I would like to follow up to my post on how the empirical world relates to mathematics, which was very conventional. This post on how mathematics relates to the empirical world is not conventional.

How does the mathematics relate to the empirical world?

Here I will build a specific relationship. It should be very intuitive.

We will let N be the cardinality of the collection of all apples.

Now we will define f such that f maps the Counting numbers from 1 to N to the empirical world as follows:

f(1) is defined to be 1’ apple *

f(2) is defined to be 1’ apple and 1’ more apple or 2’ apples

f(3) is defined to be 2’ apples and 1’ more apple or 3’ apples

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f(N) is defined to be N’ less 1’ apple (defined by the preceding line) and one more apple.

This is a standard recursive definition where each line is defined by the line before it; and the first line is defined as f(1) is 1’ apple.

A note about the terminology:

In previous posts I was using the terms one, two, three… to refer to the adjectives describing how we experience the empirical world. Now I have switched to the terms 1’, 2’, and 3’ to describe the same terms. This was simply because I did not have a good way to express the nth adjective of an empirical object.

1’, 2’, 3’ …N’ should in no way be considered numbers!

It should be clear that each number, 1, 2, …N is uniquely associated to 1’ apple, 2’ apples, … N’ apples.

We can define an association of objects in the empirical world, which we will symbolize as follows:

Without loss of generality we can assume that n < m. If not we just change the names. In the case that they are equal we know that n + n is less than N. But if n < m, then n + m < m + m < N.

We can go about showing that this association has all the properties of our standard addition.

This is the standard definition of addition in the Peano axioms.

Additionally, again assuming that m + m < N and n + n < N we have:

And assuming that n + m + l (the letter l not the number 1) < N we have:

I am pretty sure that the only people that would think that this is odd are philosophers!

The most striking way that I can put this is:

The rules of our mentality constructed abstracted mathematics (as long as the results of our operations are less than the cardinality of our empirical objects) [size=150]govern[/size] how this part of our constrained empirical world works!!!

For example, if we intellectually determine that 3 + 7 = 10, then three apples plus seven apples will be ten apples!!!

One of the straightforward questions is: are the abstracted Counting numbers finite? After all there are only a finite number of apples!

The answer, it seems to me, turns out to be cultural. (Oh yuck!!!)

Historically the empirical world was considered infinite, and if we were considering all empirical objects (the stars, the number of colors and others) the answer is clearly no - the Counting numbers are infinite.

On the other hand, our current models (after Quantum mechanics), at least in Western culture, of the empirical world indicate that it is finite. Perhaps the abstraction, if n is a number then, n + 1 is a number was premature?

Regardless of whether or not we can abstract the statement “if n is a number, then n + 1 is a number”, the rule has been codified.

A remaining question might be:

If the abstracted numbers can determine how the empirical world behaves, then can the axiomatic numbers do the same thing?

The answer is yes, in the same restricted manner as the abstracted numbers.

It turns out that the axiomatic numbers behave the same way as the abstracted numbers. Technically it requires that the axiomatic numbers to be a homomorphism** of the abstracted numbers. I sketched out this type of homomorphism in a play I wrote located at:

viewtopic.php?f=10&t=173698

The only difference is that I compared the Formal (set theoretic) axiomatic system to the Peano system. The reader can simply substitute A (representing abstracted mathematics for P to see how the Formal axiomatic system is homomorphic to the abstracted system.

- n’ apples represents all groups of apples whose cardinality is n. This is because we ultimately need to define an operation on apples that is good for any combination of apples with a given cardinality.

More rigorously n’ apples is defined as an equivalency class determined by the relationship R such that if a and b are subsets of the groups of apples then a is related to b by the relationship R if and only if the cardinality of a = cardinality of b.

To verify that R is in fact an equivalency relation we need to show:

aRa for all a. The cardinality of a subgroup of apples is uniquely defined

If aRb then bRa. If the cardinality of two subgroups of apples is equal then the order of the relationship R is not important. (This is called abelian or communitive in higher algebra).

If aRb and bRc, then aRc ‘ the relationships are transitive in nature.

Basically R is the same as equals except that the objects are being compared with regard to a specific property.

**Homomorphism Definition:

f(a* +F b*) = f(a*) +P f(b*) where f is a uniquely defined function on every element of its’ domain and range. Here the +F is addition on the formal numbers and +P is addition on the Peano numbers. It should be obvious that the domain and range can be completely different sets. Or in philosophic terms the domain of f can be ontologically different than its’ range.

Apple addition definition: