1. We know from biology that all living things are composed of cells, and we know from chemistry that all material things are composed of atoms (or subatomic particles). Because all living things are material, than all cells are composed of atoms. We know from physics that all physical things have a mathematical model, and that means all material and living things do as well (because they're physical). However, my question is whether there is any reason why this mathematical model was chosen? I have some general ideas, but, for example, why does spacetime follows a 4-dimensional Lorentzian manifold with a metric (3,1), rather than, say, a 5-dimensional Finsler manifold?

Basically, what I'm asking is what is the relationship between scientific theories and mathematical theorems?

2. Originally Posted by Ellatha
We know from biology that all living things are composed of cells, and we know from chemistry that all material things are composed of atoms (or subatomic particles). Because all living things are material, than all cells are composed of atoms. We know from physics that all physical things have a mathematical model, and that means all material and living things do as well (because they're physical). However, my question is whether there is any reason why this mathematical model was chosen? I have some general ideas, but, for example, why does spacetime follows a 4-dimensional Lorentzian manifold with a metric (3,1), rather than, say, a 5-dimensional Finsler manifold?

Basically, what I'm asking is what is the relationship between scientific theories and mathematical theorems?
The success of mathematics in physics has been discussed for a long time. Wigner has some thoughts on it that are fairly well known.The Unreasonable Effectiveness of Mathematics in the Natural Sciences

My own feeling is that there is nothing unreasonable at all about the success of mathematics in the natural sciences. Mathematics is the study of any order that the human mind can recognize. Nature appears to be explainable precisely because it is orderly. Therefore one would expect that mathematics would play a large role.

Spacetime in general relativity is a 4-dimensional Lorentzian manifold. It works out that way because Einstein was able to formulate a mathematically consistent theory that appears to explain gravitation. But that does not mean that the universe really is a Lorentzian 4-manifold. It would well be something else.

Einstein-Cartan theory is just as viable as is general relativity. In fact it makes predictions so close to general relativity that it cannot be experimentally distinguished with the current state of the art in instrumentation. In Einstein-Cartan theory one admits the possibility of non-zero torsion and thereby departs from pseudo-Riemannian geometry and metric theories.

String theories and Loop Quantum Gravity theories involve spacetime structures other than Lorentzian 4-manifolds, and may involve dimensions much larger than 4 or structures that are not manifolds at all, particularly at very small scales. It remains to be see if either such theories can be formulated to be well-defined and mathematically consistent or if they represent natural behavior.

Mathematical models are chosen because they make predictions that are consistent with experiment. There seems to be an element of beauty and elegance involved, but that is in the eye of the beholder.

Bottom line: Our theories are not complete. We know that there are shortcomings. Future theories are likely to look quite different from present theories -- but they must agree with the current theories in situations in which the current theories are known to produce accurate predictions. Nevertheless the mathematical formulations may be as different from what we have now as general relativity and quantum field theories are different from classical Newtonian mechanics.

As to Finsler geometries, I have no idea why one would attempt such a model, unless one rejects the notion that our universe is approximately Euclidean on small scales. See comment above on the need for models to be consistent with experiment.

3. Do you think mathematical theorems are stronger than scientific ones in the sense that, mathematical theories can tell us not only what's real, but also what's unreal, by disproving theorems? For example, we know it's "unreal" or impossible to square the circle. While even some of the most established scientific theories undergo revision (e.g., Newton's Laws of Motion), mathematical theorems are absolute truth. Thanks for the article by the way, I've favorited it

4. Originally Posted by Ellatha
Do you think mathematical theorems are stronger than scientific ones in the sense that, mathematical theories can tell us not only what's real, but also what's unreal, by disproving theorems? For example, we know it's "unreal" or impossible to square the circle. While even some of the most established scientific theories undergo revision (e.g., Newton's Laws of Motion), mathematical theorems are absolute truth. Thanks for the article by the way, I've favorited it
Mathematical theorems do not really represent absolute truth. A mathematical theorem is of the form "If A then B". While the entire sentence is a tautology, any "truth" eventually is traceable back to a theorem in which A is an axiom, and therefore A cannot be verified.

Scientific theories, on the other hand, are an attempt to find a description of Nature that is in fact an absolute truth. No such theory has yet been found, but that stilll the goal. Thus far all theories have been found to be good approximations, within limits, but none are absolute and ultimate truths.

Mathematics and science are different. Science seeks to describe nature, and the value of any scientific theory lies in its agreement with experiment and observation. Mathematics seeks to determine the logical consequences of a set of axioms that most people agree are self-evident. Mathematics does not seek to describe nature, and there are mathematical theories that have no known or suspected connection to natural behavior. Experiment is not a valid means of determining mathematical truth, though one counter-example can disprove a conjecture.

