How deep is math?

[wtf]

8⁲ + 1⁲ + 6⁲ = 4⁲ + 9⁲ + 2⁲ = 101

What do the squares mean?

Oh I see you mean squared. 8^2 + 1^2 + 6^2 = 64 + 1 + 36 = 101 = 16 + 81 + 4.

What does this mean to you?

[Philosophy Explorer]

8⁲ + 1⁲ + 6⁲ = 4⁲ + 9⁲ + 2⁲ = 101

What do the squares mean?

Oh I see you mean squared. 8^2 + 1^2 + 6^2 = 64 + 1 + 36 = 101 = 16 + 81 + 4.

What does this mean to you?

It means there is much to explore in math. To ask you, have you ever read Math Mysteries: the beauty and magic of numbers by Calvin Clawson?

PhilX

[wtf]

To ask you, have you ever read Math Mysteries: the beauty and magic of numbers by Calvin Clawson?

Not familiar with that one.

[Philosophy Explorer]

Here's another example many would take for granted at face value and not think further about:

8 1 6
3 5 7
4 9 2

This is the simplest magic square, a 3 x 3 without the boxes or cells, whose magic sum is 15 for its rows, columns and diagonals.

That's the way you may see it. I see much more, many multigrades (specifically bigrades).

Here are two multigrades:

8⁲ + 1⁲ + 6⁲ = 4⁲ + 9⁲ + 2⁲ = 101

8⁲ + 3⁲ + 4⁲ = 6⁲ + 7⁲ + 2⁲ = 89

So the first and third rows share a special relationship as well as the first and third columns.

Then we have this by reversing the concatenated numbers:

816⁲ + 357⁲ + 492⁲ = 618⁲ + 753⁲ + 294⁲ = 1,035,369

Next I came up with this one:

8⁲ + 3⁲ + 4⁲ + 16⁲ + 57⁲ + 92⁲ = 6⁲ + 7⁲ + 2⁲ + 18⁲ + 53⁲ + 94⁲ = 12,058

This is just a sample from this magic square which seems to readily yield multigrades which other magic squares wouldn't (different magic squares have special properties).

If you want to explore this further, check the internet and get ahold of Before Sudoku: The World Of Magic Squares by Block and Tavares (over 200 pages, besides recreational considerations, it shows practical uses for magic squares and it also covers Sudoku).

PhilX

These examples are based on rows and there are more based on columns and pandiagonals.

[wtf]

These examples are based on rows and there more based on columns and pandiagonals.

These are amusing but in no way illustrative of the depth of math.

[Philosophy Explorer]

These examples are based on rows and there more based on columns and pandiagonals.

These are amusing but in no way illustrative of the depth of math.

I disagree. At first glance, people would just see the magic square. A deeper look as I showed reveals more.

PhilX

[wtf]

I disagree. At first glance, people would just see the magic square. A deeper look as I showed reveals more.

What I'm saying is, that if you dip your toe into the shallow end of the kiddie pool ... and then after a while you dip in half your foot, and then your entire foot all the way up to the ankle ...

You have indeed gone deeper.

But you haven't gone very deep.

[Philosophy Explorer]

I disagree. At first glance, people would just see the magic square. A deeper look as I showed reveals more.

What I'm saying is, that if you dip your toe into the shallow end of the kiddie pool ... and then after a while you dip in half your foot, and then your entire foot all the way up to the ankle ...

You have indeed gone deeper.

But you haven't gone very deep.

Since discoveries are being made all the time, then there's a chance of going deeper. How much deeper no one knows since it takes time to find out. Also keep in mind as I pointed out that practical uses have been found for magic squares.

PhilX

[wtf]

Since discoveries are being made all the time, then there's a chance of going deeper. How much deeper no one knows since it takes time to find out. Also keep in mind as I pointed out that practical uses have been found for magic squares.

With your interest in numbers, have you ever given any thought to learning some number theory? The basic facts about modular arithmetic, how to solve those "What is the last digit of 999^999," and so forth. Greatest common divisors, and the great theorem that the greatest common divisor of two numbers is always a linear combination of those numbers. In other words if n and m are relatively prime, that there are always two integers r and s such that rn + sn = 1. And whatever the gcd is, you can get a linear combination to be that.

Another subject you might like is discrete math. Sets, equivalence relations, a little logic, a little computer science, a little graph theory, a little combinatorics. Modular arithmetic and the theorem about gcd's are special cases of much more general constructions.

Math starts becoming deep when you begin to study the underlying structures of the numeric patterns. For example if you take the integers mod 4, the set {0, 1, 2, 3} with addition mod 4, you have a system where you count 1, 2, 3, 0, 1, 2, 3, 0, ... forever. You can add, subtract, multiply, and divide mod 4, but there isn't any sensible notion of order.

Now think about the complex numbers. In fact think of just the complex number i. You know that i^2 = -1, and i^3 = -i, and i^4 = 1, and i^5 = i, and ... hey, they're just cycling around too! In fact the complex numbers are hiding a little copy of the integers mod 4. Specifically, each multiplication by i represents a quarter turn of the plane in the counterclockwise direction. It's like turning left at a traffic light. Four left turns leaves you pointing in the direction you started. From now on when you make a left turn in your car, think to yourself, "I'm multiplying my direction by i." That's how to understand complex numbers.

