Of that I have no doubt 
Not so. “Every” implies the entire infinite set, but at this point is not the issue.
And that is irrelevant until you proclaim that you can do that to the entire infinite set and therefore have accomplished something. The problem is that you can’t ever know if you applied it to the entire set. And in fact, you can know that you cannot apply it to the entire set, because the set has no end. And you are ignoring their relative sizes.
I gather that you are arguing from perhaps a constructivist, intuitionist, finitist, or ultrafinitist perspective. Do you have some particular philosophy of math from which you are reasoning? It would be incredibly helpful for me to know this so I can understand your points. I don’t know too much about these philosophies but from my reading I can’t imagine anyone disagreeing that we can encode a pair of integers as a single integer as I’ve shown. I do not see how anyone can argue with this but like I say, there are some alternative ontologies out there and perhaps you subscribe to one or more of them.
Suppose I said that we define an even number as an integer n such that there is an integer k with n = 2k.
And then I proclaimed: “Every even number is divisible by 2,” and proved it by starting with n, replacing it with 2k, dividing by 2, and showing that the result is the integer k, QED.
Would you object to this argument? I’m really trying to see your point of view here. Are you saying I can’t define an even number that way because it requires me to consider all of them at the same time? Then how do you define an even number?
Yes. My “philosophy” concerning math is that math is simply logic applied to quantities.
Maintain true definitional logic, and we will agree on everything.
I don’t disagree with that. Many things can be done with one thing. The issue is proving that you have done it with ALL of them inclusively.
You have defined two infinite sets and you are calling one “even numbers” which are the result of multiplying 2 times each member of the other set, called “integers”.
That is not a “proclamation”, but a tautological rewording of your prior definition.
No need for a proof for a definition.
I would only object to you saying that you are making an “argument”. You have merely declared a definition of “even number” and presumed the definition of an “integer”. There is nothing yet to argue.
You raised a number of interesting points. I’ll respond in two parts for readability.
Ok. Logic applied to quantities. That means more to you than it does to me so I’ll approach your philosophy via questions.
- First point: What kind of logic? These days there are all kinds. Classical, modal, intuitionist, first order, second order. More logics than I know about.
Reading ahead I see you object to the use of “all” applied to infinite sets. I won’t talk you out of your opinion, but the universal quantifier “for all” is a core part of standard math.
The logic used in modern math is first-order predicate logic. The universal quantifier is an essential aspect of that logic. In modern math we may freely quantify over the elements of an infinite set and make statements such as “for all” and “there exists” such and so. Without “for all” we can’t get modern math off the ground.
So, where do you come down on the use of the universal quantifier applied to infinite sets? And if you reject it, what exact system of logic are you using in your math?
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Second point. For clarification, do you accept infinite sets? You don’t have to, I just need to know your position here. Which one of these camps do you favor, if any? Or add something new, it’s all good.
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Ultrafinitism. This is the belief that there are only finitely many natural numbers 1, 2, 3, 4, but at some point you can’t keep going. For example it is not necessarily meaningful to speak of huge numbers like 10^10^10^10 or whatever. Exponents bind from the right, so this is a big honkin’ number. An ultrafinitist wouldn’t let me call it a number. Extreme position but the known universe is finite so perhaps the ultrafinitists have a point.
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Finitism. This is where each of the numbers 1, 2, 3, 4, … exists. Maybe not all at once. We do have the inductive principle so that if you have n, you also have n + 1. So you have each of the natural numbers, but you don’t have a completed set of them. As Aristotle would put it we accept potential infinity but not actual infinity. Very popular a couple of thousand years ago and in fact it’s still a powerful system for doing number theory. But you can’t define the real numbers without infinite sets so finitism isn’t very popular.
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Constructivism. You accept infinite sets as long as you have an algorithm to construct them. So you accept the countably many computable numbers, but you do not accept any bitstrings that are essentially random; that cannot be described or generated by an algorithm. This view is gaining interest lately due to the influence of computer scientists, who only know about things that are computable. It’s still too weak for mainstream math.
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Standard set theory. There are infinite sets and lots of them. They’re arranged in elaborate hierarchies. Set theorists study large cardinals that are transfinite numbers so huge they can’t even be proven to exist using the standard axioms. People get grant money for this so you know it’s respectable
Myself I accept standard set theory. To me, math has no obligation to conform to anyone’s idea of reasonableness. If an idea is logically consistent and interesting, it’s math. I enjoy learning how to reason about infinite sets, so I accept set theory. My ontology doesn’t go any deeper than that. I don’t care if any of this stuff is “real.” Being real is the job of physics. Math is not constrained by reality. Math answers only to its internal esthetic.
So, where are you at with infinite sets? That’s a key anchor point for anyone’s philosophy of math.
