Given my background of undergraduate physics, I have come to associate the word 'theory' with the description of a system which is approximate at best. For example, the theory of special relativity is called a 'theory' because no matter how many experiments we perform to test length contraction and time dilation, there's still the possibility that some experiment might turn out negative results. That's my understanding for the use of the word 'theory' to describe the concepts of special relativity and I hope I'm right. Now, I've tried to apply this understanding to the use of the word 'theory' in mathematics. The mathematical statements pertaining to sets form what is called 'set theory'; the statements relating to probability form 'probability theory'. However, these statements cannot be theories in the physicist's sense of the word, right? Because these discoveries can never be disproved or falsified. The results are eternally valid. So, why use the word 'theory' and not 'law' to describe these systems? I have been an undergraduate a long time w/o finding a satisfactory answer. I hope I can kick-start a discussion on this issue here.
A theory is a unified framework that explains or describes a phenomenon. We call it "set theory" because it involves explaining and describing sets. The word has the same meaning in physics.
So, are you implying that my understanding of the word 'theory' has been wrong all along, that a theory is not defined to be a set of falsifiable statements, but that instead it's defined to be 'a unified framework', i.e. a complete and self-consistent system of statements that explains or describes a physical or mathematical system?
Scientific laws are simple empirical relationships, usually expressed (in physics at least) as a numerical equation. V=IR (Ohm's law). F=kx (Hooke's law). F=ma (Newton's second law of motion) dU=δQ-δW (First law of thermodynamics for a closed system). None of these laws are universally true, and the equation itself doesn't say one thing about where the equation is applicable. The applicability comes from the underlying scientific theory. It is scientific theories rather than scientific laws that represent the pinnacle of science. A scientific theory typically comprises multiple scientific laws, it describes how to use those laws, and most importantly, it has somehow been tested against reality. Newton's Principia is an example of a scientific theory. There are a number of problems with Newtonian mechanics. It's only (approximately) valid, and then only in a limited domain. That scientific theories are only approximately valid and only in a limited domain is most likely the case for every scientific theory. Special relativity is only valid in a non-gravitational context, general relativity in a non-quantum mechanical context. With regard to things such as probability theory, now you are talking about theory in the context of mathematics. While mathematics is motivated by physical reality, it is not bound by physical reality. The concept of testing against physical reality by means of scientific experiments does not apply to mathematics. Example: That the universe is not Euclidean does not invalidate Euclidean geometry. Euclid's geometry remains just as true today as it did 2400 years ago. The word "theory" has a slightly different meaning in mathematics than it does in science. In mathematics, theory means "body of (mathematical) knowledge". Hence terms such as probability theory, set theory, graph theory, number theory, etc.
Hmm... That's an interesting perspective on the difference between scientific theories and scientific laws. I'd like to bring up a pointer here. As a first year undergraduate, I learnt from Young and Freedman that a scientific theory which has had an unusual longevity (e.g. Newton's laws of motion) is called a scientific law and a scientific law which has had a far greater longevity (e.g. conservation of energy) is called a postulate. I can see your take on the difference between theory and law is different. I thank you for your response, but I'm kind of torn between which interpretation to accept. :( What would you suggest I should believe? I can see now that 'theory' does not have different meanings in maths and physics. Number Nine's response suggested that 'theory' meant the same in physics and maths. His argument relied on the understanding that 'theory' has no connection whatsoever with falsifiable statements. How do you reconcile the two points of view?
The common thing between mathematical and physical theories is that they are developed logically from a small set of postulates/assumptions, albeit with varying degrees of rigor. In fact, this common thing is the entire content of a mathematical theory. A physical theory needs more than just that - it needs some connection with the physical reality, it must predict something that we could actually measure (this is not the exact statement of what it means to be a scientific theory, just a basic idea of that).
A theory cannot be promoted into a law, much less a postulate. All three are completely different things.
Um...what? How do you propose we explain or describe something without making "falsifiable statements"?
Hmm.. Thanks for your response and in helping to comment on the preceding two points of view. Now, I can see that Number Nine's perspective - that 'theory' means the same in physics and maths - is sort of incorrect. 'Theory' has an additional layer of meaning in physics, in that a theory must be tested against reality and that therefore a theory can be falsified.
It would be helpful to all of us if you could provide us with an example to illustrate your point of contention. I have always thought that a physical statement changes in status from a theory to a law to a postulate, depending on the longevity of that physical statement. No one could have guessed, I believe, in the seventeenth century that Newton's law could have survived for two centuries whilst the more derivative conservation of momentum would survive well into the twenty-first century. Also, the premise of this claim contradicts D H's point of view regarding the difference between scientific theory and scientific law, does it not?
I am sorry for having misread your point of view. You mentioned the following in your first post: 'A theory is a unified framework that explains or describes a phenomenon. We call it "set theory" because it involves explaining and describing sets. The word has the same meaning in physics.' Since you mentioned that 'the word has the same meaning in physics', I thought that you understood there to be a one-to-one correspondence between the definition of a physical theory and a mathematical theory, and this implies that there cannot be any additional layer of meaning associated with the definition of a physical theory. So, I took it to be your claim that a theory has no connection whatsoever with falsifiable statements. Sorry for the misunderstanding! But I think now we have cleared up that the word 'theory' does not have the same meaning in physics.
The only time I've ever heard theories being called "less certain" than laws or theories being "promoted" into laws after some amount of time is in discussions far, far removed from the scientific community where people have no idea what they are talking about.
No, it doesn't contradict DH. Consider gravity: Newton's law is a law, but it is wrong. Relativity is a theory, but it is right. Relativity contains postulates. Postulates are just assumptions: they don't have to come from a previous theory. Indeed, the theory they are used in is what validates them.