Mathematics is 75% Invented, 25% Discovered

flame2

I just want to get this off my chest,

We as a human species tend to do things without question while under authoratative governing principles, similar to math but I'll get to that point in a little bit. A lot of us get up and work from 9-5 because it is universally accepted to be a part of society without question of it. We read words from the dictionary and almost never question the origins of such words. We engage in religious activities without second thought of whether this is actually true or not.

The only real question is,

Why don't most people want to question these things themselves or try to understand why they do these things without question? I hope it isn't out of fear of possibly thinking for yourselves and drawing your own logical conclusions. And the same can be said for mathematics to a varying extent.

Most people on any topic automatically want to think they are right because they are afraid of possibly being wrong.
I used to be that kind of a person but I am not anymore and it has allowed me to keep an open mind and question everything in the world in which we reside in. Of course it has also allowed me to be more accepting of people's differing opinions if you are thinking. That's just a way of life.

But this all leads up to my main topic which is about math,

Math is done mostly without the question of why we accepted such things to be true if there is no distinct eveidence pointing in "that" direction, hence "axiomatically true." We accept that (a+b=c) is true. No questions or proof needed. But the problem isn't what we universally accept as true today, but was this "obviously" true 5,000 years ago before the invention of most languages? Or the invention of... numbers, notations and letters so to speak? I will discuss postulation/axioms much later again.

Before I work on that point,

Let me just state the obvious here. If you only believe math was discovered, you better have a good reason for where the numbers represeting the quantities of things in nature came from if it wasn't from humans. Not to mention you would probably get the nobel prize if you could prove otherwise. I'd be surprised to see clouds raining the number "5" when there are "5" rain clouds hovering above or seeing the number "10" growing from an apple tree once "10" apples grew from that apple tree. Yes, we discovered quantities of things but certainly not numbers.

So which leads to my next point,

Abstract concepts. All numbers and even mathematical letters, "taken from the Greek language initially," such as "Sigma" are abstract concepts. Languages aside, You cannot hear, taste, smell numbers. Plato is a great example of this very problem. AKA Plato's heaven theory. Counter argument against the argument against Plato's view by me: How can you ever be aware of numbers/notations in the universe if numbers/notations never existed in the universe until humans invented them to mean what they wanted those numbers to mean? It's sounds more like a placebo effect than anything else and it leads back straight to my very first points about the river and the apple tree. Let's pretend it's the year 1000 CE and sigma existed then. I wanted to incorporate the summation letter with another letter. Who's to say that I'm wrong for representing "Sigma" as the old scot's letter ȝ which is "yogh?"

Along with that, let's make it even more interesting,

Remember, it's still 1,000 A.D and sigma existed already.
"Assume ȝ is our sum of a series. Sum of x, from 0 to 2." Does the method actually change? No. All I did was change the notation to mean the same exact thing as sigma axiomatically. Hush, hush no questions, remember? It has to be this way now.

My next point,

Since numbers are invented and not discovered and you can't change something that is discovered for what it is regardless of before or after scenarios, you discovered it in such a way, period. Let's pretend It's 7,000 BCE and I discovered two volcanoes. My language is very limited also. But something about two volcanoes amuses me and I wanted to remember it in some form or another. In front of me I have a stick and some mud. In front of the volcanoes I draw two "Us." The "Us" are connected. It is coincidently shaped very similarly to the "3" in the future english numerals. I then write it in my primitive language as "yooyoo" because those are the words that come to this primitive mind at this time when describing these quantities. This represents this primitive human's understanding of the quantities of something and so now it must be more than a single volcano so he decides to draw in the mud for memory the single volcano... etc. Is he wrong? Is there an authoratative presence telling him he's wrong in the realm of abstract thinking? Not at all. Now eventually his civilization will build up and start representing quantities this way.

Next point,

11+5 = 16... right!?!? Well, umm... no. Why? You might ask? Because I didn't give any context for the meaning of these numbers yet. Aha. It may be universally accepted due to the way our mathematical system is logically built up today to say that
11 + 5 = 16 because axiomatically, addition is "adding" and substracting is "minusing" etc. but does this always work with any given context? I beg to differ. 11P.M + 5 hours = 4A.M. Aha. In this case the math works like this;
11+5 = 4... A.M. Because we are referring to the clock now. Something they do not teach you in school but it is not wrong given the context. Remember that.

