## Saturday, June 1, 2013

### ... on Mathematics and Semantics

Continued from a previous one:

... on reality of existence of numbers (and on Pythagoreans and Bruno)
http://assortedretorts.blogspot.fr/2013/04/on-reality-of-existence-of-numbers.html

I) Christopher K
And in any case, I﻿ think this whole discussion is really just a matter of semantics. You want number to mean countables, and from what you said earlier, "quantity" to mean "number", though personally, I think that's quite backwards because when I hear "quantity", I think "the amount of things I have". But really, I think we're just arguing over terminology. Would you have these objections if I wasn't trying to label complex and irrational numbers as numbers?
Hans-Georg Lundahl
As soon﻿ as you agree that "irrational numbers" are really things like size to size ratios rather than answers to "how many", and as soon as you agree that "complex numbers" or for that matter already "negative numbers" and "zero" (as one number) are the sci fi of maths, that is more important than the terminology you chose.
Christopher K
You say "sci fi" as if those don't have very real, very very useful applications in real life. And the whole definition of irrational number is that it isn't a real ratio of anything to anything. Asking me to agree that irrational numbers are size to size ratios, where both sizes are real numbers is﻿ like asking me to agree that orange is purple.
Hans-Georg Lundahl
Having a real, useful, application or more than one in real life does not preclude sth being scifi or fantasy. Denethor and Saruman, Weston and Devine have real useful applications in real lifen nevertheless Lord﻿ of the Rings remains fantasy and Out of the Silent Planet remains scifi.

"Asking me﻿ to agree that irrational numbers are size to size ratios, where both sizes are real numbers﻿ is like asking me to agree that orange is purple."

The phrase "where both sizes are real numbers﻿" is your interpolation.

They are size to size ratios or relations as can be proven in any case:

1) pi = circumference:diameter (any perfect circle)

2) sqrt(2) = diagonal:side (any perfect square)

3) sine x = opposite side (angle x) : hypothenuse (any rectangular triangle with angle x)
II) Christopher K
what's the difference between "minus: 1" and -1? And yes, the solution to x² = -1 is both i and -i, but it's perfectly valid for those sorts of equations to have two﻿ solutions. x² - 3x + 2 = 0 clearly has both x = 1 and x = 2 as valid solutions.
Hans-Georg Lundahl
Ah, I thought that the feeble for "negative roots" was sth like related to algebra due to equations!

The difference between operation subtraction with subtrahend 1 and number negative one is how you view subtraction. View it the right way and only the first works. Unless you deal not with numbers as such, what you are primarily counting, but in relative numbers, i e what operations you do, then sth can be simplified algebraically.﻿
Christopher K
I think you're missing the point of my question there. You're saying that -1 isn't a number and you define it to be an operation (ie, the inverse operation, which is defined by x + (-x) = 0, where -x is the inverse of x) on 1 and﻿ that you define that as "minus: 1" and I'm asking "how is that different to just saying -1?" They're functionally identical.
Hans-Georg Lundahl
It is among other things your concept of "functional identity" that I am precisely attacking.

There is naturally a science of "how many", called arithmetic, and of "how big", called geometry. They are very empiric disciplines even.

Algebra deals with functionnaly identical ways of stating operations, which is of course useful for﻿ complex ones, as in simplifying them, but which is quite as obviously not the rational basis for arithmetic or geometry.
Christopher K
But algebra and geometry have overlaps. We can write geometric formulae as algebraic ones and vice versa. They're really just two sides of the same coin. Well, to a point. Algebra can describe things that can't be described geometrically.﻿
Hans-Georg Lundahl
I do not﻿ deny that algebra and geometry have overlaps.

I am saying that geometry is a science in its own right, with an empiric basis of knowledge, and that algebra is an applied art.
III) Christopher K
i is usually defined to be the positive root for -1. And why isn't it valid if all the maths works out? If﻿ you don't get contradictions, there's no reason to say "Noooo it isn't valid!", because you're just throwing away a valid answer because you don't like it, and that's terrible science, or terrible anything in general.
Hans-Georg Lundahl
A square root is the obverse of a square number or the side of a﻿ square, and neither numbers nor sizes come in negatives.

Negatives don't come in until you ask "how many / much less than previous" and is obviously not a special kind of number but a relation of lessness measured by either number or numericalised magnitudes.

Confusing the two is terrible science and terrible logic and terrible anything (intellectual) in general.
Christopher K
A square root is something raised to the power of a half. Exponents are valid arithmetic operations, and you don't need to﻿ have square roots as the inverse function of the square function to use them.

