Math and Reality

[Purple Rabbit]

To use the color of blue as an example... it is just a hue, but upon perceiving it we turned that hue into a concept, and later into a word. Logic (upon which math is based) has been continually observed in nature... that we can use that base to consider things we haven't even seen yet is due to our trust in logic. If logic is how reality "follows" (at least "here"), then what is based off of it is probably true. However, if math is wrong at a point, it would only be because logic does not apply to the scenario. Mathematical concepts are derived from logic, which is an unprovable assumption we've made to reality. It may not be the "true" reality, but it is the reality as we can see it.

Well said Sae. But my hesitation is with all the math that was thought up with no physical correlate whtsoever. Will it ever be put to use in a model of reality?

If math is ultimately a representation of reality, then following the math would be following the reality.

How could we ever know? IOW, can this be made into a falsifiable statement? Is it not falsified already by the fact that there is a whole lot of mathematics that does not in any way apply to reality?

For all we know, infinites might be real (see singularities?).

But if we are to guess, to fabulate away, we're not better of than the supernaturalists.

Logic itself is an assumption made... but it remains that it could all be one outlandishly ridiculous coincidence. Following the math would be entirely logical given that it hasn't really let us down yet when describing, and we can see no reason for it to do so. But all of math blows up if the illogical starts occurring. Logic is simply an observation we've made of 'reality'... and there is no way we can test it's validity without using itself.

I agree it could all be a ridiculous coincidence.

But I propose the following alternative explanation for the observed coherence between nature and (parts of) math. It's just my hypothesis but I can give you a few qualitative arguments for it. And maybe it's just a matter of rephrasing your words: math and nature match so well because they are both necessarily founded on symmetries. The more we discover about nature the more it becomes manifest that symmetry is what fuels our models of nature. Symmetry is everywhere in physics. In the structure of the atom, in the standard model of fundamental particles, in the way forces act, in the relativeness of viewpoints, in the immunity for spatial translation, in rotation of objects, in the spin of particles, in the particle wave duality, in the creation of matter-antimatter. Symmetry was what Einstein drove to general relativity. Hell the LHC is rigged to find supersymmetry! We seem to finally get it: symmetry, symmetry, symmetry! Like in circles, like in the building blocks of algebra, like in the mathematical equivalents of translation, transformation, rotations. Like in balance on both sides of the equation. Like in the order of natural numbers. Like in projections from one set of co-ordinates to another one. Mathematics is imagination restricted to consistency and symmetry. And why is symmetry a necessary essence of reality? Because only symmetry can ensure some sort of stability.

[Violet]

How could we ever know? IOW, can this be made into a falsifiable statement? Is it not falsified already by the fact that there is a whole lot of mathematics that does not in any way apply to reality?

How could we ever know? IOW, can this be made into a falsifiable statement? Is it not falsified already by the fact that there is a whole lot of mathematics that does not in any way apply to reality?

It can only be made into a falsifiable statement if the illogical is physically possible... otherwise the math should lead to the correct version of reality. While we can't prove logic, we have observed that it has worked amazingly well thus far in every application/

But if we are to guess, to fabulate away, we're not better of than the supernaturalists.

It may all be a ridiculous coincidence... but it is certainly better to go off of than a lack of any positive data to support it Further, logic has formed the basis of all of our science, which can be demonstrated to work well enough to have given us computers and the like... supernaturalists have given us remarkably little positive things.

I'll leave the last paragraph for someone better suited to discuss it :S I myself wasn't aware that atoms were symmetrical except perhaps briefly (electrons). But again, i have little knowledge of any such symmetry or lack of such, thus can't adequately respond to it :S I honestly don't see why symmetry would be necessary for stability... or why "stability" would even be necessary in the universe.

[Purple Rabbit]

That mathematical concepts don't follow reality is demonstrated by the history of Euclid's Geometry (EG).

Standard or Euclidean geometry is based on five basic postulates:
1. Any two points can be connected by a straight line.
2. A finite line may be extended indefinitely in a straight line.
3. A circle may be drawn with any given center and any given radius.
4. All right angles are equal to one another.
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

From these five postulates all of Euclidean geometry can be deduced. And Euclid's attempt at it is quite convincing. Around 300 BC he put it in the 13 collected books of Euclid. For centuries mathematicians believed the oddly phrased out of place fifth postulate could be deduced from the preceding four. Over the ages many would announce a derivation of the parallel postulate (fifth postulate) that would later be disproved.

But then Girolamo Saccheri (1667-1733) decided to deny the fifth postulate and see where subsequent deductions would lead him. He was led to the outrageous result that the sum of angles of a triangle could be less than 180 degrees. It convinced him the parallel postulate should be true. As hard as was tried by him and others no logical contradictions were ever found. Non-Euclidean geometry was self-consistent. What then are we to think of the question whether Euclidean geometry is true? It has no meaning.