This works the same way Einstein disagrees with 1+1=2 in some cases. What's the implication? Well Einstein thought that just because numbers work without any physical aspects applied to them so 1+1=2 hence, abstract concepts since we cannot either touch, smell, hear numbers. Instead, When looking at physical/tangible objects, we see "2" of them right? Again, depends on the context. Einstein agrees with me. He says, sure, 2 objects but unless they have equal mass, density or volume with the same exact dimensions then it can only really equal a pair of 2, yes. So 1 brick + 1 brick with a slightly smaller dimension in the corner of the brick with equal mass, density and slightly less volume might be something like 1+0.944 =1.944. Isn't that funny. Since one of the bricks doesn't have the completely same physical qualities, then it does not equal to 2 of the same pair physically. These axioms do not work in the physical world because numbers aren't discovered and have no physical aspects to them. This is why we have physics which einstein was good at.

Which leads to my next point,

In physics there are no mathematical axioms. Why? Because in physics we explain the physical world with physical applications. Math by itself does not have an inherent trait with the real world. Once again, physical applications brings math to the scientific world view. We cannot assume anything exists without evidence. Hence, the opposite of an axiom. And because physics is binded to the scientific method. We have proved a lot of things in time thus far with physics... I'm sure you know of course. Yes, using the tool of math to explain physical phenomena is possible because it so happens to be a great tool when used in conjuction with physics. Physics must be experimentally tested and observed to be a fact. Not math. You cannot make up axioms in physics to promote a hypothesis that you cannot possibly experiment with and observe and or not understand in the first place as a starting point for a hypothesis for that matter. You cannot accept something for what it is without the burden of proof in physics. It goes against the scientific method and is anti-scientific. Both subjects have logic instilled into them but physics seems to be more logically superior considering my whole argument and I can't really at all find fault with how math is dealt with in physics if I was asked. Gödel in his incompleteness theorems acknowledged the limitations math has by itself and he wasn't a physicist.

Another point,

Since numbers by themselves sit in their own world within the human imagination, you can always make up your own rules as you go with postulations, right? Go ahead. That's pure mathematics research right there. Although applied math still falls into this trap of the numbers and notations having no meaning in the first place as I prove here by proving that starting at 0 on a ruler makes no difference compared to 2. The unsettling part is that this is supposed to be applied math. Where's the proof? Okay. Go get a ruler and find an object that is, oh, about 6 by 6 inches long. Let's pretend you want to build the same object again. Apply your ruler to it but instead of starting at 0' start at 2' instead. But start the ruler where you would start the ruler at 0' but at the 2' mark instead on the object so you get 8' that way. You obviously get 6' in length at 0' "the right way." Notice that it doesn't matter whether you started at 0' or 2' or 4 in? That's because the object never changes and you could still set up a schematic or blueprint that way if rulers were made to start at 2 in. The object will still be built the same way but the only things that change are the numbers of course. Those numbers once again prove to have no meaning in this application even with applied math. Now you just have to convince the world to start making rulers at 2' instead of
0 inches... and then 14' to end the ruler in the case of 2' of course. And... if you really cared about the smaller increments of measurements you can erase the first 30 batch of increments, start at 2 1/16 inches and build the ruler all the way up again and making sure you added that extra 30 increments to the end of the ruler which should get you to that new 14' mark. Voila.

Last point,

Math does indeed lead to discoveries due to the invention of numbers and notations. Math leads to lots of discoveries which then in return causes other people to invent lots and lots of other fields of mathematics where numbers and notations have whole different meanings. Even those invented numbers and postulations lead to some worthwhile hypothesises to possibly look into, "let's be honest, hypothesises you can't use 99.99% of the time," bust some do have that very small 0.1% chance of making a profound discovery if using an application that is of real world use. Such as Physics, Chemistry, Computer Science etc. But remember, it isn't just the math, it's the foundation of those methodologies within the fields of science that really validates those 0.1% math formulas.

A dumb illogical mathematical axiom that should be abolished,

I never understood to this day why
anything to the ^0 power is equal to 1 because anything to the power is multiplying by itself. All the math teachers I asked were stumped about why would I asked such a question? Funny thing is... they had no answer, and for a second I felt like I was standing in church getting the same look by saying I don't care for religion. Not to digress, the logical fallacy behind this is 0^0 power = undefined and not 0 even though 0×0=0 Are you kidding me? Any finite number such as 99^0=1. But wait, 99×0=0 Why not!? While 1^0=1 and 1^1=1... yet all the other finite numbers such as 98^1=98. Did I just find an illogical double standard in math pertaining to "1?" and possibly even "0" since 0 is a number also, once again? Go figure. This is another problem. Are we making up rules incandescently in the background as well? No, seriously? This is another great point in my argument then if that's the case.

