ARITHMETIC is all about numbers, and the tallying, adding, subtracting, multiplying, dividing, thereof. It is about actual *things*, masses. Two cows plus three cows is five cows. Arithmetic does not deal in abstract ideas. It is concrete.

Now let's define the next level, MATHEMATICS! It is about SYMBOLS. It is abstract, no mass, just ideas. It's not about "3+4", its about "X+Y".

I would define Mathematics as:

  Applying symbols to represent things and then by manipulating the symbols,
  coming to conclusions about the things themselves.

And beyond that,

  coming to conclusions about the things which might not be readily evident
  without going through the step of symbolization.

Here is LRH:
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Logic 21:

"Mathematics are methods of postulating or resolving real or abstract data in any universe and integrating by symbolization of data, postulates and resolutions."
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From merriam-webster.com: "integrating: 1: form, coordinate, or blend into a functioning or unified whole: unite."

IMHO: "integrating by symbolization" means integrating, coordinating, blending, not the things themselves, but their symbols.

This is what demonstration using a demo kit or clay is all about. You let little bits of stone and paper clips REPRESENT (ie, symbolize) some things you are trying to integrate. Then you move these little symbols around to see how they inter-relate and use that to understand how the "real things" would likewise move around and inter-relate.

Integrating multiple symbols into a unified whole can produce a simplicity in the unified whole where there was before a complexity of the many parts. If one doesn't see (understand) immediately how parts of a thing combine and unify, one can create *symbols* of each, eg, pictures, drawings, clay models, and then look at how these models (symbols) combine.

Logic 21 says "integration by SYMBOLIZATION OF DATA...". In other words, we are integrating, not the data, but the SYMBOLS of it.

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LRH again: 1956.08.31, The Anatomy of Human Problems, 4th lecture in the Games Congress

"Mathematics does not confront the bridge girders, mathematics confronts a piece of paper about bridge girders, doesn't it? But if the individual using the mathematics is actually capable of knowingness then the mathematics has some use.

"If the individual who is using the mathematics does not have any knowingness, then he is not capable of any use. Don't you see. If he's not capable of knowing what he's just written down as symbols, then what good are the symbols?"
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Thus, a carpenter or other skilled workman just "knows" where to place the nail. He is not using mathematics, he is using "knowingness", which is a far higher ability.

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From LRH: 1954.12.07 ACC 9.2, "Essence of Auditing, Know to Mystery Scale", track 6

"Mathematics ... is a system of vias by which you can derive the answer without having to know it."
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My friend is a carpenter. He can put a nail in just the right place. He can just "look" and then "know". Me, I'd have to measure the boards, compute the square root of the hyper-matmos, multiply by the wally-wog, and divide by 3.14, to get the "answer", and I'd put the nail in right there. I'm good at math, and my answer would probably be correct!

My carpenter friend does not use mathematics--he "just knows". I, on the other hand, have to use a bunch of vias to get there.

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More from LRH:

1952.11.00 Scientology 8-8008, chapter 12, "Differentiation, Association and Identification", page 83-4.

"Mathematics could be said to be the abstract art of symbolizing associations. Mathematics pretends to deal in equalities. But equalities, themselves, do not exist in the MEST universe and can exist only conceptually in any universe. Mathematics is a general method of bringing to the fore associations which might not be perceived readily without their use."
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I want to emphasize that last sentence, "which might not be perceived readily...". Symbolizing might be pointless without that.

For example, say I grew up in an area where it rained often. I had a stable datum: "It rained often". Then I saw that somebody had symbolized it by writing down the amount of rainfall on a daily basis, and then making a graph of that data. And when I was about 9 years old, by looking at that graph, I discovered something new: It rained far more often in the summer than in the winter. Indeed, almost all the rain happened in the summer, almost none in the winter. There was a huge difference but I had never noticed it until I saw the graph, the symbolization of it!

Rain is a MEST phenomena. Graphs of the rain are symbolizations of it. Graphs are "mathematics". Standing in the rain and getting wet is direct experience. But that experience alone didn't cause me to readily perceive the pattern that only the graphs brought to the fore. I never "saw" that pattern until I saw the graphs of it.

Graphs give management a symbolization of how an org is progressing. Comparing graphs of different stats can show associations between them. Management, being very OT, could "just know". But graphs as symbols of production are very useful as "a general method of bringing to the fore associations which might not be perceived readily without their use."
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From LRH: (1954.12.31 Unification Congress lecture 14, "Problems and Games", track 13)

The fact of the matter is that you can work out all mathematics just on the basis of A-R-C. The interrelationship of symbols, factors, figures, relationships, quantities, qualities and so forth can all be worked out of A-R-C.

We just take this triangle and we get the relative affinity of this commodity to that commodity, we get the agreement of this commodity or that commodity, you know, whether or not they have a likeness for each other. And we get in the equation itself a communication. But as we look over mathematics, we can actually evolve these.

I'm very, very sorry that in 1950 when I first ran across that-it was in September, I think, that I had a paper, the notes of a paper on this--demonstrated the conversion of the A-R-C Triangle into logic or mathematics. I didn't finish the paper. There was a lot of wild things going on of one kind or another and I had to give them too much attention and I didn't finish this paper. And when I didn't finish that paper, by the way, I turned my back on organizations, the first and foremost thing that had to be attended to. So right now I would love to have that paper. I could work it all out again, but it was quite lengthy.

And it was quite conclusive that if you don't have these three things present in a mathematical equation, if you haven't measured them to some degree, the reality of the equation is very poor. Whatever is missing in the equation, it would be missing out of the A-R-C Triangle. Reason itself derives from A-R-C.

[compare: "Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality." -- Nikola Tesla

Without experiments, they are leaving out the R. They have no A for experiments, they'd rather just "think" about it, and have no C with the physical universe they are reasoning about, to actually look at it! Thus, their mathematics may be pretty, but without ARC they are going off in wrong directions.]