Newtonian mechanics, for instance, is a solid mathematical theory and can be studied unto itself. It has in fact generated some very beautiful mathematics (Noether's theorem for instance). It simply fails the experimental test of agreeing with nature when velocities become large. It also fails to agree with classical electrodynamics, which is another solid mathematical theory.

So mathematical theorems and scientific theories are not comparable, and there is no meaning to a statement that mathematical theorems are stronger (or weaker) than scientific theories. They are entirely different animals.

5. I thought that axioms were true because their antithesis is contradictory? For example, A = A is an axiom because if A didn't equal A than it would be a contradiction. "If A then B" statements sound like postulates? E.g., if Euclid's Postulates are true in a given space, than Euclidean geometry can be conducted upon it.

6. Originally Posted by Ellatha
I thought that axioms were true because their antithesis is contradictory? For example, A = A is an axiom because if A didn't equal A than it would be a contradiction.
A=A is not really an axiom, though you may find it as such in a very formal treatment of logic. For practical purposes it is a tautology that follows from the definition of "=". "=" means "is".

The basic axioms of mathematics are the Zermelo-Fraenkel axioms, plus the axiom of choice. For most purposes one can deal with the Peano axioms (plus choice). The Peano Axioms essentially postulate the natural numbers.

Peano axioms - Wikipedia, the free encyclopedia

Zermelo

Axioms are "true" solely because one assumes them to be true without proof.

7. Just curious... When do students formally study foundational mathematics?

8. Originally Posted by epidecus
Just curious... When do students formally study foundational mathematics?
Some axioms are presented to students as early as secondary school, but I would expect rigorous study of such subjects to occur at the graduate or post-graduate level. I'm sure Dr. Rocket can provide a much better response than I.

9. Originally Posted by epidecus
Just curious... When do students formally study foundational mathematics?
If by "formally study foundational mathematics "you mean formal course work in axiomatic set theory, then most students would never study it. There is relatively little research in the area, and only a few schools would have anyone working in formal logic. However, most graduate students would pick up at least the high points of what is known, particularly Godel's theorems through independent reading or just "around the water cooler".

You will often find that logicians are in a philosophy department rather than a mathematics department.

If you mean something a bit less formal, say development of the integers, rationals, reals, and complex numbers from the Peano Axioms, then that might be encountered at the sophomore or junior level in a first course in real analysis. At that point one might also become acquainted with naive set theory to include cardinal and ordinal numbers. The point in a class like that is for students to not only see rigorous proofs but to take some baby steps in constructing proofs on their own. It is pretty easy material, if a bit dry. One might choose to just pick this up along the way as a matter of individual study.

Logic and axiomatic set theory can be really boring. There is a place for it, but it is still really, really dry. If you are looking for a sure cure for insomnia, look no further than a text on formal logic.

Some books that you might find interesting at a level of rigor a bit less than formal logic, that cover some of these topics are:

Naive Set Theory -- Halmos

Foundations of Analysis -- Landau

Both are quite short, and suitable for individual study. In fact they are best done at an individual level and can be read over a long weekend.

If you are looking for more formal material, then look to the works of Godel or Raymond Smullyan.

10. Sorry for a late reply. I've been away for a bit, but I realize I'm still interested in the thread.

Originally Posted by DrRocket
If by "formally study foundational mathematics "you mean formal course work in axiomatic set theory, then most students would never study it. There is relatively little research in the area, and only a few schools would have anyone working in formal logic. However, most graduate students would pick up at least the high points of what is known, particularly Godel's theorems through independent reading or just "around the water cooler".
That sounds a bit strange, to me at least. How is a conjecture shown to be independent of a certain 'framework' such as ZFC? I'd thought it to be on grounds of formal logic but I could be wrong.

11. Originally Posted by epidecus
Sorry for a late reply. I've been away for a bit, but I realize I'm still interested in the thread.

That sounds a bit strange, to me at least. How is a conjecture shown to be independent of a certain 'framework' such as ZFC? I'd thought it to be on grounds of formal logic but I could be wrong.
Paul Cohen received a Fields Medal for showing the independence of the continuum hypothesis, using a technique called "forcing" to construct models showing the independence of the continuum hypothesis from the postulates of ZFC.

http://www.ncbi.nlm.nih.gov/pmc/arti...00240-0135.pdf
http://www.ncbi.nlm.nih.gov/pmc/arti...00175-0117.pdf

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