In the 1800s when elliptic and hyperbolic geometry were developed the foundations of mathematics were shaken. This was a time when mathematics flew off the landscape of reality into a far more expansive plane. Before there was a confidence that Euclidean geometry was absolute and somehow real, but now the validity of one geometry over another could only be verified from experiment.

They were all true in the sense that they were all logically deduced and could be applied to certain circumstances. No one was any more insightful than another, but each provided very rich and exotic structures that could be analyzed rigorously. With Einstein’s general relativity (1915), a vastly new universe was revealed exposing Euclidean geometry for what it was: a simplification of reality. In the neighbourhood of mass the sum of angles of a triangle could indeed be less than 180 degrees.

[mannaka]

I agree it could all be a ridiculous coincidence.

But I propose the following alternative explanation for the observed coherence between nature and (parts of) math. It's just my hypothesis but I can give you a few qualitative arguments for it. And maybe it's just a matter of rephrasing your words: math and nature match so well because they are both necessarily founded on symmetries. The more we discover about nature the more it becomes manifest that symmetry is what fuels our models of nature. Symmetry is everywhere in physics. In the structure of the atom, in the standard model of fundamental particles, in the way forces act, in the relativeness of viewpoints, in the immunity for spatial translation, in rotation of objects, in the spin of particles, in the particle wave duality, in the creation of matter-antimatter. Symmetry was what Einstein drove to general relativity. Hell the LHC is rigged to find supersymmetry! We seem to finally get it: symmetry, symmetry, symmetry! Like in circles, like in the building blocks of algebra, like in the mathematical equivalents of translation, transformation, rotations. Like in balance on both sides of the equation. Like in the order of natural numbers. Like in projections from one set of co-ordinates to another one. Mathematics is imagination restricted to consistency and symmetry. And why is symmetry a necessary essence of reality? Because only symmetry can ensure some sort of stability.

I'm fascinated by your idea. This is an old topic now, but was any consensus formed on this matter? Like Vaeolet Lilly Blossom I'm not entirely sure how necessary stability is, nor was I aware of how wide-spread symmetry is.

(a related quote: )
"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning." - Eugene Wigner

[Categories+Sheaves]

I'm fascinated by your idea. This is an old topic now, but was any consensus formed on this matter? Like Vaeolet Lilly Blossom I'm not entirely sure how necessary stability is, nor was I aware of how wide-spread symmetry is.

Hooray! An excuse to rant about math!

As far as the issue about falsifiablility goes (admittedly, it's not the point being necrobumped) I'm with this guy's thinking: the 'core' statements of a paradigm (e.g. matter is indivisible) are too general to be falsifiable. It's only once we start adding auxiliary hypotheses/concepts (what other properties of matter allow us to see that it is, in some sense, indivisible?) that we can get falsifiable statements about reality, and this thing we call science.

With that out of the way: The stability or invariance of of a thing (under certain operations) is a pretty basic result of having any sort of equivalence. How do I come to say, "I have four textbooks in my backpack"? Four things in my backpack satisfy the definition of 'textbook'. In regard to the question "Is this object a textbook?", these four objects produce equivalent results, and the invariance of this result under textbook-swapping forms a "symmetry." Within the operations we can perform on the books in the world, some of these operations are "symmetries" and preserve that earlier statement about my backpack (e.g. I take Spivak's Calculus book out of my backpack and replace it with Munkres' Topology book) while some of these operations don't preserve that statement (e.g. I swap Spivak's book for the collected poems of Robert Creeley). This is all an armchair concern for now... but when we start talking about atoms or electrons instead of textbooks, this becomes a practical matter. It's not like nature and math are part of a larger conspiracy to make everything symmetric. It's just that any equivalence of things (e.g. any two electrons will behave the same way in the same circumstances) necessarily yields a sense of symmetry. In the same way a definition of "up" (or really any continuous function f: R^3 -> R) yields a sense of a "level surface."

[Violet]

* Violet was going to complain about necromancy... but then realized it was a decent response.

Purple rabbit isn't here anymore, alas

[Rhizomorph13]

Alas indeed, that guy added so much to our forums and then he split over some overblown crap that, as far as I've seen, didn't change the way our forum runs at all.

[mannaka]

It's not like nature and math are part of a larger conspiracy to make everything symmetric. It's just that any equivalence of things (e.g. any two electrons will behave the same way in the same circumstances) necessarily yields a sense of symmetry. In the same way a definition of "up" (or really any continuous function f: R^3 -> R) yields a sense of a "level surface."

Thanks Categories+Sheaves that was incredibly insightful; I'm hooked.

[Categories+Sheaves]

You're welcome.

If you want to follow up on this stuff, this is a famous result on the significance of symmetries, and this is the formalism used to describe symmetries (and the like) in contemporary math.