Why is the physical world so amenable to mathematical description?

[yougene]

so, isn't that more indicative of us than the universe? i mean, had we evolved differently to have a slightly different form of maths and logic then we'd be observing a slightly different way. the same stuff but in a different language (so to speak).

i think it's a bit egocentric to assume that we, by chance, have a stumbled onto a definitive grasp and comprehension of things. what we have is extremely useful and relatively objective, but i'd hardly consider it truely objective.

By universal I meant cross-cultural.

The point is there are probably infinitely many ways to grasp and conceptualize reality, but they are
all related by the fact that they are approximations of the same thing. The magic isn't
in any particular system, but in the ability of everything to represent everything else via self-similarity.

consider language for instance. in a time and/or place where migration between borders and language did not take much place, an inhabitant with no exposure to another tongue may be forgiven to believe that theirs is the definitive one, but as we know that would be incorrect. the same could be said of cultures and traditions. the same in terms of religion and belief causes a great deal of conflict. how is that any different to our earth-centric mode of maths and logic?

Your point is taken, but the difference here is communication is built on the foundation
of pronouns( 1st, 2nd, 3rd ). The most primal example of this meta-schema there is.

 

[L2R]

i meant to use the language example as a analogical argument, and not a literal one. the subject we are dealing with is probably the most universal we, as humanity, have. speed of light in timbuktu is the same as in pismo beach (left of albuquerque) . what i'm talking about is even though these are a consistant for humanity (and as such extremely useful to us), there is far more to the universe than us. therefore, to state that the physical world is amenable to mathematical description is misleading in its implications.

 

[bagochina]

I personally think that mathematics do hold some sort of key but fortunately that is not the whole answer. Mathematics can tell the story of universal laws and forces of nature but fall short when trying to describe Christ consciousness. Answering the question why will surely get you much farther than a prize for peace. Sacred Geometry? Interesting none-the-less.

Galileo described the universe as a large book written in the language of mathematics, which can only be read by those with knowledge of its characters--triangles, circles, and other geometrical figures. The laws of geometry are not human inventions. They are found ready-made in nature and hold a truth that is the same in all times and all places and is older than the world itself.

Peace,
Seedless

 

[CoffeeDrinker]

Because it's OUR world and we, as a species, are just so damn good at math for some reason.

 

[ebola?]

Yougene: whoa...that's really insightful.
So the content of this isomorphism is simply our tactic of parsing phenomena into parts, their relations, and then a 'reassembled' whole, the latter bearing the marks and blindnesses of our investigation (compared to some prior, unknowable whole)...

It's not a more specific schema at hand, right?

ebola

 

[psood0nym]

I'm not certain how helpful this will be, but all this talk about isomorphisms and features and configurations makes me think of categorical perception, or our tendency to functionally compress perceived within-category differences of phenomena and separate between-category differences. For example, if software is used to morph the sounds /da/ and /ga/ together, listeners tend to hear either /da/ or /ga/ rather than the inbetween sound that is not heard in language. This happens with all types of recognition: colors, animal types (cow morphed into dog), etc., and human categories can be quickly learned and discriminated by various types of animals (birds, rodents). In other words, it's pretty universal.

In fact, something like categorical perception arises even in artificial neural networks that have fundamentally different category-learning systems. From wiki:

Computational modeling (Tijsseling & Harnad 1997; Damper & Harnad 2000) has shown that many types of category-learning mechanisms (e.g. both back-propagation and competitive networks) display CP-like effects.[12][13] In back-propagation nets, the hidden-unit activation patterns that "represent" an input build up within-category compression and between-category separation as they learn; other kinds of nets display similar effects. CP seems to be a means to an end: Inputs that differ among themselves are "compressed" onto similar internal representations if they must all generate the same output; and they become more separate if they must generate different outputs. The network's "bias" is what filters inputs onto their correct output category. The nets accomplish this by selectively detecting (after much trial and error, guided by error-correcting feedback) the invariant features that are shared by the members of the same category and that reliably distinguish them from members of different categories; the nets learn to ignore all other variation as irrelevant to the categorization.

There seems to be a relationship between our concept of mathematical units, inference rules, the seemingly unchanging laws that govern the physical world, and these "invariant features" of perception that look to be the almost inevitable consequence of processing by any learning system.

 

[yougene]

So the content of this isomorphism is simply our tactic of parsing phenomena into parts, their relations, and then a 'reassembled' whole, the latter bearing the marks and blindnesses of our investigation (compared to some prior, unknowable whole)...

