Number - the brutal facts

I'm going to have to disagree too. Math is both purely theoretical and practical. I personally love the theory and understanding how it works, while dislike doing problems. One of those few people who actually enjoy proofs (of general formulas, not the "proofs" that they call geometry problems which are only particular cases that apply formulas/rules).

I disagree. There is truth to numbers outside of any application.

The identity 5=5 is not that interesting... let's make it a bit more interesting with the equation 3 + 2 = 5.

This equation is true irregardless of the application. I can get this result in any number of ways... any number of applications. I will come up with the same answer no matter what the application. This implies that there is some underlying mathematical truth.

1) I can start at 3 on a number line and move two units in the positive direction.

2) I can buy 2 apples when I already have 3 in my posession.

3) I can send two quanta of energy to an electron cloud.

In all of these very different "applications" the math is the same, the number is the same and the operation is the same.

The number 5 has meaning as its own mathematical entity. It is useful when it is given context in an "application" but this is unnecessary. Everything you can do to a five as part of an application, you can do to it as a "abstract" number.

Likewise the operation "+" has an mathematical meaning outside of any meaning thrust on it by an external context. 3 + 2 works the same in any application you give to it. And - if you don't mess with the operation (i.e. I am talking about the plus operation, not the defined symbol) the result will be the same.

The equals sign is a bit tricky, but only because there are several different mathematical operations that we have discovered, and stupidly we named them all "equals" (and use the same sign).

But each number, and each of the basic mathematical operations stand alone. They are useful in applications, but the applications don't matter.

Mathematics stands alone as intrinsic truth.

Numbers are part of a system devised entirely by humans, for humans.

We must consider that the 'pre-conceived group' of numbers is not a group of numbers, but a group of numerals. I would define these as signs without an application. As soon as we put them into an application then they become numbers, but not before.
Finally, even if we have a pre-conceived group of numbers, it is difficult to see how this can allow us to claim that mathematics 'stands alone as intrinsic truth'. For this claim to be made, numbers must be a priori and discoverable.

Numbers are part of a system devised entirely by humans, for humans. Therefore every aspect of it is entirely within our grasp and understanding. The truth of it does not rely on the system itself. It is faith that sustains it. We believe it to be true, nothing more. So numbers are digits that we have given names and value. They signify something, thus forming a rich language for exploring further.

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Numbers are part of a system devised entirely by humans, for humans.

I think this is clearly untrue.

JJ makes the correct distinction between numerals (the symbols) and numbers (the idea behind the symbols). The numeral 5, the "khamsa" (an arabic numeral that looks like a circle and means five, the Roman numeral "V" are all different numerals. But they refer to the same number that was discovered independently by countless very different cultures.

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We must consider that the 'pre-conceived group' of numbers is not a group of numbers, but a group of numerals. I would define these as signs without an application. As soon as we put them into an application then they become numbers, but not before.
Finally, even if we have a pre-conceived group of numbers, it is difficult to see how this can allow us to claim that mathematics 'stands alone as intrinsic truth'. For this claim to be made, numbers must be a priori and discoverable.

You have this backwards.

Numbers are "a prioriori and discoverable". Intelligent beings invent "numerals" in order to express the numbers they discover, and humans have invented many different types of numerals. Countless cultures have discovered the number 5, and invented different ways to represent it.

I can also give you a number that has no practical purpose in any application Let's take 2^1721443678341-63.

This string of characters is a "numeral" in that it is a way to express a number that is understandable by educated 21st century humans. But this represents a number, an idea that could be expressed by intelligent beings in any set of numerals they have devised.

This number is not part of an application... I believe it is a number bigger than the number of atoms in the Universe. But this does not make it any less of a number.

The fact is, I can do anything with this number that I can do with the number 5. I can add it to another number, I can subtract it, I can check if it is equal to another number... I can even find it's primes (which turns out to be pretty important).

Most importantly there are many ways to express these numbers, regardless of what system of numerals any strange alien culture might invent... the mathematical answers will be the same.

These numbers are the objective ideas expressed by the numerals.

I'm saying that mathmatics exists both in the realm of pure reason as a formal system and also isomorphistically in the "real world" where 5 meters is the 2 dimensional "distance" of 5 times the length of a meterstick, although according to the pure math it's only a number and has no "meaning." For example, the Pythagorean theorem describes a triangle but is not itself a triangle. It is only a specific relation. In physics, particles can be seen as mathematical probabilities that are isomorphistically "real" when they are observed, but are still only probabilities that fall within pure reason described by the Schrodinger equation.

Yes, but the rules do not prove themselves. As the Godel theorem indicates, no formal system can prove itself. We accept certain axioms and the logic we use on those, but this must always be done on the faith that this is correct.

I don't understand what you're trying to say at all.

"Mapping does not signify a relationship" - So it's done completely indiscriminately?

"Mathematics can tell us nothing about 'our' world." - It just helps us create new technology, build things, and understand physics, the study of how our universe operates?

"if mathematics is said to present our world" - You just said that it didn't.

"What could possibly count as proof for a rule?" - A derivation from the axioms.

"Only that we follow it." - So anything we follow is a rule? What?

There must be many would-be theorems in math, which have not been proven or not even formulated yet. Math system tells us new things if those new theorems are found and proven.