And I'm not sure what the difference is, apart from excluding negative numbers from the set of numbers means that subtraction is no longer a valid function because you can now perform subtraction and get something that isn't a number.
Hans-Georg Lundahl
Whole number exponents certainly are valid arithmetic operations in themselves:

"x to exponent 1/2 = y" MEANS "x to 1 = y to 2"

Subtraction is a form of division. You can divide a whole several ways, but if﻿ one is dividing it into two, three or any other number of equal parts giving possibly a remainder less than the number of parts, the other is dividing it into two parts of which one has a determined size.

Which means that "1-2= -1" is not a valid subtraction per se.
Christopher K
Subtraction using only natural numbers doesn't work﻿ in general, because subtraction is simply the inverse of addition. In other words, x - y is the same as x + (-y), and (-y) isn't a part of ℕ.
Hans-Georg Lundahl
"x - y is the same as x + (-y),"

On the﻿ contrary, it is "x + (-y)," which is a backward way of writing "x - y"

Naturally speaking subtraction as much as division means separating parts of a whole, and multiplication as well as addition means taking separate items and making them part of a whole.
Christopher K
Mathematically though,﻿ subtraction is defined as the inverse operation to addition.
Hans-Georg Lundahl
IV) Hans-Georg Lundahl
calculating pi is sth else than defining it.

pi is not a ratio of two﻿ numbers ever, though certain ratios come close (314:100 for instance) but of magnitudes
Christopher K
The way you generate it doesn't matter if you still get the exact same result. 2/4 and 1/2 are both exactly the same number, just generated in a﻿ different way.
Hans-Georg Lundahl
Point is you cannot generate﻿ an exact value for pi.

If 314:100 (=31400:10000) is an inexact nether approximation, you can generate an upper such by 31416:10000 or by 22:7 or many other ways.

Pi itself is never properly speaking generated arithmetically. Nor is sqrt of 2. And so on for other "purely geometric ratios" ak by misnomer "irrational numbers".
Christopher K
You still miss my point. I'm not saying "generate pi by keeping on adding more decimals places", I'm saying "generate pi by the limit of this﻿ infinite sum".
Hans-Georg Lundahl
That is not a generation, that is an infinitely varied approximation depending on how many times you carry out whatever the sum is that is potentially infinite. And yes, I am aware there is a so called infinite sum (which﻿ never becomes actually infinite) of incomplete operation, related to those that generate e. I recall it was related to another one as well, but have forgotten which one, was it phi?
Christopher K
Yeah, you can use Taylor series's to produce an﻿ infinite sum with the limit of pretty much any irrational number you want.
Hans-Georg Lundahl
But you can only execute a Taylor series in a finite number of steps,﻿ meaning that the "infinite sum" of it never actually exists.
Christopher K
Only﻿ if you happen to not like infinite sums. Which is possibly an issue, because they can do funny things, but they happen to work pretty well and they make good approximations.
Hans-Georg Lundahl
"infinity" in mathematics is never real infinity

that is my FIRST issue with modern mathematic terminology, and it is an old issue between theology/philosophy and mathematicians:﻿ St Thomas, when talking about the infinity of God excludes the mathematic concept (and argues only God is infinite) insofar as when geometers say "take an infinite line" they only draw out the line as far as needed, never actually to infinity

same of course with "infinite sums"
Christopher K
Well of course, we can only approximate infinite lines to however far we can draw. Clearly, nothing in the universe is infinite. That doesn't mean we can't deal with things like infinite sums, by extrapolation or other methods.﻿
Hans-Georg Lundahl
Dealing with an infinite sum by extrapolation or other methods involves not dealing with﻿ it directly, because - in this case - it does not exist as such.

I am not saying it has no useful applications in real life, I am just saying it is sth other than real arithmetic, real music, real geometry.

Appendix, on logarithms:

I said that "irrational numbers are size to size ratios or relations", but does this quite apply to logarithms?

I would tend to say, logarithms are a very special case, they are a ratio of number to ratio, of addition to multiplication (and of multiplication to exponents).

Like size to size ratios, they are however geometric in nature. They are the geometric basis of a slide rule.

Like any other purely geometric thing, it cannot be exactly parallelled in arithmetic, but it can be simulated in arithmetic.

