Mathematical Platonism: Are Group Theory Axioms Eternal?

[selfAdjoint]

Do mathematical truths subsist independently of human consciousness? Take the axioms of group theory as a lab example. There are many groups which vary in a number of ways, but they all share these properties:

A group is a set G with a function p from its cartesian product GXG to G satisfying
1. p(a,p(b,c)) = p(p(a,b),c))
2. there is an element e of G for which p(a,e) = p(e,a) = a for all elements of G. e is called the identity of the group G.
3, For every element a of G, there is an element a' satisfying p(a,a') = p(a',a) = e. a' is called the inverse of a.

Groups were discovered in the nineteenth century, largely by Galois. My question, did the properties of a group exist before Galois? Did all the many theorems of group theory derives from those proerties exist then? Were they TRUE then?

 

[honestrosewater]

Just a question - I don't have many thoughts on it. I see where you get truth from logic, i.e., you can point to something in logic and say 'this is a logical (or mathematical) truth'. But how do you get time from logic? What can you point to in logic and say 'this is a logical time stamp'? It doesn't make sense to me to even ask when (at what time) a statement is or becomes true; There is no time component to logical truth, AFAIK.
Does that make any sense? These things abstract - they don't 'exist in time'.

 

[matt grime]

I think it is a slightly ambinguously stated question.


The statement: there is a set satisfying the axioms of a group. was always true. Platonism refers not to if such statements are purely fictional but if the objects satisfying those statements may have some phyiscal existence independent of us. In a nonmathematical sense: the Island of Australia existed (without that name) before we discovered it. Well, do groups exist in that sense? Are we fortunate to have figured out a set of rules that describe these objects?

Personally I feel the question is unimportant: whatever the outcome I can still make the same conclusions about groups.

The group axioms are certainly human invention; they contain redundant information and may be give in alternate forms.

 

[selfAdjoint]

I think it is a slightly ambinguously stated question.


The statement: there is a set satisfying the axioms of a group. was always true. Platonism refers not to if such statements are purely fictional but if the objects satisfying those statements may have some phyiscal existence independent of us. In a nonmathematical sense: the Island of Australia existed (without that name) before we discovered it. Well, do groups exist in that sense? Are we fortunate to have figured out a set of rules that describe these objects?

Personally I feel the question is unimportant: whatever the outcome I can still make the same conclusions about groups.

The group axioms are certainly human invention; they contain redundant information and may be give in alternate forms.

I don't think mathematicians who have the platonist belief consider the mathematical truths physical. It's more emphasized that they are "out there, somehow". Discovered, not invented.

 

[NeutronStar]

I don't think mathematicians who have the platonist belief consider the mathematical truths physical. It's more emphasized that they are "out there, somehow". Discovered, not invented.

I personally believe that modern mathematics has gotten off-track from it’s original purpose. In the days of the early Greeks I believe that mathematics really was intended as statements about the observed quantitative nature of the universe. Therefore within that frame of inquiry all mathematical “truths” could be shown to be actual properties of the universe. In fact, if mathematics is defined in this way than any so-called mathematical “truth” that could not be shown to be a quantitative property of the universe would be highly suspect.

Today things are quite different. Today axiomatic abstraction has become the focal point of mathematics. We make up abstract rules and decide whether mathematical statement are valid only by testing whether or not they satisfy and do not logically contradict these arbitrary abstract rules of the formalism. The rules are “invented” and accepted as part of the formal mathematics, not because they have been shown to reflect the quantitative nature of the universe. But merely because they have been shown to not conflict with any previously accepted abstract rules.

Therefore modern mathematics can take any path that any modern mathematician would like to invent providing his ideas are logically self-consistent, and don’t conflict with previous axioms. No one is testing to see if any of these modern ideas have anything at all to do with the quantitative nature of the universe.

So I would have to say that modern ideas can indeed be completely independent inventions with no relationship at all to the nature of the universe.

I do believe that many of the older ideas, like the Pythagorean theorem, the proof of the 5 Platonic solids, and many other classical ideas of mathematics are indeed physical “truths” of our universe. But some of the more modern abstract group theories may have nothing at all to do with the nature of our universe.

Just my thoughts, but I do feel very strongly about this topic and I could point to specific ideas but I certainly don't have time to get into the details at this moment.

James

 

[AKG]

NeutronStar

1. Rigour is a good thing
2. Applied mathematicians worry about whether the axioms have anything to do with the natural world.

 

[vanesch]

I personally believe that modern mathematics has gotten off-track from it’s original purpose. In the days of the early Greeks I believe that mathematics really was intended as statements about the observed quantitative nature of the universe. Therefore within that frame of inquiry all mathematical “truths” could be shown to be actual properties of the universe.

This still exists ; only now we call that human activity "physics"

cheers,
Patrick.

 

[vanesch]

Do mathematical truths subsist independently of human consciousness?

You can guess that I think they do
I think it is an a priori requirement to an ontological belief in a physical world, described by a mathematical structure (the thing theoretical physicists must somehow take for granted).