Conclusion,

Math by itself is just abstract thinking in the realm of math which has no physical scope. There's no other way to put it. It has no real physical purpose until it reaches physics or any of the branches of science for that matter. And most of math by itself is subjective if you are using numbers for different contexts besides the non physical aspects of addition, subtraction etc. Remember? The clock, einstein's issues, the ruler, plato's issues, Gödel's contention of mathematical limits etc. I wrote an equation on a piece of paper and solved it, congratulations. Did it explain anything physically? No, well then it was useless unless you like doing it for fun. Michio Kaku was even more critical about math without science. Do you like doing math as a subordinate hobby? Do you like it because of the cool symbols? Do you use other symbols for the same methods because it looks cool... why not? Does it help increase your mental capacity as a whole for other things in life? If so then that's good. Although this is my opinion, I hope to spark some thoughts to some people out there. And just like my first question I could've argued for days about religion, working 9-5 for 50 years and the flaws of the english language the same way I did with the subject of math. Moral of the story, question everything in life. You'd be amazed what kind of things start being rather peculiar than you initially thought while being adamant.

~ AD/Matrix

tim wood

anything to the ^0 power is equal to 1 because anything to the power is multiplying by itself. All the math teachers I asked were stumped about why would I asked such a question? — flame2

Bad teachers. x^n / x^n = x^(n-n)=x^0=1

flame2

"I meant clandescently." I could go on about the english language too. Two mispells, yes. No biggie.

alcontali

A lot of us get up and work from 9-5 because it is universally accepted to be a part of society without question of it. — flame2

Large numbers of people do not work from 9-5. I personally only did it for relatively short period in my life. Actually, no. Even then it was flexible. People came into the office at any time in the morning and left at any time in the evening. I do not remember any contracting gig in which people were supposed to be present at any particular time.

We engage in religious activities without second thought of whether this is actually true or not. — flame2

What is the benchmark for truth in religion? Correspondence to what exactly are we talking about? The real problem is that we do not even have a definition for religion. In fact, we do not have one for philosophy either.

However, if you pick out religious law alone by itself, we have a simple benchmark for truth: A religious advisory is true in its model of religion, if it is a syntactic entailment from scripture. We are just reusing Gödel's semantic completeness theorem here. If a proposition syntactically entails from its theory, then it is semantically true in all the models for that theory.

11P.M + 5 hours = 4A.M. Aha. In this case the math works like this;
11+5 = 4... A.M. Because we are referring to the clock now. — flame2

(11 + 5) mod 12 = 4

You cannot use the standard sum operator "+" without indicating that it is distorted because it is applied to a finite set. It is merely a sum-like operator:

11 ⨁ 5 = 4

Furthermore, the factorization of the set's cardinality is not a prime power: 12 = 2²3, which means that it is not a legitimate (Galois) field. It is not allowed to do multiplication in that set, because that would lead to inconsistencies. For example, the distorted multiplication:

3 ⊗ 4 = 0

This is not allowed in a legitimate field. If a ⊗b = 0, then we expect either a or b (or both) to be zero. That is not the case for 3 ⊗ 4 = 0. Hence, general arithmetic is not permitted in a field of size 12.

I never understood to this day why anything to the ^0 power is equal to 1 because anything to the power is multiplying by itself. — flame2

There is a good explanation for the nullary arithmetic product in Wikipedia:

Let $a_1, a_2, a_3, ...$ be a sequence of numbers, and let $P_m$ be the product of the first $m$ elements of the sequence.

Then $P_m = a_m . P_{m-1}$ for all $m=1, 2, 3, ...$

Provided that we use the following convention $P_1=a_1$ and $P_0=1$ .

This choice is unique.
— Wikipedia on nullary arithmetic product

The phrase "This choice is unique" means that any other choice would lead to inconsistencies.