To me, this is what's deep. Finding the underlying patterns. Taking familiar things and seeing them from new directions to gain insight. We teach people that i is this mysterious "square root of -1," and it sounds like bullshit. If we taught them that i represents a quarter left turn in the plane, it would be very natural.

But then again, playing around with numbers is a perfectly valid enterprise. Math is deep at every level. There are mysteries and connections everywhere you look. What's deep for you is different than what's deep for me, but depth is where you find it.

[Philosophy Explorer]

Since discoveries are being made all the time, then there's a chance of going deeper. How much deeper no one knows since it takes time to find out. Also keep in mind as I pointed out that practical uses have been found for magic squares.

With your interest in numbers, have you ever given any thought to learning some number theory? The basic facts about modular arithmetic, how to solve those "What is the last digit of 999^999," and so forth. Greatest common divisors, and the great theorem that the greatest common divisor of two numbers is always a linear combination of those numbers. In other words if n and m are relatively prime, that there are always two integers r and s such that rn + sn = 1. And whatever the gcd is, you can get a linear combination to be that.

Another subject you might like is discrete math. Sets, equivalence relations, a little logic, a little computer science, a little graph theory, a little combinatorics. Modular arithmetic and the theorem about gcd's are special cases of much more general constructions.

Math starts becoming deep when you begin to study the underlying structures of the numeric patterns. For example if you take the integers mod 4, the set {0, 1, 2, 3} with addition mod 4, you have a system where you count 1, 2, 3, 0, 1, 2, 3, 0, ... forever. You can add, subtract, multiply, and divide mod 4, but there isn't any sensible notion of order.

Now think about the complex numbers. In fact think of just the complex number i. You know that i^2 = -1, and i^3 = -i, and i^4 = 1, and i^5 = i, and ... hey, they're just cycling around too! In fact the complex numbers are hiding a little copy of the integers mod 4. Specifically, each multiplication by i represents a quarter turn of the plane in the counterclockwise direction. It's like turning left at a traffic light. Four left turns leaves you pointing in the direction you started. From now on when you make a left turn in your car, think to yourself, "I'm multiplying my direction by i." That's how to understand complex numbers.

To me, this is what's deep. Finding the underlying patterns. Taking familiar things and seeing them from new directions to gain insight. We teach people that i is this mysterious "square root of -1," and it sounds like bullshit. If we taught them that i represents a quarter left turn in the plane, it would be very natural.

But then again, playing around with numbers is a perfectly valid enterprise. Math is deep at every level. There are mysteries and connections everywhere you look. What's deep for you is different than what's deep for me, but depth is where you find it.

I have a book on number theory by Beiler. It says that Gauss invented modular arithmetic, talks about multigrades, etc.

PhilX

[Philosophy Explorer]

Two more for the road:

The following can be done with the first and third rows along with the first and third columns:

8•1 + 1•6 + 6•8 = 4•9 + 9•2 + 4•2 = 62

8•3 + 3•4 + 8•4 = 6•7 + 7•2 + 6•2 = 68

I call this cyclic multiplication.

We also have:

81⁲ + 35⁲ + 49⁲ = 61⁲ + 75⁲ + 29⁲ = 10,187

PhilX

[Eodnhoj7]

One more for the road:

The following can be done with the first and third rows along with the first and third columns:

8•1 + 1•6 + 6•8 = 4•9 + 9•2 + 4•2 = 62

8•3 + 3•4 + 8•4 = 6•7 + 7•2 + 6•2 = 68

I call this cyclic multiplication.

PhilX

Interesting, but we are still left with the question of how the "cycle" provides the foundation of number. 1 moving to zero, through a series of fractals of 1, with the fractal cycling back to whole numbers, seems to be the only option I am aware of.

If this is the case, the number as "unit", may begin with division and multiplication as the foundation of number.

[Philosophy Explorer]

Here's another concept which I believe is called a magic carpet to expand the multigrades:

Repeat the 3 x 3 magic square:

8 1 6 8 1 6
3 5 7 3 5 7
4 9 2 4 9 2

Let's consider a diagonal:

456⁲ + 978⁲ + 231⁲ = 654⁲ + 879⁲ + 132⁲ = 1,217,781

If you think this can be done in general, guess again.

It's also my belief that the number of multigrades and the number of magic squares are the same: Aleph null.

PhilX

[Philosophy Explorer]

I just did cyclic multiplication. Let's examine it in connection with a 4 x 4 magic square. Consider:

16 02 03 13
05 11 10 08
09 07 06 12
04 14 15 01

where the magic sum for the rows, columns and diagonals is 34.