If you want the base ten-logarithm for two, you are feigning to ask "ten to the power of how much equals two?" which does not make sense, since the answer cannot be a whole number and since a power must be a whole number. But that in turn can be translated to the somewhat inexact equality "ten to the power of how many equals two to the power of how many?" since "xa/b=y" means xa=yb. One obvious, rough, inadequate answer is 3/10. 103 = 1000, 210 = 1024, "1000=1024". If you know that the logarithm of two on base ten is given in tables with first three decimals as 0.301 you will see that this is not far wrong.

10approx. 3/10 = 2
103/10 = approx. 2
210/3 = approx. 10

And reason for the "approx" in each of these is that a logarithm like a size to size ratio is a non-numeric ratio.

At least it is non-numeric on the one side. It would not be useful unless it also had a numeric side.

lg 2 + lg 2 = lg 4
lg 2 + lg 3 = lg 6
lg 3 + lg 3 = lg 9
lg 3 + lg 4 = lg 12

And so on. And the letters lg, which are abbreviation for logarithm, could in English equally be abbreviation for length./HGL

Update on above appendix:

If there had been no logarithms say between lg2 and lg3, of course one side of the logarithmic relation would really have been numbers as such. However, there is a logarithm for the so called "irrational numbers" - like sqrt of two (half lg2) or sqrt 3 (half lg3) and of π and of φ and therefore the "number" side of the logarithmic relation must itself be a magnitude relation. Logarithm "of three" is really logarithm of 3:1. Though that would be harder to write out each time you use it and though the results of what actually amounts to lg(3:1)+lg(2:1)=lg(6:1) work well for application on the arithmetic numbers as well./HGL

Update:

I) Hans-Georg Lundahl
Dealing with an infinite sum by extrapolation or other methods involves not dealing with it directly, because - in this case - it does not exist as such.

I am not saying it has no useful applications in real life, I am just saying it is﻿ sth other than real arithmetic, real music, real geometry.
Christopher K
If it gives real results, then what's the problem? Look at, say mathematical demonstrations of solving Zeno's Paradox, where you use a sum to infinity. Sure, sometimes we need to invent funny ways to deal with these things, but if we can show these methods to actually work, then there shouldn't be a problem.
Hans-Georg Lundahl
A method working does not mean it is without conceptual flaw.

In Zeno's paradox about Achilles and the Turtle, you need a time by distance grid with two lines, Achilles' line and the turtle's line, and you can see there is a crossing and also that Zeno was just slowing down the conceptual coverage of the space before the coverage, and neither of the lines needs to be infinite.
II) Hans-Georg Lundahl
"x - y is the same as x + (-y),"

On the contrary, it is "x + (-y)" which is a backward way of writing "x - y"

[it is "x + (-y)" that is the same as "x - y"]

Naturally speaking subtraction as much as division means separating parts of a whole, and multiplication as well as addition means taking separate items and making them part of a whole.
Christopher K
Mathematically though, subtraction is defined as the inverse operation to addition
Hans-Georg Lundahl
Christopher K
Which, I think, is a lot of where our disagreement comes from. You seem to be under the idea that the Greeks and older maths was the absolute truth and couldn't ever be replaced. Personally, I think this world-view is horribly boring. The world is an ever-changing place and we're learning new stuff all the time and we need to update our world-view to adapt to this. Yes, we've changed the definitions of some things, but this isn't a bad thing. It helps us grow as people.
Hans-Georg Lundahl
It does not.
III) Hans-Georg Lundahl
Having a real, useful, application or more than one in real life does not preclude sth being scifi or fantasy. Denethor and Saruman, Weston and Devine have real useful applications in real life nevertheless Lord of the Rings remains fantasy and Out of the Silent Planet remains scifi.
Christopher K
Fantasy isn't sci-fi. When you call something sci-fi, you imply that it's something that might maybe exist in the future and you try to justify with reasons why it could work or maybe just technobabble. We might develop a Warp Drive some day. We won't develop the Five Wizards of Middle-Earth ever. By saying that zero and negatives are sci-fi, you're saying that they're not﻿ anything that exists in reality. The computer you're using disagrees.
Hans-Georg Lundahl
fantasy and sci fi imply you make thought experiments about what could be possible, even if clearly it is not real

wizards in LotR come very much closer to being really possible than "infinite improbability generator" of HHGG

zero and negative how-manys* do not exist in reality, and a computer is not in a position either to agree or to disagree

they have applications, so does - on your saying - i, but so have Saruman and Denethor: they apply to certain attitudes about tradition and politics.
tekhiun
complex numbers don't exist, the concept of them exist same thing with any number, some you can map to the real wolrd, others you cant.
Hans-Georg Lundahl
numbers like 3 exist

size to size ratios like pi exist

i does not exist
* footnote
If zero and negatives do not exist as "how-manys", they do exist as "how-many-more-or-less-than" in relation to "how-manys", and they do exist as "how-much-more-or-less-than" in relation to "how-muches".

