Here it means "keep going like this..." — Banno

In other words you're never done.

That doesn't follow.

OK, suppose you tell someone "keep going like this". At what point have they completed (are done) with that command? When they reach infinity?

If they are anyone except you, they will have understood the process and see how to construct any element, and realise that there is no need to do so in order to proceed.

Regardless of whether or not they understand the process, unless they "keep going like this" they cannot be said to have done the task. Understanding the task to be done, and doing it are two distinct things.

And, if someone thinks that understanding the task constitutes doing the task, as you apparently do, then that person actually misunderstands.

Regardless of whether or not they understand the process, unless they "keep going like this" they cannot be said to have done the task. — Metaphysician Undercover

The task is to sum the sequence. The answer is that the sequence diverges to infinity. Hence the task is completed.

Your answer is not a sum. The task has not been done.

Face it Banno; you're trying to avoid doing the task by giving some answer which amounts to "it can't be done".

Your answer is not a sum. — Metaphysician Undercover

You were expecting what? An integer? What is it you see as missing from the answer?

The sum! Wasn't that the task, to sum the sequence?

Why is infinity always discussed here ?
1. Infinity is not a number.
2. Being undefined means we cannot assign any value to something.
3. Lim x->0 ( 1/x) is undefined
4. Lim x-> inf ( 1/x) is zero.

I think 3,4 will clarify all that confusion going on here.

...you're trying to avoid doing the task by giving some answer which amounts to "it can't be done". — Metaphysician Undercover

That's a much better critique.

The sum! — Metaphysician Undercover

The sum diverges to infinity. What more is there to say?

But "the sum diverges to infinity" is not "it can't be done"!

I'm 101 percent confident

Take the other example - 1-½+⅓-¼...

It converges to two.

You disagree?

But "the sum diverges to infinity" is not "it can't be done"! — Banno

Well, it's not a sum. And to say that the sum "diverges to infinity" says I can't give you that sum. So if you're not saying "it can't be done" then why can't you give me the sum?

Take the other example - 1-½+⅓-¼...

It converges to two.

You disagree? — Banno

I'd need a definition of "converges" before I'd agree to what you're saying, but if you mean comes closer and closer to, as you continue on the unfinished process signified by "...", then I'd probably agree. But this implies that it's not ever done.

it's not a sum — Metaphysician Undercover

Why not?

A sum is the total. "Diverges" signifies a direction. It would be a category mistake to say "diverges to infinity" is a sum.

And yet that is what mathematicians do.

f its uncountable/infinite, then that suggests that 1/0 is legitimate — Devans99

Wow. This goes on forever, doesn't it?


$\underset{x\to {{0}^{+}}}{\mathop{Lim}}\,\tfrac{1}{x}\text{ }=\text{ }+\infty$

1/0 undefined. Suggest you move on to another topic. "infinite sum" is OK amongst professional mathematicians.

This passage is directly from a book of wittgenstein
N(0) is used for the cardinality of the set of natural numbers. Wittgenstein shows that the technique we learn in writing the usual whole numbers and writing N(0) are different and he concludes that we cannot say we have written N(0) numeral.

We have all been taught a technique of counting in arabic numerals. We have all of us learned to count-we have learned to construct one numeral after another. Now how many numerals have you learned to write down?

Turing: Well, if I were not here, I should say N(0)

Wittgenstein: I entirely agree, but that answer shows something. There might be many answers to my question. For instance, someone might answer, " The number of numerals I have in fact written down." Or a finitist might say that one cannot learn to write down more numerals than one does in fact write down, and so might reply, "the number of numerals which I will ever write down". Or of course, one could reply "N(0)" as Turing did. Now should we say, "How wonderful-to learn N(0), numerals, and in so short a time! How clever we are!"?-Well, let us ask, "How did we learn to write N(0) numerals?" And in order to answer this, it is illuminating to ask, "What would it be like to learn only 100,000 numerals?"

Well, it might be that whenever numerals of more than five figures cropped up in our calculations, they were thrown away and disregarded. Or that only the last five figures were counted as relevant and the rest thrown away.-The point is that the technique of learning N(0),numerals is different from the technique of learning 100,000 numerals. Take the biggest numeral which has ever been mentioned. What is the difference between learning a technique of counting numerals up to that numeral and learning a technique which did not end at that numeral?
Well, it might have been that one's teachers said, "This series has no end." But how did you know what that meant? They might have said that and then when one reached the numeral six billion, they might say, "Well, now we have got here, I need hardly say . . ." and shrug their shoulders with a slight laugh.-So how did you know what they meant? Simply from the way in which the series was treated. I did not ask, "How many numerals are there?" This is immensely important. I asked a question about a human being, namely, "How many numerals did you learn to write down?" Turing answered "N(0)" and I agreed. In agreeing, I meant that that is the way in which the number N(0), is used. It does not mean that Turing has learned to write down an enormous number. N(0) is not an enormous number. The number of numerals Turing has written down is probably enormous. But that is irrelevant; the question I asked is quite different. To say that one has written down an enormous number of numerals is perfectly sensible, but to say that one has written down N(0), numerals is nonsense.

As I said, ellipsis means unfinished. So using the ellipsis and claiming "it's done" is a false claim. — Metaphysician Undercover

The ellipses aren't necessary. We have an increasing sequence of partial sums that is not bounded above. This sequence has no limit in the real numbers. 'Diverges to infinity' indicates more than merely a failure to converge by specifying that the sequence of partial sums is increasing. This is basic real analysis.

