set theory, in which case the crows don't have a say.

comment by KyriakosCH · 2019-06-10T13:03:33.321Z · score: 2 (2 votes) · LW · GW

Indeed, crows are a good example of non-human creatures that use something which may be identified as math (crows have been observed to effectively even notice the -its practical manifestation, obviously - law of displacement of liquids :) )

I used human as a synecdoche here, that is chose the most prominent creature we know that uses math, to stand for all that (to some degree) do. Even if we accept that crows or other creatures have a similar link (itself debatable) it still would link math to dna found on our planet. My suspicion is that what we identify as math is a manifestation of relations, sequences or outcomes of dna, more easily observable in human self-reflection and sense (which is why I mentioned the shells we see in the form approximating the golden ratio spiral).

In essence my suspicion is that math is tied to specific dna-to-conscious animal logistics, and serves as a kind of interface between the deep mind and consciousness, parts of which are occasionally brought up and examined more rigorously. (humans being the species which is more apt to self-reflection, makes us likely the main one here to be conscious of math concepts). I am not of the view that math is cosmic. Approaching this philosophically, it basically connotes that the external world is not mathematical, but because human examination of phenomena in scientific manner presupposes use of the human mind it inevitably is examined through math. One could hypothesize the existence of some other field, non-human, which is equally applicable to the study of the cosmos, and possibly some intelligent species of alien uses that, with compatibility with math being probably non existent.

comment by wizardcheetah · 2019-06-10T15:27:48.832Z · score: 2 (2 votes) · LW · GW

I'm interested to know if you think this is an argument for cosmic math. Even if you are not convinced by it I am still interested to know if it is arguing against your position of anthropic math.

Consider 'momentum'. It is a concept that comes straight out of math. The only reason momentum is named is because it is conserved. You have a system, you do some direct measurements, and combine those measurements into a derived measure of momentum, m1. You keep the system closed, but otherwise do a bunch of whacky stuff to the system and you meaure the momentum again, m2. You will find that m1-m2 = 0. If we go anywhere in the universe we will find the same observation holds true.

If we made contact with ET's that have progressed to primitive farmers however far humans made it before doing math. If we teach them to do the observations they will find the same phenomenon. If we left them to develop on their own then is there some part of that process that they would be incapable or unmotivated to do?

This may be my last contribution.

comment by KyriakosCH · 2019-06-10T17:36:14.098Z · score: 1 (1 votes) · LW · GW

If those aliens are able to understand notions such as momentum, it would be because they can (in whatever way; sense or other) understand more fundamental notions which may be non-cosmic. Some good examples of such notions, from Eleatic philosophy (Parmenides, Zeno etc) are size, form, position, movement (change) and time. To a human, those ideas tie to something evident. An alien may not have them at all. An alien closely resembling humans may have them (as well as math).