The isomorphism(s) are intangible "forms" that are signified by the idea of "isomorphism". Our capacity to parse/integrate, and the tendency of the world to differentiate/integrate into parts/wholes are both instantiations of this form. (there's a reason platonism is popular amongst mathematicians despite being outdated)

It's not a more specific schema at hand, right?

Right, but there are more specific variations.
circuits( e.g heterarchal processes ), concentric circuits( e.g hierarchal processes ), trees( e.g processes of divergence/convergence ), etc...

 

[satricion]

^^
Interesting, can you clarify a bit more so I can make sure I understand?

So you said earlier that you think that our consciousness projects schemas onto perception. But you also say that the universe is structured isomorphically...which to me implies that different processes operate in analogous ways, and that these analogies are somehow ontologically given, ie, real or built into existence as it relates to schemas of perception. Then you say that isomorphism(s) are like platonic forms of wholes, parts and relations, which to me implies that these are also ontologically real in some way (ie they are concretely functioning abstractions or principles).

I'm seeing an issue here in terms of the relationship between subject/object. Ie, you say that schemas are imposed (so they come from the subject), but you also say that isomorphisms are built into the world and that our schemes of perception (like mathematics) are isomorphic with the world because the principle of isomorphism or the operation of isomorphisms operate as fundamental ontological principles. It seems to me that we can't impose schemas and have them be outcomes of a kind of fundamental isomorphism at the same time.

Unless, that is, you're saying that the specific schema we come up with is contingent, but the fact that it is isomorphic with the world is not? That would make sense, do I have that right?

In a sense, that would imply that while the schemas we use to do things in the world, and which are outcomes of a practical relationship with the world, are contingent, they are guaranteed to be of some kind of utility, because they are not arbitrary but rather reflect at least some element of the actual world. Is that right?

Because then that would mean that maths is not a perfect a priori, but an outcome of the subject's practical relationship with the world. Which I think is true. Anyway do I have this right?

 

[yougene]

Interesting, can you clarify a bit more so I can make sure I understand?

I'll post some concrete examples later.

I'm seeing an issue here in terms of the relationship between subject/object. Ie, you say that schemas are imposed (so they come from the subject), but you also say that isomorphisms are built into the world and that our schemes of perception (like mathematics) are isomorphic with the world because the principle of isomorphism or the operation of isomorphisms operate as fundamental ontological principles. It seems to me that we can't impose schemas and have them be outcomes of a kind of fundamental isomorphism at the same time.

We could have just as easily delineated mind and body to the side of object, in which case what is the difference? This is a territory where the parameters of subject/object need to be formally stated to signify which ontology we are currently speaking through. Otherwise this becomes a mindboggling game of semantics.

Unless, that is, you're saying that the specific schema we come up with is contingent, but the fact that it is isomorphic with the world is not? That would make sense, do I have that right?

That's what I'm saying.

In a sense, that would imply that while the schemas we use to do things in the world, and which are outcomes of a practical relationship with the world, are contingent, they are guaranteed to be of some kind of utility, because they are not arbitrary but rather reflect at least some element of the actual world. Is that right?

That's right. Perspectives aren't right or wrong, just partial approximations. I think the interesting implication here is that there is an ordering of perspectives from less inclusive to more inclusive. Probably countless other orderings as well, it is another ontology after all.

Because then that would mean that maths is not a perfect a priori, but an outcome of the subject's practical relationship with the world. Which I think is true. Anyway do I have this right?

Math isn't apriori but what it signifies is.

 

[RedLeader]

how so? the universe is incredibly structured. from the way chemicals and molecules work to the speed of light and sound.. the universe is not a series of chaotic and unpredictable forces but behaves in very predictable ways.

Of course, but that was not your original claim. Your original claim was that for the universe to exist, it must exhibit non-random behavior:

Whether there is a god or no god or whatever is irrelevant because what maters is that for the universe to exist, it must follow certain principals and rules. It cannot be random/chaotic but rather very structured.

That is a claim about necessity, not actuality. You're making a modal error.

I think it's a sign of a universe that shows self-similarity/structural isomoprhisms across every level of complexity.

The mapping of math onto the world works the same reason metaphor works.
The types of relationships, objects take on with one another are the same from one ontology to the next. For example, I can tell you that a computer is organized like a nested Russian doll, or I can tell you that a computer can be seen as a partially ordered set of circuits. You can get a good description of any 2 of these via isomorphisms of the 3rd one.

The question could be reworded as "why do all phenomena map so well onto the ontological schema of wholes, parts, and relations?"

The schema of wholes, parts and relations is a necessary platonic structure; it would exist in any possible universe, it would apply to any possible universe.