Let's check the rows for multigrades:

1) 16⁲ + 2⁲ + 3⁲ + 13⁲ = 438
2) 5⁲ + 11⁲ + 10⁲ + 8⁲ = 310
3) 9⁲ + 7⁲ + 6⁲ + 12⁲ = 310
4) 4⁲ + 14⁲ + 15⁲ + 1⁲ = 438

So we can see there are two multigrades here. Let's check the rows for cyclic multiplication:

1) 16•2 + 2•3 + 3•13 + 13•16 = 285
2) 5•11 + 11•10 + 10•8 + 8•5 = 285
3) 9•7 + 7•6 + 6•12 + 12•9 = 285
4) 4•14 + 14•15 + 15•1 + 1•4 = 285

How about that? Let's check the columns:

5) 16•5 + 5•9 + 9•4 + 4•16 = 225
6) 2•11 + 11•7 + 7•14 + 14•2 = 225
7) 3•10 + 10•6 + 6•15 + 15•3 = 225
13•8 + 8•12 + 12•1 + 1•13 = 225

Wow! Let's rotate the quadrants 180°:

11 05 08 10
02 16 13 03
14 04 01 15
07 09 12 06

Let's dot multiply the rows and columns by (1,2,3,4):

1) 1•11 + 2•5 + 3•8 + 4•10 = 85
2) 1•2 + 2•16 + 3•13 + 4•3 = 85
3) 1•14 + 2•4 + 3•1 + 4•15 = 85
4) 1•7 + 2•9 + 3•12 + 4•6 = 85
5) 1•11 + 2•2 + 3•14 + 4•7 = 85
6) 1•5 + 2•16 + 3•4 + 4•9 = 85
7) 1•8 + 2•13 + 3•1 + 4•12 = 85
1•10 + 2•3 + 3•15 + 4•6 = 85

All 85. Try dot multiplying by (4,3,2,1).

Another pattern from the top magic square is:

16•3 + 2•13 = 74

5•10 + 11•8 = 138

9•6 + 7•12 = 138

4•15 + 14•1 = 74

You will find a similar pattern for the columns (in general, due in part to symmetry, any pattern you find for the rows
will also occur in the columns).

I just barely scratched the surface with magic squares.

PhilX

[Philosophy Explorer]

Let n = 0 to 3 and y becomes 2, 3, 5 and 9. Compare with the 2nd column and you will see they are the same.
Coincidence?

A sample size of four data points doesn't count for much.

Of course no finite set of data points can ever be sufficient. There are many numerical patterns that go on for millions of datapoints then fail.

But even if we can never be certain a pattern holds for all integers when we can only test finitely many, if there are a LOT of datapoints, at least it makes for an interesting story. T

With four data points, you haven't got anything compelling. There's no story there. Nobody's going to sit up and go, Wow that's amazing!

I would like to illustrate this point with a mathematical anecdote. You don't have to follow every detail but there is a point at the end.

Let Primes(n) be the number of primes less than or including n. For example Primes(3) = 2, because 2 and 3 are the only primes less than or equal to 3. Likewise Primes(6) = 3, because the smaller primes are 2, 3, and 5. Note that the Primes() function gives us a sensible answer for any positive integers, prime or composite.

Now it's easy to calculate Primes(n) for a given n. You just count them! You could write a program. The problem is that it would be a SLOW program. As n gets large, the computation becomes intractable.

It would be good if we could somehow approximate Primes(n) with a more tractable function that grows more slowly. In the early 1900's people found such a function to approximate Primes(). The function is called the logarithmic interval. Nevermind what Li() is, that's not important. What's important that as n gets large, P(n) and Li(n) get as close together as you like. Li() approximates Primes().

It was noted that Primes() always seems to be less than Li() no matter how many values of n they could calculate at the time. Everyone believed that Primes(n) < Li(n) for every n.

Then in 1914, Littlewood (the guy played by Toby Jones in The Man Who Knew Infinity) proved that there must be SOME n for which the number of primes was greater than the output of the formula for that n. In other words the difference switches signs. But he had no idea how large such an n must be.

In 1933 one of Littlewood's former students, Skewes, proved that (if you assume the Riemann hypothesis) the n that Littlewood predicted could be no larger than a certain specific number. There's a big number, called Skewes' number, such that there is SOME value of n less than Skewes' number with Primes(n) > Li(n). The inequality flips.

The value of Skewes number is 10^10^10^34.

How big is this number? Exponentiation associates from right to left. Reading from the right, 10^34 is 1 followed by 34 zeros. And 10 to the power of that, is 1 followed by 10^34 zeros, And finally there's one more level up. Skewes' number is a 1 followed by 10^10^34 zeros.

That's how big this number is. We can't imagine such a number.

And the point of all this is, that for all we know, the claim "Primes() is always smaller than Li() holds up for every single value of n less than Skewes' number.

Now there have been much sharper lower bounds since then, but the point remains. A proposition can hold for a huge, unimaginably large number of datapoints; and then fail.

Four datapoints ... that just ain't gonna cut it.

I'm glad you brought up Littlewood who was an associate of GH Hardy. Hardy had correspondence with Ramanujan (from India) and was so impressed with this math genius that he arranged for his passage to England.
Together they took math to new heights and contributed much (Wikipedia has more details).

PhilX