Updating again:

Hans-Georg Lundahl
fantasy and sci fi imply you make thought experiments about what could be possible, even if clearly it is not real

wizards in LotR come very much closer to being really possible﻿ than "infinite improbability generator" of HHGG

zero and negative how-manys do not exist in reality, and a computer is not in a position either to agree or to disagree

they have applications, so does - on your saying - i, but so have Saruman and Denethor: they apply to certain attitudes about tradition and politics.
Christopher K
The problem is that negative numbers do exist in reality, or do the electrons in your computer﻿ not carry a negative charge that aren't floating around freely because they're being attracted to the positively charged protons in the nuclei of the atoms of the metals used?
Hans-Georg Lundahl
positive and negative in electric charges or for that matter credit and debt (which also cancel if﻿ brought in contact) are not really

"electric charge times minus one"

but

"negative electric charge times one"

meaning that this is no example of negative numbers existing
Christopher K
And while you're technically correct about credit and debt, pretending﻿ that negative numbers don't exist makes the maths more difficult. Allowing negatives makes everything flow much more easily.
Hans-Georg Lundahl
I do not think so.

In the example of credit and debt, do the sums of each account side, then subtract lesser sum from greater sum is a very﻿ good and useful and practical and easy way of dealing with it
Christopher K
Well sure, if you ignore all the evidence that says that electrons have the exact same charge as a proton, but multiplied by -1, Coulomb's law which relies on that fact and requires negative numbers, not to mention anti-particles, such as positrons which are identical to electrons except having the opposite charge (ie, multiplied by -1), then yes, there is no examples of﻿ negative numbers. Well, apart from all the others.
Hans-Georg Lundahl
Oppositeness of charge is a physical concept rather than a mathematic one.

It can of course﻿ be described as "multiply by -1", but if there is no such number that is clearly not what it really means.

Coulomb's law could certainly be restated using no negative numbers. In a more correct fashion.

Just as one can restate

(a-b)^2

=a^2-2ab+b^2

in a more correct but less useful way as:

= a^2 - ab - b(a-b).

Exactitude of concept is as useful for its purposes as ease of calculation for calculating
Hans-Georg Lundahl
numbers like 3 exist

size to size ratios like﻿ pi exist

i does not exist
Christopher K
i is as real as any other number. Just because it doesn't exist on the real plane doesn't make it any less real. Wikipedia has a nice section on real-life applications of complex numbers, which it might be a good idea to look at.

And again. You can say "pi is just a ratio" but it needs to be a ratio of at least one irrational number to another﻿ number. Look at the ratio diameter:circumference. One of those numbers is an irrational number.
Hans-Georg Lundahl
I never﻿ ever said of either pi or sqrt of two that it was a ratio of two natural numbers, I said it was a ratio of not numbers, but lengths - have you forgotten that?

Now, number has a discrete range of values, meaning there is nothing between 1 and 2, nothing between 3 and 4.

In lengths or any other quantities with continuous ranges this is not so. In any of these number is only assigned through ratio.

"3 cm" = "3:1 in relation to the cm"

"4 cm" = "4:1 in relation to the cm"*(continued again below)*

and of course:

"pi﻿ cm" = "circumference:diameter in relation to the cm"

"sqrt of two cm" = "diagonal:side (of perfect sqr) in relation to the cm"
Christopher K
And you don't see how that makes doing maths with either of those numbers, both of which have a number of applications outside geometry, more difficult﻿ and tedious?
Hans-Georg Lundahl
Writing it out as a ratio would certainly make it more tedious and difficult. REGARDING IT AS ESSENTIALLY a ratio does not.

And admitting that a certain part of say physics uses the same constant as﻿ was known from geometry is not in any way demeaning to physics, indeed, it poses physicians the question "why is it exactly the same"? Can it be because movement is in geometry, for instance?
*(continued from further up)* Christopher K
And we represent those lengths by numbers, exactly as you're﻿ doing there.
Hans-Georg Lundahl
Representing a length by a number does not make it a number.