Structure of whole and part is not mathematics, though. In the simplest case, one could have a universe of one atom. Dividing it in half, though the language sounds mathematical, isn't really math. It's a necessary logical proposition, and as shown by Godel, math cannot be reduced to logic. Adding more objects does not make the whole and part relationship suddenly mathematical.

Note that this is not the same as saying that every possible universe is non-random (meaning I am not contradicting what I wrote to aanallein).

You cannot reword Heuristic's question like this. If you could reword the question that way, then that would mean that all of mathematics is reducible to talk of wholes, parts and relations between wholes and parts. Please try to prove that, for example, the square root of two is irrational using nothing but these three things (hope you all the time left in the universe to do it ).

I will grant you the benefit of the doubt that you are actually thinking of this in terms of more complex ideas from engineering topics, but still you must acknowledge that such terminology is not a working proxy for mathematics. With that said, everyone else in this thread who is borrowing from your perspective should instead be asking "Why is the physical world so amenable to computational description?" That's fine for dialogue about a question posed by Wolfram (for example), but kind of a jump tangent from dialogue about a question posed by Heuristic.

so, isn't that more indicative of us than the universe? i mean, had we evolved differently to have a slightly different form of maths and logic then we'd be observing a slightly different way. the same stuff but in a different language (so to speak).

i think it's a bit egocentric to assume that we, by chance, have a stumbled onto a definitive grasp and comprehension of things. what we have is extremely useful and relatively objective, but i'd hardly consider it truely objective.

consider language for instance. in a time and/or place where migration between borders and language did not take much place, an inhabitant with no exposure to another tongue may be forgiven to believe that theirs is the definitive one, but as we know that would be incorrect. the same could be said of cultures and traditions. the same in terms of religion and belief causes a great deal of conflict. how is that any different to our earth-centric mode of maths and logic?

edit: nb. "earth-centric" meaning perspective from our capable yet limited sensory limits and within our common "dimensional" place (if there are any others) .

Well put.

To defend against the claim that the human mind projects a mathematical structure on the universe, the universe actually has a mathematical structure and that the two might be isomorphic, we have that if we simply evolved with no theistic intervention, then whatever mathematical structure that our minds project upon the world was fashioned by non-rational forces. Therefore, there's really no reason to suspect that the structure in our minds is the same as (or even isomorphic to) the actual structure of the universe. It might be useful, it might be practical. But we have no way whatsoever to evaluate whether it is correct.

Though not the most pleasant train-of-thought for us non-religious/spiritual folk...

 

[yougene]

Structure of whole and part is not mathematics, though.

It's the foundation of mathematics.

think Set Theory

To clarify, we're talking about mathematical structures here.

In the simplest case, one could have a universe of one atom. Dividing it in half, though the language sounds mathematical, isn't really math. It's a necessary logical proposition, and as shown by Godel, math cannot be reduced to logic.

Not sure what the problem is here.

What you said is isomorphic to saying: divide NATURAL number 1 by 2 which equals 0 with a remainder of 1.

Makes perfect sense, you can't divide an atom( for this thought exercise ).

You implied an ontology of RATIONALS, why?

It's a necessary logical proposition, and as shown by Godel, math cannot be reduced to logic.

This capacity for delineation precedes logic. There's nothing logical about creating signifiers like 1, 2, 3 or I, You, It.

You cannot reword Heuristic's question like this. If you could reword the question that way, then that would mean that all of mathematics is reducible to talk of wholes, parts and relations between wholes and parts.

Mathematics IS completely mappable to relations between wholes and parts. Take any field of mathematics and you can represent it as relations from one ontological set to another. That's the whole point of set theory.

Please try to prove that, for example, the square root of two is irrational using nothing but these three things (hope you all the time left in the universe to do it ).

You need to specify a starting ontology to apply these relations to. I believe that would be the natural numbers in this case.

To give a broad layman description.
Define the NATURALS and enumerate all relations from NATURAL to NATURAL
Construct the INTEGERS from NATURALS and enumerate all relations from INTEGER TO INTEGER
Construct the RATIONALS from INTEGERS etc....
PROOF HERE USING DERIVED ONTOLOGIES

I will grant you the benefit of the doubt that you are actually thinking of this in terms of more complex ideas from engineering topics, but still you must acknowledge that such terminology is not a working proxy for mathematics.

That's cool. I'll give you the benefit of the doubt that you're thinking in terms of applied mathematics.

whatever mathematical structure that our minds project upon the world was fashioned by non-rational forces. Therefore, there's really no reason to suspect that the structure in our minds is the same as (or even isomorphic to) the actual structure of the universe.