The initial state for the multiplication result's accumulator must be initialized to 1, even before carrying out any repetitions at all. If you happen not to carry out any repetitions at all, then you must still return the accumulator as a result, initialized as it is to 1:

// function implementation to raise a to power n

function raise_to_power(a, n) {

    // initialized to the only legitimate value possible
    accumulator = 1

    for k = 1 to n {
      accumulator = accumulator * a
    }

    return accumulator

}

The argument in Wikipedia is that this algorithm will only be consistent when choosing to initialize the accumulator variable to 1. Any other choice will lead to trouble.

jgill

Since numbers by themselves sit in their own world within the human imagination, you can always make up your own rules as you go with postulations, right? Go ahead. That's pure mathematics research right there. — flame2

If you say so. :roll:

fishfry

I never understood to this day why
anything to the ^0 power is equal to 1 — flame2

There are a lot of good mathematical reasons why this is true. One way is to write down the integers on the top row of a table, and directly underneath each integer $n$ , write $2^n$ , like this:

... -3      -2      -1      0     1     2     3     4  ...

... 1/8     1/4     1/2     ?     2     4     8     16      ...

Now what should go underneath 0 for the pattern to make sense? It's clear that each time we move one position to the right, we multiply the number on the bottom row by 2. The only sensible definition of $2^0$ is 1.

In other words this definition is not arbitrary, it's natural.

Related to this idea, we want this exponent rule to be true: $2^n \times 2^m = 2^{n+m}$ . So what is, say, $2^5 \times 2^{-5}$ ? On the one hand, it's $32 \times \frac{1}{32} = 1$ ; and if we want the exponent rule to be true, we must have $2^0 = 1$ .

If you want to think of it this way, the definition of $2^0$ is arbitrary, in the sense that we could make the notation mean anything that we wanted it to. But we also want to make a sensible, natural definition; and the most sensible and natural thing is to define $2^0 = 1$ .

The same reasoning applies to any nonzero number raised to the 0 power. Defining it to be 1 makes the most sense.

GrandMinnow

x^y may be defined as the number of functions from y into x.

The empty function is the only function from 0 into x, so the number of functions from 0 into x is 1, so x^0 = 1.

Gregory

Good job flame2, this is great. I told my geometry teacher in college "if an Indian yogi says one plus one is four, something is going on in his mind, and it could be the truth sensation." Now we DO take things as numbers. Isn't three pizzas more than two? Actually , that is culturally determined too! Amazing

Gregory

If 2 has the exponent 0, the answer can be 2 or 0. Whether we say it means 2 zero times which would equal zero, or 2 multiplied a zero number of times which equals 2, we can see that these things are arbitrary. -2 × -2 can be -4 or +2. Saying it's positive 4 is bringing philosophy into math. And I know mathematicians can hate philosophers :)

GrandMinnow

If 2 has the exponent 0, the answer can be 2 or 0 — Gregory

2^0 = 1

The number of functions from 0 into 2 is 1.

Or, for natural numbers, we have the inductive definition:

x^0 = 1
x^(n+1) = (x^n)*x

Gregory

↪GrandMinnow

Indeed, mathematics is 75% invented, 25% discovered. Whether the exponent is 1 or 0, it clearly means something different from having an exponent of 2 or greater. Stringing random numbers together that look beautiful to you isn't math dude.

jgill

And I know mathematicians can hate philosophers :) — Gregory

Not really. We are quite tolerant, and chuckle as we do when a cute puppy barks. :smile:

Indeed, mathematics is 75% invented, 25% discovered — Gregory

The controversy among mathematicians whether math is discovered or created has finally been laid to rest. Thank you, Gregory! :cool:

SophistiCat

x^y may be defined as the number of functions from y into x. — GrandMinnow

What do you mean by that?

GrandMinnow

What do you mean by that? — SophistiCat

x^y = the cardinality of {f | f is a function & domain(f) = y & range(f) is a subset of x}.

Gregory

Strictly speaking, math is just like theoretical physics: it's a branch of philosophy. The only truth is that there is no truth and that this is true. And also that it is true that it is true that there is no truth. On to infinity? Maybe. (It's only maybe true that it might go to infinity! Godel's snake) I think mathematicians should consider philosophy thru considering substance. Why does a substance have to be one? Can a substance be 9? Then, apply this consideration to numbers.