"3 cm" does not MEAN "3 pieces, one centimetre each, glued to each other", what it means is "3:1 in relation to length known as cm".

It is very practical﻿ to forget what a certain set up of numerals mean when you are counting with them, but generalising the forgetfulness to total amnesia of real mathematical semantics does not add to that kind of usefulness.
Christopher K
Which, I think, is a lot of where our disagreement comes from. You seem to be under the idea that the Greeks and older maths was the absolute truth and couldn't ever be replaced. Personally, I think this world-view is horribly boring. The world is an ever-changing place and we're﻿ learning new stuff all the time and we need to update our world-view to adapt to this. Yes, we've changed the definitions of some things, but this isn't a bad thing. It helps us grow as people.
Hans-Georg Lundahl
It does not.﻿
Christopher K
A﻿ very well thought out reply there. Mind elaborating on that?
Hans-Georg Lundahl
Growth does not mean leaving qualities behind but adding to them.

If number has a valid definition, changing it will not make us grow.

However, discovering that despite examples like 3-4-5 triangle, lengths are not really numbers and that there is beside the discrete range of values known as number also a continuous range of values in other types of quantity (bigness, weight, etc) is real growth.

This step was taken between Hipparcus﻿ and Aristotle/Euclid. Not by Cantor, Gauss etc.
Christopher K
This is why I still get the feeling we're arguing semantics here. You like the word "number" to mean "natural number" and I think it should be any form of quantity. But I don't see it as leaving qualities of numbers behind. Everything you can do with natural numbers,﻿ you can still do with complex numbers. What's being left? We're just adding to it so that every mathematical operation will work and give a valid answer (with the exception of division by 0) and we don't get bogged down in semantics
Hans-Georg Lundahl
"You like the word 'number'﻿ to mean 'natural number' and I think it should be any form of quantity."

Why not use "quantity" as the general concept then?

"Everything you can do with natural numbers, you can still do with complex numbers. What's being left?"

The fact of being a real multiple, conceived by one being added to one.
Hans-Georg Lundahl
Dealing with an infinite sum by extrapolation or other methods involves not dealing with it directly, because - in this case - it does not exist as such.

I am not saying it has no useful applications in real life, I am just saying it is﻿ sth other than real arithmetic, real music, real geometry.
Christopher K
If it gives real results, then what's the problem? Look at, say mathematical demonstrations of solving Zeno's Paradox, where you use a﻿ sum to infinity. Sure, sometimes we need to invent funny ways to deal with these things, but if we can show these methods to actually work, then there shouldn't be a problem.
Hans-Georg Lundahl
A method working does not mean it is without conceptual flaw.

In Zeno's paradox about Achilles and the Turtle, you need a time by distance grid with two lines, Achilles' line and the turtle's line, and you can see there is a crossing and also that﻿ Zeno was just slowing down the conceptual coverage of the space before the coverage, and neither of the lines needs to be infinite.
Christopher K
I was more thinking of the other paradox, of the arrow going to a target because I find it's an easier one to visualise. And yes, I do realise the flaw in his thinking but it doesn't make it less of﻿ an interesting paradox.
Hans-Georg Lundahl
I had only heard Achilles and the turtle﻿ in any elaboration.

Mind elucidating?
Christopher K
It's more-or-less the same paradox, but instead, you shoot an arrow at a target. After some time, it halves the distance to the target, then after some more time, it halves it again, and so on,﻿ never reaching the target. I think Numberphile's done a video on both versions of the paradox.
Hans-Georg Lundahl
That version can be solved with a diagram of distance and time, and the paradox can be shown﻿ to be a mental retarding before delays artificially put up.

Update with Christopher again:

I) Christopher K
And while you're technically correct about credit and debt, pretending that negative numbers don't exist makes the maths more difficult.﻿ Allowing negatives makes everything flow much more easily.
Hans-Georg Lundahl
I do not think so.

In the example of credit and﻿ debt, do the sums of each account side, then subtract lesser sum from greater sum is a very good and useful and practical and easy way of dealing with it.
Christopher K
That's still longer than one subtraction.﻿
Hans-Georg Lundahl
You still have to add up before you﻿ subtract one from other, exactly same length, wizeacre.
Christopher K
Depends on what exactly you're trying to do. Say, I've got \$6 in my bank account and I buy something for \$10. I've now got﻿ -\$4 in my bank account (Plus some ridiculous bank fee for going into overdraft). It's much more simple than having to deal with two separate accounts.
Hans-Georg Lundahl
Whether it is that for the computer system or not, it is surely by doing:

- \$10
+ \$6
____
= -\$4

same as would have﻿ been done with opposite signs, a simple subtraction of lesser amount from greater amount only that the subtraction meaning cancelling of opposite "account forces" you end up with a result carrying opposite "account force". Precisely as with two accounts measuring their forces.
II) Hans-Georg Lundahl
Representing a length by a number does not make it a number.

