The greatest failure of the human species is the inability to effectively teach mathematics.
Clearly, it should be the easiest subject: the most elaborate theorems and their applications can be instantly learned by any computer, and the human brain is allegedly much more advanced than any computer.
Most humans instinctively reject the claim that the human brain is overrated.
And yet: there is not one person on this planet who can explain the meaning of the mathematical term 'e' in normal language. This will be the new definition of intelligence: the ability to explain 'e' understandably. By this definition, there is no intelligent life on Earth.
Of course there isn't any noticeable demand for mathematical insights. Most people are afraid of thinking too deeply or too far ahead.
Some insights are not meant to become widespread. They would interfere with the mundane concerns of ordinary life.
Like chess or tennis, you can't bluff your way through math, unlike many social fields. But the basic rules should be clear to anyone. Assuming they want to, humans should be able to grasp the procedures and sequences of any mathematical principle, after finding useful metaphors to demonstrate those principles.
It could be a drawing, a thought experiment, or perhaps an interactive animation. The novel Flatland introduced a brilliant way to visualize higher dimensions. A four-dimensional object can be visualized by imagining how an inhabitant of two-dimensional space would see a three-dimensional object passing through.
In some ways, 2-D space is as hard to understand as 3-D space or above, but higher dimensions can be distorted in many more ways. A sheet or a string can only bend and fold in so many directions before closing back on itself. This is fortunate for protein synthesis and other biological functions.
When too many possible directions exist, as in universes with four or more dimensions, space can easily curve without ever folding back on itself.
An inhabitant of an 81-dimensional universe would have less in common with an inhabitant of a slightly differently folded and compacted 81-dimensional universe than with us.
Math can solve seemingly intractable problems beyond the limits of human intuition.
Imagine a gambling game where the player has a 51% chance to double their money or a 49% chance to lose all the money they bet.
That looks like a good gamble, with an expected gain of about 2% every turn. But a smart player wouldn't want to risk all their money at once.
How could they maximize their expected winnings over the long term? Should they bet 50% of their capital the first time they gamble? 10%? 75.508733%?
It turns out long-term capital growth is maximized when the player bets 2% of their total capital each turn. This provides a slow but steady 2/10.000 growth rate, which at 250 turns leads to a 5% annual return.
Math is everywhere and always true, like God only for real.
Nature is stitched together (if it is unified at all) by logic alone.
That's all that really exists. All the other things are just ghosts in the numbers.
And the numbers involved are necessarily huge. Numbers like 8.4 x 10^75 or 1/50^50 look deceptively accessible, small enough to write on the back of your hand.
By the same reasoning, prime factoring looks like it should be easy: the big number can only be divided by two smaller numbers. How hard can it be to find them both?
However, a problem called the combinatorial explosion means that usually every single possibility has to be tried to ensure finding the only correct solution.
And there are many, many possibilities to try out.
To write down the number of ways that the energy in the universe could be rearranged would take a piece of paper so big that to write down the size of that piece of paper you would need a piece of paper too big to fit inside the universe (assuming ordinary notation).
The quantum computers of the future will merely accomplish what already seems intuitively simple, by hiding their own complexity and the complexity of the problem they're solving.
One of the biggest unsolved mysteries in mathematics is known as 'P=NP'.
If P=NP is true, then it will be possible to generate a universal problem-solving method that can solve any mathematical problem whose solution can be easily checked, but which would normally take an extremely long time to solve. This new method would be able to solve the problem in roughly the same time it takes to check the answer.
"Roughly the same time" means it could still take trillions of times longer, or vastly more than that, but mathematically speaking that's not too bad.
For example, using a code, it's easy to turn any text into gibberish, but it's very hard to decode that text if you don't know the code. Even if you do know the code, it can be a lot harder to decode the text than to encode it.
It's a lot harder to solve a sudoku puzzle than to generate one.
This is true for almost all problems with many variables, where a change in one variable immediately alters all the others. The most famous example is the traveling salesman problem.
The only sure method is to test all possible solution combinations. P=NP would mean there must always be a shortcut, a form of entropy reversal. It appears too good to be true.
The biggest known math problem is infinity.
The ancient Indian manuscript Surya Prajnapti (400 BC) identifies three types of numbers: enumerable, innumerable, and infinite. Each type is further divided: lowest, intermediate and highest enumerable; nearly, truly, and innumerably innumerable; and nearly, truly, and infinitely infinite.
Then things start to get complicated.
50% of infinity is identical to infinity itself. The second, smaller quantity can be derived from the first one merely by changing the last in an infinite string of digits, which is no change at all. 50% of any finite number (like 1 or Flannigan's Number) seems like a much greater change.
Transcendental numbers can't be generated by any polynomial equation (such as ax+by+cz^n) at all.
There are countless different types of infinities. Some may even explain human existence.
Cantor's famous diagonal proof lined up two infinite sets of numbers, demonstrated a perfect correspondence between them, and then showed that it was possible to create still larger sets of transfinite numbers.
Zoom in forever on a fractal, and it soon becomes clear the numbers involved must exceed the complexity of the physical universe. The spectacular diversity only keeps expanding. This insight alone implies the existence of other universes.
Human existence however, could be described with strictly finite numbers (though still much too large to define exactly). In fact our minds are too simple; almost as simple as they can be.
Small, human-level minds are easier for mathematical reality to generate in vast quantities. There must exist an infinite number of copies of each human-level mind, and very rapidly decreasing percentages of all higher minds.
Any human-level mind represents a small but finite fraction of everything that exists.
Minds can be generated by mathematical equations, or exist as random islands of order in a sea of chaos, yet almost all minds are components of the larger universe equations which created them.
There are an infinite number of equations that can generate universes, that could generate an infinite number of human-level minds. Equations are everywhere: obvious, hidden, and implied. Theoretical universes are sprouting forth from all points of reality.
Humanity is marked by simplicity in size and time, one of the first planetary civilizations in the observable universe. Our minds are as small as they can be while still allowing self-awareness.
Being easier to describe, smaller and simpler minds are also more likely to be consistent.
Since our minds are so constrained, our universe must appear relatively simple and predictable too, or humans couldn't survive here.
In other ways this universe is almost as complex as it can be, though almost no one has remarked on this fact. The hidden molecular, atomic, subatomic and especially quantum complexity is beyond absurd. All the supercomputers in the world can't calculate the interactions of one electron with itself.
Maybe this hidden chaos is necessary to balance out our supremely unlikely mind patterns.
Or the unseen collisions and particle interactions around us may be subtly changed by our brain patterns, duplicating and amplifying our awareness on a vast scale.
According to some quantum interpretations, the hidden complexity could also be shared by many parallel universes. The most productive mathematical universe equations would also generate the most observers.
Human-level minds are spread out throughout the 'top level', the basic outline, of an endless transfinite fractal, which only gets more complicated zooming in.
Our existence can't be fully simulated by any finite method. Merely by existing, humanity has access to what can only be described as transcendental knowledge, beyond the limits of finite math to analyze.
Turing called any process that could generate such ineffable knowledge an Oracle.
Human minds are indeed Oracles, but only because they are so simple.
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