In what ways has the invention of the computer changed mathematics?

>>8532628
It allowed us to get away from the analytical standard that before was a thing.

Now with computers approaching things numerically could even be called the preferred approach.

>>8532652
could you give me a concrete example of an analytical vs numerical approach?

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>>8532665
Solving a problem stupidly through brute-force going through every possibility vs. analyzing the problem in its general form

>>8532669
Numerical solutions are not brute forcing nor going through every possibility.

>>8532665
The simplest example is that of an integral.

If you had a curve and wanted to find the area under it from say 0 to 5 then you could

analytically: Find the antiderivative of your curve and then use the fundamental theorem of calculus to calculate the exact area

numerically: Use a Riemann sum but just break down your function in like 100 intervals and then ask a computer to sum all the rectangles, this will give you an area that will be like a couple of decimals off.

Or you could use better methods found here https://en.wikipedia.org/wiki/Numerical_integration

the riemann sum was never meant to be a tool for approximation really but if one day you are out in the wild with an integral and you do not memorize methods then obviously the riemann sum is your friend.

>>8532628
it has given discrete """""""mathematicians""""""" bragging rights on how their field was important to CS/ IC design
[spoiler]protip:it wasn't[/spoiler]

have computers helped mathematics achieve much that it might not have had the capability to do so before?

>>8532696
https://en.wikipedia.org/wiki/Four_color_theorem

I have for some reason taken a huge recent interest in mathematics. I have a very rudimentary understanding of the basics from doing two years of physics at university but I'd like to know more - can anyone recommend some texts which are challenging but not instant deep-end impossible tier stuff?

>>8532696
>>8532718
To expand on this. Some mathematicians first proved that every map could be broken down and classified into a finite set. Now to prove the four color theorem all you have to do is check each of these types of maps. The problem was doing this by hand would take years since there were so many cases to check. With a computer we can check these much faster.

>>8532776
Axler's linear algebra.
Armstrong's groups and symmetry.
Rosenlicht's analysis.
Spivak's calc on manifolds.
Roughly in that order. Spivak should definitely be last. Also you don't need to read all of each book.

>>8532665
Consider a tennis tournament. There are 12 players. One player must be eliminated in each match between two players. How many matches took place?

Numerical:
6 matches at first. 6 players are eliminated so we have 6 players left.
Now 3 matches take place. We are left with 3 players.
Now one match between 2 of the 3, and one final between the final 2.

So in total 6+3+1+1=11 matches took place.

Analytical:
Since one player is eliminated per match, except for the winner, the total number of matches is 12-1=11.

The issue is that for more complicated problems finding an analytical solution is not entirely obvious, but a numerical method nearly always is obvious. Computers assist here in the actual computations. Say if there were a trillion players, then you can see why the numerical method would be impractical. In the past we would have to find an analytic solution, since computing it would be impossible. But with computers numerical solutions become much more viable.

There are several proofs that involve proving that if a solution exists then a solution must exist below some particular upper bound, then using a computer to check the billions of cases up to that.

>>8532628
It has turned studying mathematics into a meme, because now if you have a difficult calculation you just type it into your calculator.

Thank >>8533283 for the recommendations Anon.

>>8533624
Yeah to make a computer do something you just enter the commands!

The critical problem solving skills in mathematics are not the part where you actually calculate, it's selecting what and how to calculate the answer.