Mathematical proofs and an understanding of them for scientists

[DissenterK]

To what extent, if any, is an understanding of mathematical proofs required for a scientist? I can empathize with a need for an understanding of the general machinery of the tools you are using (understanding, for example, how it is the chain rule came about, ie, how it was derived) but, using the aforementioned example, does a scientist need to know that the chain rule applies to all x in R and how it is so? In other words, when prompted, would a good scientist be able to formulate a proof for the validity of the chain rule for all x in R? I ask because I consider the study of proofs to be monotonous and time consuming and I'd like to circumvent the problem if I may. By the way, I don't mean to suggest that it is a waste of time to learn how to prove. I think it a good exercise in problem solving but I have a friend (a student of mathematics) who becomes infuriated when I use a tool in study without knowing the proof for the theorem. Must I know how to prove every theorem I use?

 

[davidbenari]

I'm guessing you are a student. I also am. My goal is to know how to prove these things, but I usually give up whenever I recognise that to prove some of these things requires some specific course that only math majors take. I would suggest that at the very least you should be able to prove things in PHYSICS even if you're simply assuming the validity of your mathematical toolset.

I'm very interested in the question you just made, because this topic has been bugging me too.

 

[DissenterK]

I'm guessing you are a student. I also am. My goal is to know how to prove these things, but I usually give up whenever I recognise that to prove some of these things requires some specific course that only math majors take. I would suggest that at the very least you should be able to prove things in PHYSICS even if you're simply assuming the validity of your mathematical toolset.

I'm very interested in the question you just made, because this topic has been bugging me too.

You make the same suggestion my friend makes but you add no justification for it. Why not make the assumption that the mathematical tool-set at your disposal is valid, being that you're, say, a physicist? Why not leave the proofs to the students of mathematics? I stress this point even further if you do nothing to contribute to the development of a proof for some theorem. I mean to say that if all you do is read a proof written by another and begin to know and understand it, what purpose does it serve beyond the encouragement of problem solving skills I mentioned above? Here's an analogy. The carpenter may not know how it is his tools were designed but so long as he is aware of how to use them and for what applications they were designed (or as long as he could make the necessary inferences) haven't all the conditions for his position been met? Suppose he does understand how it is that his tools work. How would this aid him in his carpentry? Now this analogy has one obvious flaw in that someone could suggest that the carpenter, by knowing his tools beyond the required extent, could engineer superior ones. But with respect to mathematics and proofs of theorems there exists no similar relationship. I don't see how the physicist, by attaining an understanding of how to prove the theorems he works with, could design superior ones. After all, he has only demonstrated the capacity for his current tool (theorem) to be applicable for a set of values. Where is the room for innovation here? Am I missing something?

 

[davidbenari]

What I suggest is that you should at least be able to prove things which are only related to physics even if you don't know how to prove everything in the mathematics you'll be using to do that.

Now, I think knowing math proofs is important for many reasons. One might be, as you say, to hone your mathematical or problem solving skills. But I fear that at some point, if I ignore many of the proofs, I'll get lost in confusion or some course-material will be harder to understand given that I don't have a profound understanding of the mathematics involved. Also, in my experience, for every proof you learn you also learn a lot of new mathematics, or begin to interpret stuff you already knew in different ways.

Finally, (I might be wrong here because I just started college), I think there will be courses in physics that will require that we provide some sort of proof. I think mathematical physics courses will require this.

 

[DissenterK]

What I suggest is that you should at least be able to prove things which are only related to physics even if you don't know how to prove everything in the mathematics you'll be using to do that.

Now, I think knowing math proofs is important for many reasons. One might be, as you say, to hone your mathematical or problem solving skills. But I fear that at some point, if I ignore many of the proofs, I'll get lost in confusion or some course-material will be harder to understand given that I don't have a profound understanding of the mathematics involved. Also, in my experience, for every proof you learn you also learn a lot of new mathematics, or begin to interpret stuff you already knew in different ways.

Finally, (I might be wrong here because I just started college), I think there will be courses in physics that will require that we provide some sort of proof. I think mathematical physics courses will require this.

I empathize with you when you write "...I'll get lost in confusion or some course-material will be harder to understand..." and I have those same feelings myself. I need to give your response some thought, in particular, because of the last sentence in your second paragraph. I'll respond later if I come up with anything new or if I concede and am satisfied. Thank you for the help you've given.

 

[nayanm]

Say you made it a point to prove every theorem and concept in physics before you every apply it yourself. Ultimately, you'd find that you'd gone far from the realm of science into the realm of theoretical math and eventually into the realm of philosophy.

Imagine your burden of proof! Not only would you have to prove to yourself the current theory of some subject matter, but you'd also have to follow the complete historical progression of the subject through the ages to convince yourself what is not valid.

Heck, think of a subject like thermodynamics. You'd ask yourself about the workings of heat. How exactly do we know it isn't a fluid called 'phlogiston' or 'caloric', as was previously believed? In quantifying heat, you'd have to eventually employ multiplication, and for that you'd have to also prove to yourself the completeness of the real numbers. Before you know it, you'd be debating whether numbers themselves are pure abstractions formulated in our brains or whether they have some root in the physical world.

So regardless of how deep into a subject area you're willing to go, at some point you're going to have to draw a line. Exactly where you want to draw that line is up to you.

 

[enorbet]

I submit that the language of Science is Mathematics. How fluent do you wish to be? How far "out of the loop" are you willing to accept?

I further submit that a carpenter who knows the fundamentals about his tools will make superior choices in tools, adapt to different work loads better, and apply his tools in better form to a superior result. That said I do recognize that analogies expressed in words are subject to considerable limitation... but then, I suppose that is why Math is the language of Science.

 

[UltrafastPED]

I further submit that a carpenter who knows the fundamentals about his tools will make superior choices in tools, adapt to different work loads better, and apply his tools in better form to a superior result.

And to drive this a bit further: a machinist who fails to understand the fundamentals of his tools won't be able to do the work at all!

 

[mathman]

As a mathematician my advice to scientists is to fully understand the statement of the mathematical theorems, so that when you use them, you can be confident that they apply in the given situation.

 

[UltrafastPED]

As a mathematician my advice to scientists is to fully understand the statement of the mathematical theorems, so that when you use them, you can be confident that they apply in the given situation.

Amen!