[By the way it occurs to me that my little example with the even numbers (about which I’ll have much more to say in the next post) can be fully carried out in the finitist system. That proof (or “bedtime story” if you don’t like the word proof) does not require a completed infinity, it only requires each of the numbers 1, 2, 3, … In other words the bedtime story can be carried out in a universe with no infinite sets at all.]
In the next episode I will argue that this is actually a proof and not just a bedtime story, but I didn’t want to get hung up on that right now so call it a bedtime story if you like.
I appreciate your optimism that we might come to agreement. Or course we will have to agree on a definition of “true definition logic.” So let’s nail down
a) What logic you are using,
b) Whether you allow quantifiers, and
c) What is your position on infinite sets? Ultrafinitist, finitist, constructivist, or standard set theory, or other.
d) Also … which is troubling you more? The fact that we assume infinite sets? Or the fact that we can quantify over all the element of a set? Because if it’s the former, I don’t actually need any infinite sets for the even number example. But I do need universal quantification.
I’m working on responding to your other points so let me leave the rest to later.
I know James well enough to answer this…
He agrees with my proof …
If existence is ever destroyed, we wouldn’t exist !!
He believes in infinities.
Part 2.
Ok I referred to this earlier. The ability to universally quantify over the elements of an arbitrary set is a core part of the logic used in set theory, namely first-order predicate logic.
I don’t have to “do” anything to the elements. I don’t understand your objection. If I say for example that that “every nonempty finite set of positive integers has a largest element,” that is not a controversial statement. It’s provable even in the finitist system where there are no infinite sets. I am still stating a property possessed by each and every one of the infinitely many finite sets of positive integers, but I don’t actually need any infinite sets to do so. I only need the basic properties of numbers.
You can’t be objecting to basic number theory but that seems to be the stance you are taking relative to my mapping function from pairs of integers to single integers. Basic number theory does not require infinite sets.
I should wait till you explain your position on the “for all” operator before I say any more. I’m not sure if it’s the infinite sets or the “for all” that’s bothering you more.
That is actually a very good point. I plead guilty! You are right. A definition can never add any truths. In other words if you have some axioms and you logically derive some theorems, making definitions does not add any more fundamental truths to the system. All a definition is, is a shorthand for some other concepts.
So in fact all mathematical proofs are tautologies and anything you prove about a definition is also just a tautological rewording. My little proof about even numbers is like that; and so is Wiles’s proof of Fermat’s last theorem.
The creative part of math is coming up with the right definitions. But once you have a finished proof, in theory you could translate it to pure logic and a machine could check its correctness line by line.
Every mathematical proof is a tautology if you want to take that point of view. Once you choose your axioms and your rules of inference, the production of a theorem is deterministic. The theorem was already inherent in the axioms. And definitions don’t add any new information at all.
So you are right about my little proof about even numbers; but in fact that’s no knock on my proof. You could say the same thing about Wiles’s proof or the best work of Newton and Gauss. If you broke it down to pure logic, it would have to be a tautology. The creativity is in which tautology you write down; whether, when humans apply meaning to the symbols, it tells us something interesting about the world. Whether the definitions you make are interesting.
As an example, there are people and cats whether there are biologists or not. But when biologists come along and make definitions: “An ape has two legs, a feline has four legs,” the definition adds no new information. But it helps us understand the world. A definition is a means of identifying and classifying what we regard as important.
There is absolutely a need. Any math major would start out on proofs like that and work their way up. They make the students do exactly this type of proof in “Intro to proofs” classes and also in Discrete math class. Of course the even number proof is a baby proof, a simple proof, a beginner proof. But structurally it is no different than any other mathematical proof. You need to read the definition and be able to construct a chain of logic that leads from the premises to a conclusion.
That’s why I am using it. It has all the features we care about. It uses a “for all” clause and it says something about an infinite class of objects. You object to it but I claim it doesn’t even involve infinite sets and is totally standard mathematical reasoning. I am not buying your objections because in fact there are actually no infinite sets involved.
I’ve addressed this point already and in fact it appears that you made the same point again. But it’s worth repeating. The even number argument is the very paradigm of what a mathematical argument looks like. It’s different only in length and importance to the greatest published mathematical proofs.
So now that you understand that the even number proof is in fact a proof – and heck, even if you still want to call it a bedtime story, I don’t care – the question is whether you accept it. “All even numbers are divisible by 2.” Do you reject that for its use of “all?” Well then we can’t do any math. What is your solution to that?
He believes in infinities but doesn’t believe in the “for all” statement? I’m having a tough time understanding that line of reasoning.
Wtf, you’re trying to talk to 4 people at once!!
Who the fuck are you???
I can tell your good at proofs!!
Not many people are like that
That’s a deal breaker… Don’t worry about it, I already said James doesn’t understand!
Even if there are magnitudes of infinity, that is an all set statement!!
It’s absurd!
Obviously !
I’m done for the night!
Just another discussion forum addict.