The weakness of math is its philosophy of truth. The weakness of philosophy is that it "would render us entirely Pyrrhonian, were not nature too strong for it." (Hume)

Or is that it's strength?

jgill

Strictly speaking, math is just like theoretical physics: it's a branch of philosophy. — Gregory

Oh no, no, no . . . anything but that! :scream:

I think mathematicians should consider philosophy thru considering substance. — Gregory

There's always the danger of abuse. :gasp:

Gregory

Hume found doubtwithin mathematics:

"No priestly dogmas, invented on purpose to tame and subdue the rebellious reason of mankind, ever shocked common sense more than the doctrine of the infinite divisibility of extension, with its consequences; as they are pompously displayed by all geometricians and metaphysicians, with a kind of triumph and exultation. A real quantity, infinitely less than any finite quantity, contained quantities infinitely less than itself, and so on in infinitum [uncountable]; this is an edifice so bold and prodigious that it is too weighty for any pretended demonstration to support, because it shocks the clearest and most natural principles of human reason. But what renders the matter more extraordinary, is, that these seemingly absurd opinions are supported by a chain of reasoning, the clearest and most natural".

"Yet still reason must remain restless, and unquiet, even with regard to that skepticism, to which she is driven". (Hume)

Heidegger would have asked: "What is the wisest way to live in the world, with doubt or with faith? Which makes us live in the world best and, most importantly, helps us to die?"

Skeptical questions "admit of no answer and produce no conviction. Their only effect is to cause that momentary amazement and irresolution and confusion" says Hume. Is this amazement pleasurable? And is it useful?

If physically 2 feet can somehow by the laws of physics (even if in another dimension) equal 3 feet, than physics has prove that you can do whatever you like with numbers. 2=3

fishfry

If physically 2 feet can somehow by the laws of physics (even if in another dimension) equal 3 feet, than physics has prove that you can do whatever you like with numbers. 2=3 — Gregory

Numbers aren't lengths. That's exactly the distinction between math and physics. In math, 2 = 2 and 2 isn't 3. In physics, our measurement the length of an object is a function of our motion relative to that object. So in physics, length is a relative term. That's why they call it relativity. In math, numbers are absolute.

Of course we use math as a tool in physics. But the essential nature of mathematical objects is that they are abstract and conceptual; in no way "true", but only logically consistent, interesting, and perhaps useful. Those are the only criteria for mathematical existence. In fact we may even dispense with logical consistency, as we don't know for certain that our modern math is consistent; and we may dispense with utility, because it's the appliers of math, and not the mathematicians themselves, who care whether math is useful.

Whereas length, mass, electric charge, and so forth, are descriptions or models of the physical world. The models use math but are not the same as math.

tl;dr: Math isn't physics.

Gregory

In math, numbers are absolute. — fishfry

I appreciate this. In my opinion, though, a number is an idea that is rubbery. You can say 9 is the same as 2 in my mind. Maybe we need to just step back and accept that others have their own truth

fishfry

I appreciate this. In my opinion, though, a number is an idea that is rubbery. You can say 9 is the same as 2 in my mind. Maybe we need to just step back and accept that others have their own truth — Gregory

But of course. You are entirely free to do that.

The only problem is that your system wouldn't work very well and wouldn't have the usual properties we want numbers to have. Such as multiplication distributing over addition, and multiplication and addition being commutative, and so forth. If you just say 2 = 9 and that's one of your axioms, then so be it. But by the time you have fleshed out that idea and its ramifications, you'll see that you have a system that's not very interesting, mathematically. For example you have an even number equal to an odd number. So you won't be able to do much number theory.

But there's no rule that says you can't make things up. As I say, the criterion is mathematical interest. Is it something mathematicians would be interested in? In this case, probably not. But a lot of mathematicians work on things other mathematicians are not interested in. So really, I have no objection to what you want to do.

Now the burden is on you to work out a system that is interesting, and not just nihilistic, as in "everything's equal to everything else." What good is that? You couldn't count beans with it. So, I'll grant you that 2 = 9. Now what?

SophistiCat

2^0 = 1

The number of functions from 0 into 2 is 1. — GrandMinnow

x^y = the cardinality of {f | f is a function & domain(f) = y & range(f) is a subset of x}. — GrandMinnow

OK, but by the same token 2^2 = 1, because the number of functions from 2 into 2 is 1. What am I missing?

GrandMinnow

the number of functions from 2 into 2 is 1 — SophistiCat

The number of functions from 2 into 2 is 4.