"3 cm" does not MEAN "3 pieces, one centimetre each, glued to each other", what it means is "3:1 in relation﻿ to length known as cm".

It is very practical to forget what a certain set up of numerals mean when you are counting with them, but generalising the forgetfulness to total amnesia of real mathematical semantics does not add to that kind of usefulness.
Christopher K
I never said the length is a number, I said﻿ length is represented by a number and then we perform maths on those numbers.
Hans-Georg Lundahl
There are geometric facts that are salient enough even without the numbers﻿ attached to it.
Christopher K
I know that.﻿ Numbers just make various operations easier, and applicable to more situations.
Hans-Georg Lundahl
Some of them yes.

That is why assigning numbers to length (via arbitrary length unit) is done at﻿ all.

But if I have an A7 paper held horizontally to text in, I do not measure either height units or parts of height, I make letters a certain height in a certain height of the height, without bothering to assigne number to the height.
III) Hans-Georg Lundahl
You are assuming today's professional mathematicians are all that count.

Euler did not use number for any mathematic quantity. He used the Latin word quantitas, which means quantity.

You cannot translate quantity with number, since﻿ Latin for number is numerus. And while we are at Latin, you cannot bypass the distinction Boethius made about quantitas subdividing into numerus, which is the subject of arithmetic and magnitudo which is the subject of geometry.
Christopher K
You're assuming that classical mathematicians are all that count. Yes, they did use those words back then, but since then, people have decided for whatever reasons that the more modern descriptions are more accurate, or at the very least, they just used the word "number" until it stuck. Meaning of words change over centuries. Awful no longer means﻿ "filled with awe".
Hans-Georg Lundahl
Awful probably never meant "filled with awe" but "filled with what is awe inspiring" [or filling with awe] ,﻿ and that awful still means.

The modern words are not more accurate, and they take more syllables (unlike in German, where Zahl for quantity or number is short and Anzahl for number precise meaning is also short).

Number - Quantity (1 less in modern)

Natural number - number (3 more in modern)

Real number - magnitude (equal).

Modern usage makes it much more awkward to speak of "natural" number.
IV) Christopher K
This is why I still get the feeling we're arguing semantics here. You like the word "number" to mean "natural number" and I think it should be any form of quantity. But I don't see it as leaving qualities of numbers behind. Everything you can do with natural numbers, you can still do with complex numbers. What's being left? We're just adding to it so that every mathematical operation will work and give a valid answer (with the﻿ exception of division by 0) and we don't get bogged down in semantics
Hans-Georg Lundahl
"You like the word 'number' to mean 'natural number' and I think it should be any form of quantity."

Why not use "quantity" as the general concept then?

"Everything you can do with﻿ natural numbers, you can still do with complex numbers. What's being left?"

The fact of being a real multiple, conceived by one being added to one.
Christopher K
For one thing, because there's no reason to use a different word when number is working just fine and it's what every mathematician uses. [answered above in III]

"The fact of being a real multiple, conceived by one being added to one."

I'm not﻿ entirely sure what you mean by that.
Hans-Georg Lundahl
One is the metaphysic ground for both arithmetic and geometry.

In arithmetic its integrity is assumed and you add another to get two, another yet to get three and so﻿ on. Those are the definitions of numbers.
Christopher K
Ahh, I get you now. But again, you don't lose that quantity by expanding number to include irrationals and complex numbers. Every arithmetic operation can be brought back to the successor operation﻿ (ie, increasing by 1). Addition is just repeated successors. Subtraction is just the inverse of addition. Multiplication is repeated addition. Exponentiation is repeated multiplication, and each of those has a well-defined inverse.
Hans-Georg Lundahl
If this "inverse" of successor operation is limited by "one by one" it stays within arithmetic.

Whenever it includes dividing wholes into parts, we are already dealing with geometry or at least music.

And reversing successor operation of number building will not take one down to zero nor into negatives. Each item added to one can be﻿ taken away leaving one.

When a net result can be zero or negative, we deal with cancelling forces rather than with subtraction. One arbitrarily dubbed minus.