If you take consciousness as that which grasps "reality" than you have every reason to SUSPECT isomorphisms. It's a valid induction.

You can't consistently say something like that, before stating:

The schema of wholes, parts and relations is a necessary platonic structure; it would exist in any possible universe, it would apply to any possible universe.

 

[yougene]

^^
Interesting, can you clarify a bit more so I can make sure I understand?

I'm presenting these examples in the context of graph theory.

Typically nodes represent some sort of entity/action and lines represent a relation.

Different types of graphs can emerge depending on the properties of the relation.








Cyclical relationships are called circuits/cycles.

NSFW:

Trees are schemas depicting "parent-child" relationships

NSFW:

This is a syntax tree.



Evolutionary Tree

Tree is a particular variation of partially ordered sets.

NSFW:

Here one is used to depict a containment hierarchy ( nested russian dolls ).
Each parent set contains the child set within itself.

Some ancient examples.

 

[yougene]

human categories can be quickly learned and discriminated by various types of animals (birds, rodents). In other words, it's pretty universal.

Fascinating, where did you run into this?

 

[aanallein]

Of course, but that was not your original claim. Your original claim was that for the universe to exist, it must exhibit non-random behavior:

Well we don't have any evidence of universes that don't follow strict rules. At least that I'm aware of.

 

[solistus]

Well we don't have any evidence of universes that don't follow strict rules. At least that I'm aware of.

What would that even look like? It's basic philosophy of science that nothing can be absolutely proven, only absolutely disproven. Physical properties can only be established through induction, not deduction. We can measure a property 8 billion times and it could change unexpectedly before measurement # 8,000,000,001. It probably won't, but for all we know the apparent consistent rule is just one iteration of some more complex pattern or algorithm, or is just a high likelihood with any other result being an outside possibility. On the quantum level, we can't explain a truly coherent system under which all the rules we have observed make sense. Or, there could be random fluctuations in spacetime, perhaps dimensions we cannot yet even measure, that would cause both the object we are measuring and the device measuring it to warp in a way that caused our equipment to report no net change (measuring devices are inevitably based on assumptions derived from the laws of physics - if some unknown phenomenon occurred at the quantum level something like this could happen). All we can say with absolute certainty is that certain patterns have never been confirmed as having been broken.

Of course, there's really not much practical application to all that beyond to caution against basing one's philosophic outlook on the premise that science can explain literally *everything*. Science is not very good at answering why questions, only at describing observed patterns of cause and effect, of interactions of matter and energy over space and time. In short - physical laws are great for all practical issues relating to manipulating objects in space, but they're a shaky foundation at best for philosophy.

At any rate - math is just a set of linguistic tools, a system of signifiers to express the types of relationships we actually observe. I like how you word the title here because it's wrong to say math is "in" nature - rather, nature is, as you put it, amenable to mathematical description. That description is, directly or in the case of machines like computers, indirectly, the result of a human being deploying linguistic faculties to process information. Numbers themselves have abstract meaning in relation to each other (just like words in the abstract have meaning based on how they relate to the meanings of other words), but the numbers "in" nature are just expressions of an observed relationship. The real marvel is that observed relationships in nature seem to follow such definite patterns as found by the physical sciences. The reason math seems to 'work' so elegantly is because spatial relationships follow such consistent rules - math only seems remarkable when you notice similarities between things through the mathematical information you collect about them, because it is possible to establish stable units of measurement for various physical properties (size, mass, etc.) and measurements in those units display a series of rational and consistent causal relationships to other knowable factors.

 

[psood0nym]

Fascinating, where did you run into this?

I've read a lot about categorical perception. I've got a few pdfs on my computer I see right away I can take titles from. Check out the ref sections of these when you find the full text and the wiki refs on the categorical perception page for more.

This one's visual with monkey's:

Categorical Representation of
Visual Stimuli in the Primate
Prefrontal Cortex
David J. Freedman,1,2,5 Maximilian Riesenhuber,3,4,5
Tomaso Poggio,3,4,5 Earl K. Miller1,2,5*
The ability to group stimuli into meaningful categories is a fundamental cognitive
process. To explore its neural basis, we trained monkeys to categorize
computer-generated stimuli as "cats" and "dogs." A morphing system was used to systematically vary stimulus shape and precisely defined the category boundary.
Neural activity in the lateral prefrontal cortex reßected the category of
visual stimuli, even when a monkey was retrained with the stimuli assigned to
new categories.

