I contend that what we commonly call “math” is really three subjects:
1) Arithmetic calculation
2) Quantitative reasoning
3) “Real” Math which is made up of
…..a) Pure math
…..b) Applied math
and that these really have not all that much to do with each other.
We all know what arithmetic calculation is: Addition, subtraction, multiplication, division, maybe some exponentiation. In the “old days” you might learn how to calculate a square root by hand, but nowadays, not. Some mathematicians are good at calculating (Gauss, for example, was a prodigy at this) but others are bad at it, and one, at least, could not remember what 8×6 was.
We all need to know a little arithmetic, but not that much – far less than is taught. We *might* need to multiply by 1/2 or 1/4; we are unlikely to need to multiply by 2/5 or 4/7 or what not. That’s why there are calculators. I suppose it’s of some utility to be able to calculate quickly and accurately, but it’s not a key thing.
It’s much more important to know what calculations to do – which buttons to push on your calculator.
Quantitative reasoning is ‘number sense’. It is highly developed in people who are good at things like accounting and engineering. I have a friend who is an accountant, and he says he can look at a balance sheet and sense that something is wrong, and then he would look closely.
But this is important more generally. If you accidentally push the wrong buttons on your calculator, you need to know that you messed up. If you have a problem like:
>>> If I make $843 each week, what do I make in 52 weeks?
and you accidentally push + instead of X and get $895, it’s important to know that that is wrong, even if you don’t know the exact right answer.
“Real math” is the most interesting and the least known. I daresay that most people, even most adults, have very little notion of what mathematicians do all day; they’d be surprised (maybe shocked) to learn that much of math doesn’t involve numbers. I divided 3) into two subcategories, because pure math and applied math are different, but people who are good at one are often good at the other, and they interact a lot.
Applied math involves using math to solve real problems in all sorts of fields – all of the sciences, engineering, architecture, and many others. I am not going to talk much about this, as I know relatively little about it, outside of statistics, which I know fairly well.
So, what is pure math?
>>>> Mathematics is the search for structured beauty. <<<
G.H. Hardy, a very famous mathematician of the 20th century, said that
>>>> A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more
>>>> permanent than theirs, it is because they are made with ideas.
>>>> The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. Beauty is the first
>>>> test. There is no permanent place in this world for ugly mathematics.
Beauty? Patterns? In math?
But a certain kind of beauty, made with a certain set of tools and a certain set of rules. All the different arts have sets of tools – whether they are musical instruments, paints and brushes, words, clay, or what have you. Math’s tools are axioms and theorems and logic and proof, but the key is the beauty.
Paul Erdos, the most prolific mathematician of the 20th century, said
>>>> A mathematician is a device for turning coffee into theorems.
When Carl Frederich Gauss, (perhaps the greatest mathematician ever, certainly in the top 5) was asked why his proofs were difficult to follow, he said:
>>> When a sculptor is finished with his creation, he removes the scaffolding
One problem for those who wish to appreciate the beauty of math is that much of it hides behind high layers of abstraction, and the math currently being worked on often requires many years of study even to understand. But not all. Here is a proof, from Euclid, that anyone who remembers grade school math can understand.
The question: Is there a largest prime number?
The answer: No.
Some background … a prime number is a integer (or whole number) larger than 1, which can only be evenly divided by 1 and itself: 2, 3, 5, 7 ….
There are ways to test if a number is prime. There are very efficient ways of doing this, but we needn’t bother with them; we can simply go from 2 to p, testing each number. And each test can be simply involve trying to divide the number by all the numbers smaller than it. For example, if we want to see if 25 is prime, we test
2 into 25 – doesn’t go evenly
3 into 25 – doesn’t go evenly
4 into 25 – doesn’t go evenly
5 into 25 – does go evenly
so 25 is not prime.
On the other hand, if we test 11
2 doesn’t go evenly
3 doesn’t go evenly
similarly for all numbers up to 10.
11 is prime.
(again, I know the above is very inefficient).
We shall prove this by reductio ad absurdum , a common method of proof. We assume that what we wish to show is NOT true, then we deduce something impossible. Therefore, what we wish to show must be true.
1). Assume there is a largest prime number. Call it p .
2) Find all the prime numbers smaller than p. Multiply them all together.
3) Add 1.
Call this number q . q is certainly bigger than p.
There are now two possibilities:
1) q is prime. But then p is not the largest prime.
2) q is not prime. But, in this case there must be some number larger than p that is prime, because q cannot be evenly divided by any prime smaller than p, or by p itself.
Hence, p is not the largest prime; we can repeat this process for any number, so no number is the largest prime.
QED (which is a Latin abbreviation for quod erat demonstrandum – “which has been shown”; it’s commonly put when we finish a proof).
Now, you may find this beautiful, or not. If you are a mathematician, it is virtually certain you find it beautiful, but I can’t *prove* it to you, any more than I can prove to you that a piece of music is beautiful, or a painting.
Some people think that “Math is applied logic”. I think this is incorrect. Logic and axioms and so on are the tools of math. Just as paint and brushes and canvas are tools for painting. But one would not say that “painting is applied coloring”. A further example of what I mean is given by the MU puzzle in the book Godel Escher Bach, by Douglas Hofstadter.
Here it is, in brief:
Your task is to make the string MU.
You start with the string MI
You may apply any of the following rules:
1) If you have a string that ends n I, you can add U
2) If you have Mx you can make Mxx
3) If you have III anywhere in a string, you can substitute U for that
4) If UU occurs anywhere in a string, you can drop it.
from MI you can make MIU by rule 1
from MI you can make MII by rule 2
if you get MIIIU you can make MUU by rule 3
If you get MUUI you can make MI by rule 4
you can apply them in any order, as many times as you like.
Fool around a bit. Try to get MU. Don’t work at it too long, just get the idea. I’ll give the answer at the end of this article.
One last thing:
Much of mathematics is written in language that is obscure, requiring years and years of study. I don’t even understand the titles of some journals (never mind the titles of the articles!)
There are two fields where this isn’t so:
In number theory, the problems are easy to state, but hard to solve – and they seem hard to solve
In probability, the problems are easy to state, but hard to solve. But they seem easy. That’s dangerous, as you’ll see if you look at my article on the Monty Hall problem.
It is not possible to get MI
Think of how many I there are in each string, whether that number is divisible by 3, and how the rules affect the number of I’s. E.g. in
>>>> MI there is one I
>>>> MU there are zero I
Rule 1 If you have a string that ends n I, you can add U
does not affect number of I
Rule 2 If you have Mx you can make Mxx
Doubles the number of I
If you have III anywhere in a string, you can substitute U for that
Subtracts 3 I
If UU occurs anywhere in a string, you can drop it.
does not affect it.
Now, we have to go from a 1 I situation to a 0 I situation. If you divide 1 by 3, you get a remainder of 1. If you divide 0 by 3, you get a remainder of 0.
If you double the number of I, you never get to a situation that is divisible by 3.
1 2 4 8 16 etc. never divisible by 3
and subtracting three leave the remainder the same.
So, it’s impossible to get MU from MI.