Maths

Just as Philip Wadler used to repeat, Mathematics is a candle which illuminates everything (in the darkness of ignorance at the sea of screaming bullshit).

One has to understand not how add or multiply some numbers, or even integrals or derivatives, but how do we know what they are.

How do we know that something we talk about is not bullshit?

Well, there are previous results, usually in a form of explicit rules, which we have been taught in a school, to memorize and then use (without understanding).

This is, however, not mathematics itself. Proper mathematics begin which observations, generalizations, and propetly abstracted out recurrent patterns.

After we got some abstractions, the main questions is “how do we know“ something is true about them.

We have to make ourselves sure, then others will check our reasoning and become convinced. This is how all the previous results came to be.

A written down formal reasoning (a sequence of verifiable discrete steps from a set of premises to a conclusion) is called a proof. There are lots of subtleties, of course.

The beginning of proper mathematics is to contemplate and try to zoom-through the abstractions and conceptual layers encoded in the statements like:

\[\forall a,b. a + b = b + a\], or \[\forall a,b,c. a + (b + c) = (a + b) + c\], or \[\forall a,b,c. a(b + c) = ab + ac\]

to actually realize and internalize why these are necessarily true.

These above, for example, require familarity and deep understanding of the following notions:

• addition is a generalization of the universal notion of putting or coming together (in the same locality) (think of concatenation of two segments of a Number Line and couning the “nothes”)
• the Natural Numbers or \[\mathbb{N}\] as a generalization of associative counting “on fingers”.
• multiplication in the context of \[\mathbb{N}\] is a repeated addition to itself (\[n\] times).
• multiplication produces the same result no matter which factor is chosen as “times” (think about the notion of an area of a rectangle).
• in other contexts (\[\mathbb{R}\]) the similar operation is scaling or even application (matrices).
• the notion of a Sum which can be expressed as parts put (added) together
• the notion of multiplying (scaling up) each part of a /Sum by the same number of times.

There are a lot of mathematics in the simple arithmetic and algebra.

These generalizations has been proven valid (true) for all Natural Numbers. Thus they become results.

The results are universal, so they have been called “laws” (of algebra) and given unique names.

• The Commutativity Law (of addition).
• The Associativity Law (of addition).
• The Distributive Law (of multiplication over addition).

and so on.

The key point is to realize why they are necessarily true, given what the nature of addtition and multiplication operations are.

It is even better to realize (zoom through) that it all comes from the properties of addition.

There are, of course a lot more, but the principle and mental techniques of formal reasoning are exactly the same.

There are notions of a triangle (one more than two dots), and of the right triangle and a whole set of relations between its sides and angles.

There are notions Sets, of Sets together with (a small Set of) related Operations, and so on.

These higer level /abstractions capture and generalize high-level patterns which have been discovered while observing the common properties of /operations.

So, we study properties of our captured abstractions and operations and, when there is no contradictions, generalize to even higher-level notions.

Contrary to what Chuds are saying, these “higher realms” are not infinite, and there is literally nothing “higher” that a Monoid (Categories are artificial, “empty” abstractions).

Lets come back to Earth. The main skill is to zoom through and to collapse these abstractions back to What Is, and to follow through (and validate) all the reasoning which produced them.

This is should become a habit (of thinking) to be used with everything one hears, reads or sees.

The fundamental principle is that all the proper mathematical notions and concepts could be traced back to What Is.

A prime number, for example, is just a “pile” which cannot be constructed by any other number being added to itself \[n\] times, except \[1\]. So it is just \[1\times n\] or just \[$n\].

This is also means that you can only reach there with “steps of size one”. Any other factor would either overshot or undershot the location (on a Number Line).

The implication of this is the notion of a factorization and that a product of two primes cannot be factored out by any other means but exhaustive enumeration.

This is how one understands math deeply - from the ground up and from the first principles, not just memorizing the formulae, as they “teach” you in a school.

Last but not least, the (only) proper philosophy would postulate that everything in the mind is (must be) reducible (im principle) to What Is. What isn’t (cannot be traced back or unfolded back to reality) is called bullshit.