This one's auditory with chinchillas (couldn't copy and paste from pdf, but you should be able to find it with the title):

Speech perception by the chinchilla: voiced-voiceless distinction in alveolar plosive consonants. (Kuhl)

 

[Pythagoras]

What would that even look like? It's basic philosophy of science that nothing can be absolutely proven, only absolutely disproven. Physical properties can only be established through induction, not deduction. We can measure a property 8 billion times and it could change unexpectedly before measurement # 8,000,000,001. It probably won't, but for all we know the apparent consistent rule is just one iteration of some more complex pattern or algorithm, or is just a high likelihood with any other result being an outside possibility. On the quantum level, we can't explain a truly coherent system under which all the rules we have observed make sense. Or, there could be random fluctuations in spacetime, perhaps dimensions we cannot yet even measure, that would cause both the object we are measuring and the device measuring it to warp in a way that caused our equipment to report no net change (measuring devices are inevitably based on assumptions derived from the laws of physics - if some unknown phenomenon occurred at the quantum level something like this could happen). All we can say with absolute certainty is that certain patterns have never been confirmed as having been broken.

Of course, there's really not much practical application to all that beyond to caution against basing one's philosophic outlook on the premise that science can explain literally *everything*. Science is not very good at answering why questions, only at describing observed patterns of cause and effect, of interactions of matter and energy over space and time. In short - physical laws are great for all practical issues relating to manipulating objects in space, but they're a shaky foundation at best for philosophy.

At any rate - math is just a set of linguistic tools, a system of signifiers to express the types of relationships we actually observe. I like how you word the title here because it's wrong to say math is "in" nature - rather, nature is, as you put it, amenable to mathematical description. That description is, directly or in the case of machines like computers, indirectly, the result of a human being deploying linguistic faculties to process information. Numbers themselves have abstract meaning in relation to each other (just like words in the abstract have meaning based on how they relate to the meanings of other words), but the numbers "in" nature are just expressions of an observed relationship. The real marvel is that observed relationships in nature seem to follow such definite patterns as found by the physical sciences. The reason math seems to 'work' so elegantly is because spatial relationships follow such consistent rules - math only seems remarkable when you notice similarities between things through the mathematical information you collect about them, because it is possible to establish stable units of measurement for various physical properties (size, mass, etc.) and measurements in those units display a series of rational and consistent causal relationships to other knowable factors.

The poblem with Humean induction, and the epistemological dead-end to which it leads, is that as far as we can tell induction is a fine tool of reason, if one keeps in mind your caveat of desequenced induction.

I also agree in part with your conclusion that Maths, whenever used, by anyone is a heuristic for expressing a relational ontic-mereology. It is through mereology that numerical abstraction is possible, It is interesting to note that ancient greek syllabuses focused on geometry and inter-cardinal relationships (hence the discovery of pi, phi etc) and an oft overloooked point, that geometry is open to two human senses (touch, sight), unlike arithmetic which is visual and highly abstract, never mind higher maths that uses few real numbers but has a host of 'imaginery' numbers to keep it happy.

As solistus mentions maths is a great heuristic for understanding and buiding material objects, but to the next person who elevates maths to epistemic sovereignity ask them "In what way does Godel's ideas prove/disprove the continuum hupothese'? Or more simply is 3 a member of the set 4? Beyond Godel's (or rather Cantor's - thans to Azzazza for ppointing this out) Alephnul we are no closer to knowing infinity as an object of inquiry.

One of the most telling concepts is

Because it's OUR world and we, as a species, are just so damn good at math for some reason

Coffee Drinker

When it comes to higher maths the fact that it is non-computational hints that our minds are non-computational, and to give proofs for the answerrs to Hilbert's remaining problems will come from a human mind - one could conclude from this that the transition from noumena to phenomena and back again (higher maths reasoning) is that the information of that noumena is 'categorised' by the human mind imbueing it at the outset as 'human-theory-laden).

Just a thought.

PAX

 

[yougene]

Maybe someone here can explain what the hell this is supposed to mean.

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

In contrast to set theory, which takes the set–member relationship as fundamental, the core notion of mereology is meronomic, which means based on part–whole relationships.

I'm not seeing the difference here. Mereology seems to be a language to grasp partially ordered sets. What makes it different from set theory then?

 

[rachamim]

Jews have long believed (generally speaking) that it is part and parcel of the Design. Mathamatics equate to definative parameters,rules. If there was no deity we would see a much more haphazard randomness. That is the Cliffnote version, do not have the time to get deep into it.

 

[Roger&Me]

The universe is a giant web of interrelationships. Math is the study of interrelationships.