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lately I have been thinking about the terms theory, thesis, hypothesis and theorem. I am quite sure about the differences between a thesis and a hypothesis, but I did not really get what a theorem is. I would like to ask you to confirm my definition of each term or don' t. If you disagree with my definition, please explain why and give - if possible - a good example.

As an example I will use the axiom "Nothing is faster than light" of the Theory of Relativity.

Thesis:
As far as I have figured out, a thesis is an assertion which a controversy is immanent to. So it needs a argumentative reasoning.

My first question, in reference to my example, is if every axiom is implicitly a thesis or not.

I am not sure about it, because if I took the axiom from the example, I would not know if there is any controversy about this assertion or if it is just an undisputed assertion, although there could be a controversy, because nobody can know if there is not something faster than light. On the other hand, if I have the assertion "There is a God", its clear to me that there will be a controversy. So it must be a thesis.

Further on I will assume the example axiom to be a thesis for further illustration.

Hypothesis:
A hypothesis is a derived form of a thesis. It asserts a correlation between - at least - two factors (where a thesis is simply an assertion).

So in reference to the example I could derive the hypothesis "If there was something faster than light, then there would be a God" from my thesis "Nothing is faster than light". It asserts a correlation between the factors "speed of light" and "existence of a God" (although there is no causal relationship).

Theory:
A theory is an abstract concept which tries to explain a discipline of reality. It consists of axioms/theses.

Theorem:
This is something that I was not really able to figure out. I found a pattern in theoremes: It always seems to be based on some theory and tries to solve some paradoxon or other problem the theory has, but it does not change the theory at all. It is some lego that is put on the theory to fill a hole.

I don' t know if there is any truth in what I have written about the theoreme.

So I hope you can help me understanding :)

  • The words have different meaning in difference contexts, i.e. you don't prove theorems in physics you prove theories but a theory in physics is more like a conjecture in mathematics. A theorem is something that is proved deductively using logical reasoning or inference (think about what you do in a formal logical system like propositional logic). Mathematical theorems, like the fundamental theorem of arithmetic, are theorems because they can be proved in a completely deductive way. Physical laws are not proved deductively, they are proved inductively through observation and experiment. – Not_Here Jul 29 '17 at 10:42
  • From wikipedia's article on theorem: "In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental." In mathematics, conjectures are a lot more like scientific laws than theorems because they are things that we do not have a deductive proof for yet, but every test we've ever done to the conjecture came up positive. Consider Goldbach's conjecture, it is a conjecture because it doesn't have a proof yet, but we have yet to find a single counterexample, despite massive testing. – Not_Here Jul 29 '17 at 10:44
  • The biggest issue you're having, I think, is that you're mixing how the words are used in science and how they're used in math. Obviously there is over lap but the words have different meanings when they're used very strictly in those two contexts. For example, your definition of theory (besides the use of the term axioms because those also have different meanings in different contexts) is correct when applied to science but theory has a different definition in mathematical logic where it's a set of axioms and theorems in some formal language. – Not_Here Jul 29 '17 at 10:51
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The biggest trouble that you will face when thinking about these ideas is that you are mixing terminology from two distinct fields into one. Some of those words are used very differently in science than they are in mathematics. It can become even more confusing when you consider how much math is used in science, but an explanation of how the words are used in their specific contexts will help illuminate the delineating line.

In mathematics, "thesis", "hypothesis", and "conjecture" are all used synonymously. From Wolfram's Mathworld:

Hypothesis: "A hypothesis is a proposition that is consistent with known data, but has been neither verified nor shown to be false." "In general mathematical usage, 'hypothesis' is roughly synonymous with 'conjecture.'"

Conjecture: "A proposition which is consistent with known data, but has neither been verified nor shown to be false. It is synonymous with hypothesis."

Additionally, if you go to the disambiguation of the word "thesis" on wikipedia you see that in the subjects of mathematics and logic it links to "hypothesis" and "conjecture" as well. An example of a mathematical thesis is the Church-Turing thesis which, as you can see, is also sometimes called the Church-Turing conjecture and is described in that article as being a hypothesis. The reason that the Church-Turing thesis is a thesis is because it tries to take an informal idea (the idea of an algorithm) and give it a precise mathematical statement. Due to the fact that it starts with an informal idea, there isn't a purely deductive way to prove that the idea is true, therefore it's left open as a hypothesis and would be proven untrue if a counter example is shown.

As such, mathematical conjectures, theses, and hypotheses are statements in mathematics that seem probable and no counter example has yet been shown. This means that conjectures have yet to be proven, which delineates them from theorems.

A theorem is something that is not a conjecture, it is something that has been proven true. From Mathworld:

Theorem: "A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof."

Examples of theorems are proven mathematical statements, things like the fundamental theorems of arithmetic, algebra, and calculus. Other, much simpler theorems, are things like the Pythagorean theorem. The picture of a theorem as something that solves a paradox or other problem is incorrect, a theorem is just a provably true statement. The Pythagorean theorem doesn't resolve a paradox in geometry, it is just a statement that has to be true, is provable, given the initial rules of geometry (the axioms).

Those are how the words are used in mathematics alone and in science they are sometimes used very differently. One important thing to understand is that science does not deal with theorems, scientific theories don't prove theorems. This is because science relies on inductive and abductive reasoning to learn about the world through empirical observation. A scientific theory is a general description of the world that is testable and has withstood repeated testing. In this way, they're usually made up out of scientific laws.

Scott Aaronson, a quantum information scientist who works on both fields, has highlighted the differences in the terminology on multiple occasions:

"I've learned from working in quantum information that there's a difference in terminology between fields. What mathematicians and computer scientists call a conjecture is typically what physicists would call a law."

"I think my word is 'theorem'. It's something that you say is true and then you explain why it's true and then you put a box."

The context for the second quote is that Leonard Susskind asked Aaronson to explain a word from his mathematical background that people with a physics background might not understand or use in the same way. The reason everyone laughed is because the other physicists and computer scientists in the room understood how tongue in cheek Aaronson's pick for the word was: physics doesn't deal with theorems, in a mathematical sense, it deals with conjectures. That's why Susskind sarcastically says "A theorem, what's that?" And of course the "box" comment is a reference to what's put at the end of a proof of a theorem.

Ultimately, theorems are things that are deductively proven and as such exist in mathematics and logic. Theorems are used in science as well, you can use the Pythagorean theorem to help you solve a mechanics problem, but science doesn't prove theorems. Science tries to create theoretical models that help explain physical phenomena and those models can always be revised via new information. So if you are trying to apply the word theorem to a physical theory you are going to fail, the only parts of the physical theory that are theorems are the purely mathematical parts.

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