Is Arithmetic Circular?

[Eodnhoj7]

The subtraction of subtraction is addition:

(-1 - 1 = -2) = (-1 + -1 = -2)


Division is further the subtraction of subtraction, as the number of times x may be subtracted until 0 is reached:

(6/3 = 2) = (6 - 3 - 3 = 0)


The addition of addition is the number of times x may be added together:

(3×2 =6) = (2+2+2=6)

[Impenitent]

circular until you start squaring numbers to find the sides of the triangle

-Imp

[Walker]

Back to topic …

[Eodnhoj7]

circular until you start squaring numbers to find the sides of the triangle

-Imp

Squaring is the multiplication of multiplication thus is circular.

[alan1000]

Arguments from within arithmetic itself cannot resolve this question since, if you are right, they will necessarily be circular. The question can only be resolved by careful analysis of the Peano axioms. I would say that that your chances of detecting a circularity there, after the world's best intellects have failed to so do in 150 years, will corrrelate 1:1 with a temperature of 32 Fahrenheit in Beelzebub's fireplace.

[alan1000]

Arguments from within arithmetic itself cannot resolve this question since, if you are right, they will necessarily be circular. The question can only be resolved by careful analysis of the Peano axioms. I would say that that your chances of detecting a circularity there, after the world's best intellects have failed to so do in 150 years, will corrrelate 1:1 with a temperature of 32 Fahrenheit in Beelzebub's fireplace.

[alan1000]

You'll have to explain in more detail how this is all circular. You are perfectly correct to assert that division, for example, is logically reducible to a process of iterative subtraction, and your own example provides a sufficient demonstration. But you'll have to explain in more detail how this represents a "cirularity".

[Eodnhoj7]

You'll have to explain in more detail how this is all circular. You are perfectly correct to assert that division, for example, is logically reducible to a process of iterative subtraction, and your own example provides a sufficient demonstration. But you'll have to explain in more detail how this represents a "cirularity".

The number of times something is subtracted from another as in the repetition of subtraction; this repetition is circular as all repetition is circular.

[Eodnhoj7]

Arguments from within arithmetic itself cannot resolve this question since, if you are right, they will necessarily be circular. The question can only be resolved by careful analysis of the Peano axioms. I would say that that your chances of detecting a circularity there, after the world's best intellects have failed to so do in 150 years, will corrrelate 1:1 with a temperature of 32 Fahrenheit in Beelzebub's fireplace.

That which is resolved from without requires the resolution of that which is without with that of within. If Peano axioms are justified through arithmetic and arithmetic is circular then the Peano axioms share this same nature of circularity.

Either way multiplication is the addition of addition and addition is multiplication as x added to y is the multiplication of 1.

The below are circular:

"For every natural number x, x = x. That is, equality is reflexive.
For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality."


"Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0."

https://en.wikipedia.org/wiki/Peano_axioms

S(0),S(S(0)),S(S(S(0))) is the circularity of S

[wtf]

S(0),S(S(0)),S(S(S(0))) is the circularity of S

Starting at 0, this gives you the sequence 0, 1, 2, 3, 4, 5, 6, ...

I see no circularity. Where do you see circularity?

[Impenitent]

I see circularity at equidistant points around the center

-Imp

[wtf]

I see circularity at equidistant points around the center

What is the center of the sequence 0, 1, 2, 3, 4, 5, ...? Or if that's not what you mean, what do you mean?

[Skepdick]

Starting at 0, this gives you the sequence 0, 1, 2, 3, 4, 5, 6, ...

I see no circularity. Where do you see circularity?

Why have you assigned numerical meaning to a generator? e.g Why are you stratifying s(0) and s(s(0)) as numbers?

Code: Select all

In [1]: def s(n):
   ...:     yield s(s(n))
   ...:
In [2]: s(0) == 1
Out[2]: False
In [3]: s(0) == s(0)
Out[3]: False

[wtf]

\
Why have you assigned numerical meaning to a generator? e.g Why are you stratifying s(0) and s(s(0)) as numbers?

Can you explain why you think I did any such thing?

Are you complaining about something I said? Or something Giuseppe Peano said?

1 = S0, 2 = S2, 3= S2, etc. are definitions, not stratifications. Perhaps you're not aware of that.

[Skepdick]

Can you explain why you think I did any such thing?

Are you complaining about something I said? Or something Giuseppe Peano said?

Are you not agreeing with, and therefore repeating/amplifying Peano's definitions?

Are you not repeating, and therefore saying the same thing as Peano?

1 = S0, 2 = S2, 3= S2, etc. are definitions, not stratifications. Perhaps you're not aware of that.

Why do you constantly keep tripping up over notation/syntax?

In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists.

= is the assignment predicate. The expression "x = y" can be re-written as define(x,y); or assign(x, y).

You are assigning the number 1 to the predicate symbol S(0). You are invoking define(1, S(0)) which stratifies S(0) as 1.
You are assigning the number 2 to the predicate symbol S(1). You are invoking define(2, S(1)) which stratifies S(1) as 2.
You are assigning the number 3 to the predicate symbol S(2). You are invoking define(3, S(2)) which stratifies S(2) as 3.

Peano's definitions ARE stratifications by definition of what stratification is. None of the above is controversial.

What's controversial is the following missing definitions:

define(0, ???)
define(S, x, ???)

Which is why I gave you the generator s() which is not stratified - it has no interpretation in Peano's framework.

Code: Select all

In [1]: def s(n):
   ...:     yield s(s(n))
   ...:
In [2]: s(0) == 1
Out[2]: False
In [3]: s(0) == s(0)
Out[3]: False