Is mathematics built on the first concept: division, or analytics?

[Kharakov]

Are there 2 schools, synthetic and analytic math?

Obviously Peano's axioms are synthetic. But they rest on the foundation of division. Division includes infinities and wholes. You can't have wholes without division from a greater whole (you already have apples divided from the continuum if you count them).



Do you know of any good, easy to understand without years of symbolic training (explained by non-lazy, non divisive, non Trumplike people), explanations of the analytic foundations of math?

 

[AdamWho]

Division isn't part of the operators in the field of real numbers. You only have addition and multiplication (and even multiplication is slightly redundant)

 

[Speakpigeon]

Are there 2 schools, synthetic and analytic math?
Obviously Peano's axioms are synthetic. But they rest on the foundation of division. Division includes infinities and wholes. You can't have wholes without division from a greater whole (you already have apples divided from the continuum if you count them).
Do you know of any good, easy to understand without years of symbolic training (explained by non-lazy, non divisive, non Trumplike people), explanations of the analytic foundations of math?

This one seems good to me: Introduction to Synthetic Mathematics (part 1) by Mike Shulman.

https://golem.ph.utexas.edu/category/2015/02/introduction_to_synthetic_math.html

In general, mathematical theories can be classified as analytic or synthetic. An analytic theory is one that analyzes, or breaks down, its objects of study, revealing them as put together out of simpler things, just as complex molecules are put together out of protons, neutrons, and electrons. For example, analytic geometry analyzes the plane geometry of points, lines, etc. in terms of real numbers: points are ordered pairs of real numbers, lines are sets of points, etc. Mathematically, the basic objects of an analytic theory are defined in terms of those of some other theory.

By contrast, a synthetic theory is one that synthesizes, or puts together, a conception of its basic objects based on their expected relationships and behavior. For example, synthetic geometry is more like the geometry of Euclid: points and lines are essentially undefined terms, given meaning by the axioms that specify what we can do with them (e.g. two points determine a unique line). (Although Euclid himself attempted to define “point” and “line”, modern mathematicians generally consider this a mistake, and regard Euclid’s “definitions” (like “a point is that which has no part”) as fairly meaningless.) Mathematically, a synthetic theory is a formal system governed by rules or axioms. Synthetic mathematics can be regarded as analogous to foundational physics, where a concept like the electromagnetic field is not “put together” out of anything simpler: it just is, and behaves in a certain way.

(...)

There, I've learned something!
EB

 

[steve_bank]

Explain what you mean by synthetic and analytic.. To me analytic referring to math has a specific meaning. Analytic functions. In philosophy analytic has a specific meaning on the basis of knowledge regarding science and arrival at truth.

Are we taking philosophy or math? I had a thread on philosophy on analytic and modern philosophy.

Historically math began with counting followed by subtraction and addition. Multiplication and division being addition and subtraction.

All numerical math boils down to addition and subtraction. In calculus a limit of sums and a limit of differences. Areas abd derivatives.

The basis of math has always been the need to solve problems, necessity the mother of invention.

 

[steve_bank]

The thread I mentioned was analytical vs continental philosophy.

So analytic philosophy is concerned with analysis – analysis of thought, language, logic, knowledge, mind, etc; whereas continental philosophy is concerned with synthesis – synthesis of modernity with history, individuals with society, and speculation with application.