I studied math but didn’t do it as a career. My math skills are quite modest by the standards of mathematicians. Even by the standards of math majors. I’m also interested in the philosophy of math but certainly no expert. Just a Wiki dabbler as so many of us are these days. I enjoy trying to add some mathematical clarity to online forums, especially around infinite sets. I always seem to end up talking to people who believe in “alternative” math and trying to get them to clarify their ideas. I must admit I’ve never talked to anyone before who accepts infinite sets but not the “for all” operator. I await further clarification from James on that topic.
Thanks for asking.
You’re much more polite than me wtf…
Here’s a simple disproof for James …
Let’s have some fun!!!
Compared to zero any non zero number is infinite in magnitude!!!
This means that 1 is an objection to James as an all set!!
LOL!!
You’re going to figure out, if you already haven’t that you’re talking to someone who deserves every math prize on earth right now! Whatever… There’s other more pressing serious shit happening on earth though!
But math is like music, it makes life beautiful!
Logic, as specified by Aristotle, also called “dialectics”, is in reality, merely the coherent (consistent and unambiguous) use of language (ordered concepts, relationships, and their communication).
It CAN be applied. But you must apply it coherently. Presumption is the seed of all of your sins.
That is very likely true.
That would be the closest on your list, but Hyperreals is a little closer.
Which could explain their willingness to be dogmatically flawed at times.
We’ll see. 
I was hoping that you weren’t done with it. 
A bit too short of a “bedtime story” so far. But if sleep is your aim…
I have covered those points. Definitional logic is merely the rigor of defining precisely what you mean by the words/symbols that you use and you remain coherent in your presentation/communication using them with one cardinal rule: Don’t Presume (not as trivial as you think).
I take it that you have a habit of becoming hyper-manic when triggered by the sight of a presumed flaw in some else’s reasoning, especially someone you perceive as intimidating (the rabid dog syndrome).
I said only that if you are going to presume that an operator has been applied to an infinite set, you must prove that you actually have done so. The example you gave in an effort to create a counter proof to mine was dependent upon that issue, which you now skirt. If you stick to ensuring your proof properly, your flaws will become evident (even to you).
All a definition is, is a coherent, unambiguous, and comprehensive explanation of the intended meaning of a word or symbol to be used within a communique.
Learning?
No. “All” proofs are not really tautologies. They are only tautological if they don’t actually add anything significant, any formerly unrealized relationship, to the argument as they merely repeat the same concept, perhaps using different words.
I have used exactly that in a critical way.
You didn’t prove anything … ? What was your hypothesis? You merely stated a definition (of “even numbers”) and then stated it again with a few words changed. What were you trying to prove? 
There absolute is not. And there is no proof for definitions other than to prove that someone else was using the same as yours. A definition is a declaration of intent. There isn’t anything to prove about it: “My cat’s name is Booty, prove me wrong.” or "Given the constant k… But wait, first prove to me that k is really a constant as you say!" or "X is a number between 1 and 10, so … But wait. How do we know that x is really a number between 1 and 10??!! Maybe you just think it is."
Then let’s hope that you can improve. Math professors often have a lot to learn about actual logical proofs. Russell apparently enjoyed showing them up on a regular basis.
It’s statements like that which give me concern for your mental state.
You are the one who called it a “bedtime story”, not me.
I said that it was merely a tautological definition, you agreed, yet now proclaim that it is a valid proof that “all even numbers are divisible by 2” (when in fact, you merely stated that the integers that are divisible by 2 were the numbers that you call the “even numbers”).
Your hyper-mania can become an issue as it distorts your perception of who is really saying what. Let’s hope you gain a little control over it.
True. I haven’t had a serious challenge in a long time. It is a bit refreshing. Could get fun. If I could just get him on topic and a little less presumptuous, he could actually be be useful.
He has already proven you wrong twice. You’re not paying attention.
Whatever dude. If the mods are happy with your style of discourse, that’s on them. You really haven’t a clue and you’re a rude asshole as well.
You are the one with 23,000 posts on a handle registered only six years ago. That’s over 10 posts per day, day in and day out, 365 days a year, for six years running. And I do mean running, like diarrhea.
No wonder you accuse other people of “mania.”
Like I say, if the mods are happy with your ignorant, bullying, and insulting style of conversation, that’s on them. I’m done here.
You are the one who picked the fight, “dude”.
“If you can’t take the heat … don’t pick a fight with the chef.”
I get hyper manic a lot James …
That’s true. We get used to it.
Ok, let’s assume there is a big whoosh going on here James …
Care to explain?
He said he thinks, just thinks a set might contain itself… I let it slide because that may or may not be identity itself… Which is trivial to the point I was making about types and orders of magnitude !
What’s the second one??
Please enlighten me!
Ahh… I went back James …
Computability!!
If a string of numbers has no pattern… Then there is no pattern to discern that it has no pattern, that’s a basic law… I brushed it off because I was thinking of other stuff …
A good trick would be to count all of the numbers that have no pattern. 