 

[matt grime]

I do believe that many of the older ideas, like the Pythagorean theorem, the proof of the 5 Platonic solids, and many other classical ideas of mathematics are indeed physical “truths” of our universe. But some of the more modern abstract group theories may have nothing at all to do with the nature of our universe.

so, you've made a platonic solid have you? or gotten a perfect 3-4-5 triangle? the greeks thought that you couldn't construct various things with straight edge and compass, and by group theory we konw that is indeed the case. oh, group theory has far more application to the real world (physics) than these idealized models you are talking about. (incidentally there aren't such things as "abstract group theories", there is group theory and there are abstract groups as well as concrete ones, many beign important in the study of platonic solids. By cayley hamilton every abstract group in the proper sense is a concrete one). One may even say the the existence of the group Z/2Z is a phyiscal truth of our universe since it is the idea behind symmtery. befoe you continue in this inaccurate representation of group theory you perhaps ought to go away and learn about gauge groups, spectra of buckminsterfullerene, particle phsyics, crystallography, mathematical physics, statisical physics (hecke algebras and magnetic monopoles via serre duality?). do you understand how the homology and cohomolgoy groups of manifolds are important? or why the n=1 superconformal neveu schwartz lie algebra is of interest to (topological) quantum field theory (no, neither do i though i once wanted to), what about the existence of nondegenerate symplectic 2 forms? they're elements of the exterior algebra, that's a group you know.

 

[NeutronStar]

1. Rigour is a good thing

Agreed.

2. Applied mathematicians worry about whether the axioms have anything to do with the natural world.

Evidently they aren't worrying very hard.

 

[fuzzyfelt]

How about both, relying on each other?

 

[fuzzyfelt]

As a muddly example, you may recall my obsession with symmetry.
Dennet mentions it as an evolutionary advantage in sexual selection, I think in hunting and defense (recognising whether another animal is looking straight at you), I don't know if he mentions it socially, but in portraiture the most basic step is to draw and imaginery line down the middle of the face to note the greatest differences to make a recognisable representation. A bit of a leap perhaps, but maybe symmetry, first allowed us and to some extent other animals to distinguish between things, spaces for a start (that we evolved to see space as we do), past and future (that we evovled to see time as we do), to distinguish our own self conciousness and to distinguish between reality and symbols. Further, measuring all relationships - social, artistic, mathmatical, etc.. Mithen mentions symmetry as an inidcator of a degree of consciousness. I wonder if awareness of symmetry, combined with complexities it allows, was central to consciousness, and whether it dictated the way we evolved to percieve, and think about, things.
Perhaps there are many ways other than via symmetry, in which things could work, but symmetry was the approach that evolved, and which works because it was chosen. This idea is slanted toward the platonic with pre-existing possibilities, but it allows our consiousness the choice of which of those rules would work. To almost creat the rules.
I don't like the slant. Idealistically, I'd like to think there was a reason why it works, if it were to work, and that the human mind can use these rules (like using theoretical supersymmetry), maybe to take us beyond them.

 

[sd01g]

Do mathematical truths subsist independently of human consciousness?

Another way of asking this question is: Where, how, and in what form or substance did mathematical truths subsist/exist before there was human consciousness? Best guess: they did not, unless 'human consciousness' existed in some form or substance before it was transferred to humans.

Those who think that MATHEMATICAL TRUTHS subsist/exist independently of human consciousness (this includes books, tapes, CD's and computer hard drives) have the probably impossible task of trying to locate WHERE and in what form or substance these truths subsist/exist, or existed prior to their entering human consciousness.

 

[CrankFan]

Those who think that MATHEMATICAL TRUTHS subsist/exist independently of human consciousness (this includes books, tapes, CD's and computer hard drives) have the probably impossible task of trying to locate WHERE and in what form or substance these truths subsist/exist, or existed prior to their entering human consciousness.

No one is claiming that these things existed in written form of some sort prior to human existence. What's being claimed is that they were true, independent of human experience.

It's not such a radical concept. Do you think that facts POP into existence at the moment you learned them? Do you think that the moment that you first accepted that the moon orbits the Earth is the very same moment that the moon started orbiting the earth? Couldn't it have been TRUE, before you became aware of it and you merely discovered that it was true.

It's a trivial theorem of arithmetic that: There are no integer solutions for the equation 2m^2 = n^2

At the moment in time that you learn that this statement is true (by a proof), how do you reflect on the status of the proposition at any time prior to your acquaintance with the proof?

After you've been enlightened do you, in thinking back, regard the proposition as not being true until you discovered it was true? If that's the case then how come you couldn't have instead proved that it is false?

If the truth value of these propositions aren't independent of the mind then why can't we mentally determine their truth value through sheer will? Why can't you find integers m and n such that 2m^2 = n^2?

 

[CrankFan]

It's a trivial theorem of arithmetic that: There are no integer solutions for the equation 2m^2 = n^2

Duh... change that to "no non-zero integer solutions"...