Why Should You Avoid Assumptions in Math?

[airbusman]

So I started my first ever university course, at U of T. (It's a summer course.) The class textbook, which is called "An Introduction to Proofs and Problem Solving", should give you an idea of what the class is about. Anyway, the first thing that my Professor taught us is that you should never make assumptions in math. For instance, the first question that we took up together was "how many squares are there on an ordinary checkerboard?" At first, I thought "64, it's simple." But we soon learned that we have to take into account the fact that there are 1 x 1, 2 x 2, 3 x 3, etc squares. So the real answer is actually

(1/6)n(n + 1)(2n + 1)

This was a real eye opener. In high school math, we were taught that assumptions were normal, and the textbooks we used assumed that we made assumptions. It's quite different in uni, I learnt. It's all about generalization. In the first class, I learned so many things that I didn't know about. Now I have to get into the mathematician state of mind, and read my suggested reading in the textbook, do the problems, finish the problem set, etc etc...

I'm going to have a lot of fun in this course!

 

[Psinter]

That answer is incomplete. What does the n stands for? If I cannot make assumptions, then I need someone to tell me what is the definition of n.

Have fun at uni btw!

 

[verty]

You need to be a little bit careful. I had an exam once that asked, 6 poles are placed in a circle, how many gaps are there between the poles? So of course I said 15 but the answer was 6. It's all about what the teacher expects.

 

[1MileCrash]

Really, don't assume in math? Math is all about beginning with an assumption and then studying the consequences. Without assumptions math doesn't exist.

 

[Borek]

Making assumptions is perfectly correct way of dealing with math. Basic proof of the fact $\sqrt 2$ is not a rational number starts with "Assume it is".

 

[airbusman]

Wow...I guess I got some stuff mixed up. Maybe I should go over the first lesson with my professor the next time we meet.

On a side note, I always knew that university was going to be much more fast paced than high school. But I still think it's cool that on the first day, we covered things from the basics (geometry, trigonometry, algebra, etc) and in the end we finished a discussion on truth tables, tautologies, and general topics in logic. It's great. (Y)

 

[lisab]

I think a more nuanced approach is: know what assumptions you're making, and define them properly.

 

[Nick O]

EDIT: As pointed out by 1MileCrash, I am being stupid here. Read my later post (#14, so long as none of our posts are deleted) to see what I meant to say without the stupidity.

Making assumptions is perfectly correct way of dealing with math. Basic proof of the fact $\sqrt 2$ is not a rational number starts with "Assume it is".

[STRIKE]It's worth mentioning for the OP that an equally valid first step is "assume it isn't". From there, you examine the consequences of the assumption until you find a contradiction.

As lisab said, the important thing is knowing and defining your assumptions, not necessarily having the correct assumptions.[/STRIKE]

 

[1MileCrash]

It's worth mentioning for the OP that an equally valid first step is "assume it isn't". From there, you examine the consequences of the assumption until you find a contradiction.

I assure you that you'll find no contradiction in assuming √2 isn't rational. Furthermore, it is not an equally valid assumption because it is what you're trying to prove!

 

[Nick O]

I assure you that you'll find no contradiction in assuming √2 isn't rational.

[STRIKE]Oh, sorry, when I skimmed his post I read that as "assume it is irrational".

(Final edit. I need to get back to the book where I left my brain.)[/STRIKE]

 

[1MileCrash]

Oh, sorry, when I skimmed his post I read that as "assume it is irrational".

Even if that's what he had said, it would just make both of you wrong. You cannot assume what you are trying to prove. You can assume the negation of what you're trying to prove. The two are never "equally valid steps."

 

[Nick O]

[STRIKE]FIXED. I should stop typing while studying.

Gah, I'm jumping back and forth between this and a textbook for one of my classes. The assumption I meant to say was also valid was "assume that it isn't [irrational]", and the assumption I thought he made was "assume it is [irrational]". I misspoke (mistyped) in my most recent post, and was inconsistent due to a foggy mind.

The point is that either assumption is fine. One might just be easier to work with.[/STRIKE]

 

[1MileCrash]

FIXED. I should stop typing while studying.

Gah, I'm jumping back and forth between this and a textbook for one of my classes. The assumption I meant to say was also valid was "assume that it isn't [irrational]", and the assumption I thought he made was "assume it is [irrational]". I misspoke (mistyped) in my most recent post, and was inconsistent due to a foggy mind.

No.

I understand that you made a simple mistype/misread, I'm not talking about that.

What I am saying is that:

The point is that either assumption is fine. One might just be easier to work with.

Is completely false. This is your confusion I am addressing, not the typo.

Assuming that √2 is irrational to prove that √2 is irrational is not fine.

 

[Nick O]

You are correct, my statement was false. When I was writing it, I was thinking of my main point (state your assumptions) rather than what I was actually saying (assumptions don't matter in proofs). They do matter in that you can't assume what you are trying to prove, so my statement was wrong. After that, I assumed(!) that I knew what I was talking about when I wrote that, and that it could be represented through bidirectional proofs.

I will try to redeem myself by saying what I was thinking about before I stupidly composed that post:

Making an "incorrect" assumption doesn't make incorrect proofs. It simply describes a mathematical system where your assumptions are true, which may be useful. For example, assuming that the parallel postulate did not contain any new information (which was incorrect for Euclidean geometry) resulted in the the discovery of other geometries.

 

[1MileCrash]

You are correct, my statement was false. When I was writing it, I was thinking of my main point (state your assumptions) rather than what I was actually saying (assumptions don't matter in proofs). They do matter in that you can't assume what you are trying to prove, so my statement was wrong. After that, I assumed(!) that I knew what I was talking about when I wrote that, and that it could be represented through bidirectional proofs.

I will try to redeem myself by saying what I was thinking about before I stupidly composed that post:

Making an "incorrect" assumption doesn't make incorrect proofs. It simply describes a mathematical system where your assumptions are true, which may be useful. For example, assuming that the parallel postulate did not contain any new information (which was incorrect for Euclidean geometry) resulted in the the discovery of other geometries.

Agreed

 

[HallsofIvy]

Perhaps what your professor said, or meant, was "do not make unstated assumptions".

 

[Ibix]

Agree with HallsOfIvy. Maths if one good way of realising that people bring a lot of assumptions to the table (e.g. that the squares in the chessboard and the pieces of plastic making it up are equivalent concepts) and that you need to be very, very specific about what you mean if you want to be rigorous.

The painful bit is that 64 is the right answer given the colloquial definition of "squares on a chessboard". In the game of chess, the 64 regions where you can place one and only one piece are important. To a mathematician, however, the rules of chess aren't relevant and the answer is the more complex one you gave. In many an applied situation, the maths is the easy bit. The trick is working out what definition of "square" the people you're talking to are using...

 

[Nick O]

If we ignore the colors completely, there is an infinite number of squares.

 

[Ibix]

...give or take the atomic nature of matter.

Both your comment and my response rather neatly illustrate my point about assumptions.

 

[Borek]

You don't have to ignore colors for that.

Which just restates the same thing about making assumptions...