Computers in Math

[Jagella]

I make use of computer a LOT in my math studies. My PC helps me to reduce errors, research questions if I'm having difficulty, crunch numbers, create geometric figures, graph data and functions, and interact with a lot of smart, like-minded people like you.

 

Anyway, can any of you think about the down-side of computers in math studies?

 

Jagella

[Bignose]

I have some, but I am curious why you are asking. I'll just be blunt, is this for a homework question or similar? It just seems a little too pre-packaged to me...

[ajb]

Anyway, can any of you think about the down-side of computers in math studies?

In my experience students can sometimes be too quick to trust calculators and computers over some simple reasoning. For example, I remember a group of engineering students claiming my answer to some question was wrong because their calculator said something different. I forget what it was we were calculating but it was something physical, like the classical radius of an electron or something. There answer was 10 of metres or something equally silly. All that had done is put brackets in the wrong place when using their calculator. We have all done things like that, but they failed to use other knowledge to quickly realise their calculation must be incorrect.

 

Similar small coding errors can course trouble. Not understanding what the code is actually doing rather than what you want it to do is a general problem, I include myself as a person who has problems like that.

 

There must be other problems. What do you thing about it?

[Jagella]

I have some, but I am curious why you are asking. I'll just be blunt, is this for a homework question or similar? It just seems a little too pre-packaged to me...

I'm not in formal schooling, so no, this is not a homework question.

 

When Andrew Wiles proved Fermat's Last Theorem, I understand that he did not use a computer. So computers are not always indispensable in math studies. I'm wondering why computers might not always be helpful in mathematics.

 

Jagella

What do you thing about it?

I think that in most ways computers have been a tremendous boon to learning and applying math. Often, though, I've found that scribbling on a piece of paper can be faster and easier than using a computer. Using pen and paper frees me from the strictures of the PC allowing me to think about the math rather than which keys to press on the keyboard or mouse.

 

Jagella

[ajb]

Often, though, I've found that scribbling on a piece of paper can be faster and easier than using a computer.

That may depend on what you are trying to do. But generally, the best way to be sure that you understand what is going on is to do it by hand rather than reach for the 'black box' of a computer. That said, using computer algebra systems and similar is now common in research. One does not want to waste time with boring calculations that are part of something bigger, or are straight forward but cumbersome and long.

Edited October 3, 2015 by ajb

[Jagella]

...using computer algebra systems and similar is now common in research. One does not want to waste time with boring calculations that are part of something bigger, or are straight forward but cumbersome and long.

I often use pen and paper for the basic ideas--ideas that don't require rigorous computational accuracy. This method helps me to think about the underlyng reasoning without the many distractions inherent in computers. I then often move on to the computer to check my work. I use software to do the calculations that are too complex to do manually and to draw precise geometric shapes if need be.

 

Recently I was studying logarithms. I wanted to get an idea about how to use them to do calculations. I then checked my answers using my computer. I think it was a good way to get a feel for how math evolves and what a boon computers are to the study and application of math.

 

Jagella

[Keen]

I come from pure mathematics background and my philosophy when it comes to using computers in mathematics is to avoid them as much as possible, so I believe I can provide you with some downside of computers.

 

When it comes to pure calculations, computer is an invaluable tool, but it should just be a means to gain time. I never put calculations into a computer, that I would not be able to do myself, because precomputed functions are kind of a black box and it may become problematic when you simply trust your computer without knowing what it is doing. If I understand well all the algorithms implied, then I am not against using computers to gain time in calculations.

 

However I refuse to use computers when doing some rigorous mathematical proofs. The problem is: computer does the calculations just fine, but does not help you to understand what is happening. When a proof is done by some brute force calculations I consider it incomplete in a sense, that it does not make people any wiser. I don't know what your mathematical background is, so it's hard to give specific examples.

[Jagella]

I agree that it's important to know what a computer is doing. It's entirely possible to make errors with a computer.

 

Would you agree that it may be best to start out with pen and paper and then check your results with a computer? I often take that route.

 

My background in math involves algebra, geometry, trigonometry, linear algebra, abstract algebra, calculus, and statistics.

 

Jagella

[overtone]

Computers, at least so far and in ordinary use, restrict one to distant visual registration of pattern. This handicaps memory and readiness of technique, so that things like factoring polynomials or rotating shapes or "feeling" the explosive difference between logarithmic and exponential growth become alien and unintuitive instead of automatic. This costs time, at a minimum, and may screen from insight.

 

Deaf people dance more by feeling the rhythm through the floor and the passing bodies, than by watching other people's feet.

[ajb]

However I refuse to use computers when doing some rigorous mathematical proofs. The problem is: computer does the calculations just fine, but does not help you to understand what is happening. When a proof is done by some brute force calculations I consider it incomplete in a sense, that it does not make people any wiser.

Computers can be useful for proofs that require checking a finite list of examples.

 

You still of course have to be careful that the computer is doing what you think it is.

 

https://en.wikipedia.org/wiki/Computer-assisted_proof

 

I do not use such proofs, but sometimes I use a computer algebra system to do calculations such as integrals and sometimes algebra that I could do, but don't want to spend the time doing.

Edited October 8, 2015 by ajb