Gregory

What good is that? You couldn't count beans with it. So, I'll grant you that 2 = 9. Now what? — fishfry

Well in my very first post on this forum I tried an argument against science itself. I supposed two objects that are materially identical to each and asked what distinguished them. Well, individuality of course! But maybe individuality counts for more than science is willing to grant. Being "over there" instead of "here" might make more of a difference in doinig science than science wants to grant. More recently I applied this to numbers and wondered if 1 and 1 really are so similar, and than whether 2 and 9 are really so dissimilar.

But I see your point, I think. IF numbers work as they did when we all first learned math (rigidly), then normal mathematics is objectively true. I think maybe that refutes the relativism I keep getting in. I'm not dumb and I admit when I am wrong. Thanks fishfry

SophistiCat

The number of functions from 2 into 2 is 4. — GrandMinnow

I don't understand.

$\biggl(2\biggr) \rightarrow \biggl(2\biggr)$

How is this four functions?

(FWIW in calculus texts the exponential function is usually defined generically as a power series.)

GrandMinnow

four functions — SophistiCat

{f | f is a function & domain(f) = 2 & range(f) is a subset of 2}

=

{
{<0 0> <1 0>}
{<0 0> <1 1>}
{<0 1> <1 0>}
{<0 1> <1 1>}
}

which is a set having 4 members.

/

The context of the discussion did not include consideration of exponents other than natural numbers. Of course, with exponents other than natural numbers, the definition must be expanded.

Nagase

↪SophistiCat

Notice that

↪GrandMinnow

is working with the set-theoretical representatives of the natural numbers, i.e. the finite von Neumann ordinals, in which each ordinal is the set of its predecessors (so 0 is the empty set, 1 is the singleton of the empty set, 2 is the set consisting of the empty set and the singleton of the empty set, etc.). In this context, (cardinal) exponentiation is defined in that way, as the cardinal of the set of functions from one set to the other. (The parenthetical aside is only meant to mark that it is also possible to define ordinal exponentiation, but this generally faces additional complications, since ordinals generally have more structure than cardinals and we want to preserve some of that structure when defining arithmetical operations)

GrandMinnow

set-theoretical representatives of the natural numbers — Nagase

To add to that comment, we note that it works with any sets, not just von Neumann ordinals.

Take any set S that has x number of elements and any set T that has y number of elements. Then x^y is the number of functions from T into S.

And, for natural numbers, this is equivalent to the inductive definition:

x^0 = 1
x^(y+1) =( x^y)*x

SophistiCat

↪GrandMinnow

This wasn't very helpful.

↪Nagase

It's still not clear to me how the use of this set theoretic representation explains "the number of functions from 2 into 2 is 4". Whatever representation you use, you still have one number (in this case, one set) in the domain and one number (one set) in the codomain. I mean, I can see (after your explanation) what he is doing, but I am not even sure how to formulate that correctly without specifically referring to the internal structure of a von Neumann ordinal.

GrandMinnow

This wasn't very helpful. — SophistiCat

It is exactly correct, precisely clear, and states exactly how it works in terms of von Neumann ordinals or even using any sets. And

x^0 = 1
x^(y+1) =( x^y)*x

is also utterly clear.

you still have one number (in this case, one set) in the domain and one number (one set) in the codomain — SophistiCat

Wrong. As Nagase so clearly and generously explained. 2, as a von Neumann ordinal, is a set with two members. And as I explained, it does not even depend on 2 itself being a von Neumann ordinal, or having itself 2 members, as we could take any set with 2 members. So (1) von Neumann ordinals are the usual means for an exact set theoretic mathematical formulation, and (2) we could also use any sets, and (3) I also gave you the inductive definition, which is the standard definition, that is equivalent with using von Neumann ordinals.

This is basic finite combinatorics. You can read about it not just in many a textbook on the subject, but even online by doing an Internet search on 'exponentiation'.

SophistiCat

↪GrandMinnow

Being correct and being clear is not the same thing. I took ten or so different courses of mathematics in college. It's been a long time and I have forgotten much of it, but I still understand basic combinatorics, thank you very much.

GrandMinnow

The explanations are both correct and clear.

I said it's basic combinatorics. I have no opinion on what you know about it otherwise.

I have offered you several explanations. Those explanations depend on knowing what a function from a set into a set is. And even if one does not have familiarity with the von Neumann ordinals (which is the standard set theoretic explication of natural numbers), I even explicitly listed the set of functions concerned. I have also mentioned that you can take any sets with two members and the definition works. And I gave you the standard inductive definition also. And I've given you a search term on the Internet where you can see this approach. There is no lack of clarity.

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