– what does it mean for a theory to be “proven” mathematically?
In: Mathematics
My sister is a mathematician and likes to joke that philosophy has conclusions and no rules while math has rules and no conclusions.
Now, that’s not really true, but it’s a good joke. What it means to “prove” something mathematically is to demonstrate that the logical link exists between two things. Without asking us what you’re trying to prove, we can’t do anything more than… link you to the wikipedia article on mathematical proofs or to some examples of proofs.
Could you be more specific?
Unlike most things in our world, math is bound by its own rules. We call these basic rules axioms. This is different from most sciences, where we develop theories that describe what we have observed. These theories can change as the evidence changes. But in math, axioms don’t change. We decided on the basic rules and we can use them to solve problems.
A mathematical proof is a logical argument that uses these axioms to show that something else is true. These proofs can follow different approaches, but they all boil down to using the rules we have established to demonstrate something.
If you are familiar with the Pythagorean Theorem (a^2 + b^2 = c^2 for right triangle lengths), [here](https://www.cut-the-knot.org/pythagoras/index.shtml) are some proofs of it
The theory has been represented with mathematical objects (numbers) and we have used known universal facts about numbers to show that the theory is always true (or at least always true within some defined limitation)
If you are talking about pure mathematics, a proof is a sequence of logical steps that get you from a starting point which is already known to be true to an ending point that is often suspected or hypothesized about. The issue usually lies in the logic part, it has to be completely iron clad.
So for a famous example was Fermat’s last theorem. If you remember the Pythagorean theorem which says a²+b² = c², there’s a more general question on whether or not you can do this for other exponents like ³ or ⁴ and the rest. The answer is no and a famous mathematician claimed he could prove it hundreds of years ago but never did. It took several major breakthroughs in other fields of math that needed combining to get to the heart of the problem but it was eventually proven that he was right and those equations don’t work. But not without controversy, right before publishing, one of the most important logical steps actually had an error in it that took ages to fix and find new logic for.
Now if you’re asking about proving say a physics theory with math, this is more a question about theoretical vs experimental physics. A similarly famous example would be the higgs boson. It was theorized through a whole lot of math and quantum mechanics that this particle and phenomenon should exist. It serves as an excellent explanation for the things we observe and predicts several other things we hadn’t quite observed yet, one of which was a particle. It took another 30 years for experimentalists to build a large enough apparatus (and by large I mean a several mile diameter particle accelerator under the mountains in Switzerland) to confirm the theory and the math done. Higgs and his collaborators work was not a mathematical proof, it was a set of equations that made predictions that were found to be true thus we take the equations as accurate.
By contrast, string theory is a similar set of math predictions that has never once been verified in any degree (in fact a few parts have actually been proven impossible). As a result, it’s not given the same level of credence and remains firmly in the realm of theoretical physics.
Depends on what you mean by “theory” (as in, what a scientist means, or what is commonly called a theory). And what you mean “mathematically” (like, that the result of an equation is proof?). This is a really vague question.
In physical sciences, you cannot really prove anything mathematically, you prove with data that your math describes reality. Of course that data will be likely analysed with maths too, especially statistics. But at the end of the day the data is the proof, not the math. So you don’t really prove theories mathematically, but math is involved.
Math itself doesn’t have “theories” as such (unless you mean theorems, which is a term), but here the proof is one of logic. What you’re proving is that the result you’re proposing is a direct consequence of applying the rules of math to some initial statement. I.e. You propose some initial assumption, apply previously proven math to it and show that either your result has to be true or (because some things can’t be proven true directly) there is a contradiction in maths itself if your initial statement isn’t true/false (depending what you’re proving). If you can show that “if A is true then B is true” or “if A is true/false then math breaks”, you have proven something in math.
We can postulate many things about math, without being able to prove them. We make observations of the world, and note certain patterns and we might ask “Does this pattern occur in all cases?”. We can assume it does, but until we actually have a way to prove it holds, its merely a theory.
A classic example of a mathematical proof goes as follows. Given a number x in the set of all integers (meaning given a number without a decimal), if that number is even than x times itself will also be even. This makes sense, you can test it with a few numbers, but we want to prove its always true. Thats a bit harder, but we can do it.
We can start from somewhere we know, how we define an even number. Its a number that can be divided by 2. So we say that x = 2n where n is any number. Lets square both sides and see what happens.
`Square both sides`
`x^2 = (2n)^2.`
`x^2 = 4n^2`
`rewriting to be of the form x=2n`
`x^2 = 2(2n^2)`
Look at that, in our last step we see that our right side is of the from 2 * some n. This is the same as our definition of even number! We have used whats called a direct proof to show that an even number, times itself is also even. There are other ways, but this is one of the easiest examples I could think of.
You prove theorems with mathematics, not theories.
A proof can be done mathematically a variety of ways. We used to prove the pythagorean theorem by a rearrangement proof that was taught in 6th grade geometry. We lost the will to properly teach geometry in American schools (if you are American) because everyone became obsessed with getting kids into calculus (where understanding geometry is crucial) so we replaced geometry with algebra.
Here is an example of a rearrangement proof:
[https://www.youtube.com/watch?v=mijd9BWVF40](https://www.youtube.com/watch?v=mijd9BWVF40)
Here is an example of a proof by similar triangle:
[https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-pythagorean-proofs/v/pythagorean-theorem-proof-using-similarity](https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-pythagorean-proofs/v/pythagorean-theorem-proof-using-similarity)
Basically, you take well known axioms that are long established, and you prove your conjecture by applying these axioms in creative ways. Once you have proved it using axioms, it moves from conjecture to theorem. You don’t have to prove axioms in your proof. Like, with using the similarity proof, you have to demonstrate that you have properly created similar triangles, but you don’t have to prove the axioms of similar triangles, you can simply wield them.
Using one of those proofs, among many available for this theorem, proves the theorem mathematically. We end a proof by saying Q.E.D – quod erat demonstratum, or roughly “that which was to be demonstrated.”
Have you ever played a board game where you stacked a bunch of the rules in one turn to do something cool? A math proof is the showing of how the rules stack to get to an outcome
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