Consider also that proofs are finite objects. These finite objects and the things we do with them are inspired by intuitions of the so-called 'infinite.' And mathematicians aren't allergic to intuitive ways of talking. But in the end we have finite proofs that use a finite number of symbols. Such proofs can be (tediously) translated into dead symbols (bits if you like) and checked mechanically (by a computer, for instance).

This makes mathematics a prototypically 'normal' discourse, and perhaps explains the mixed feelings that metaphysicians have toward it. As I see it, the old dream of metaphysics is to do 'spiritual math' about matters of ultimate concern. Proofs of god, etc. But non-mathematical language seems caught up in time to a much greater degree. 'History is a nightmare from which I'm trying to awake.'

A believe in infinite past time is therefore akin to a belief it is possible to count 'all the numbers'. — Devans99

That only holds if one is a realist about the past who believes that the object of history is an unknowable reality in being unobservable and transcending present and future information.

In contrast, according to anti-realism the very truth of a past-contingent proposition reduces to present information. if the present state of information is ambiguous with respect to two historical possibilities, then according to anti-realism there is no matter of fact as to which historical possibility is true. Moreover, since the anti-realist never wants to claim 'perfect' knowledge of the past, he must insist that the past is infinitely extensible in a literal sense as and when new information becomes available.

For example, suppose Archaeologist A at time T unearths evidence E implying the existence of a fact F at time T', where T' < T. In stark contrast to the realist, the anti-realist considers A and E to constitute part of the very meaning of F, such that the truth of F is a function of T.

I did not ask, "How many numerals are there?" This is immensely important. I asked a question about a human being, namely, "How many numerals did you learn to write down?"

This is the category difference which makes Banno's claim of "done" false.

Consider also that proofs are finite objects. — softwhere

Continuing with the Wittgensteinian perspective, the finitude of the proof would be dependent on the definitions of the terms. The definitions create the boundaries of meaning, required for the proof. If there is any vagueness, or undefined terms in the proof, then the proof cannot be considered as a finite object. Therefore it is very unlikely that we actually have any truly finite proofs, because definitions are produced with words, which themselves need to be defined, etc., ad infinitum. Vagueness cannot be removed to the extent required for the production of a finite object.

This makes mathematics a prototypically 'normal' discourse, and perhaps explains the mixed feelings that metaphysicians have toward it. As I see it, the old dream of metaphysics is to do 'spiritual math' about matters of ultimate concern. Proofs of god, etc. But non-mathematical language seems caught up in time to a much greater degree. 'History is a nightmare from which I'm trying to awake.' — softwhere

Yes, we can class mathematics as "normal discourse", but to characterize "normal discourse", as working with finite objects of meaning, is what Wittgenstein demonstrates as wrong. This is why we must work to purge the axioms of mathematics from the scourge of Platonism, To consider proofs as finite objects is a false premise.

The cardinality of a set depends on the notion of bijection — softwhere

Bijection/one-to-one correspondence is a procedure that produces paradoxes like Galileo's Paradox, or the cardinality of the naturals is the same as the cardinality of the rationals. It is therefore to my mind an unsound procedure. Cantor did nothing to help our understanding of infinity IMO; he has lead us down the wrong path entirely.

Hence the measure of an uncountable union of points... — softwhere

My (and Galileo's) point exactly - you fundamentally cannot measure something that is
uncountable/infinite - you would never finish measuring it - it is impossible to measure and claiming that bijection can provide a sound measure is ignoring the evidence (of paradoxes).

1/0 undefined. Suggest you move on to another topic. "infinite sum" is OK amongst professional mathematicians. — John Gill

The infinite sum concept in maths has definite problems, see here for an example:

https://en.wikipedia.org/wiki/Thomson%27s_lamp

I know very little about anti-realism. From your anti-realist perspective, what exactly is the past?

To me the past is a deducible concept without referencing external realities - I have thoughts, these thoughts from a causal chain. The present exists, there are thoughts that I am no longer having, so the past exists. There are thoughts that I will be having so the future exists. I can label each thought with an integer. Assuming a past eternity, then the number of thoughts would be equal to the highest number. But there is no highest number, so a past eternity is impossible?

:smile:

I think if we all started reading Wittgenstein's Lectures on the foundation of mathematics, a lot of issues that come up here can be addressed in a good manner. As you have pointed at rightly. Wittgenstein regarded mathematics as a human invention, a finite calculus at most. But due to platonism, we sometimes give answers that answer different questions.

Wittgenstein further on even challenges the proof by induction used in mathematics.

Let's consider a proof by mathematical induction.
@ is a mathematical property. If it is known that @ (1) and it is known that [n] : @[n]. --> .@[n + l], then [n] . @[n].
It's misleading because it's not known how
[n] : @[n]. --> .@[n+ 1] is proved.

That's simply an indication that we can do logic without knowing how the logic works. To know how logic works is a completely different issue. This question is Socrates' claim to fame. The artists and skilled craftsmen would claim to "know", because they had a technique which produced the desired results. This attitude extended into all fields, science, mathematics, even ethics and sophistry. Socrates demonstrated that these people who know how to do something do not know how it is that their activity brings about the desired end. Therefore their own claim to "knowing-how" is not grounded in anything, the activity is just a habit, and so is not real knowledge at all..

In other words you're never done. — Metaphysician Undercover

Say, $\sum_{n \in \mathbb{N}} f(n)$ is not a process like going shopping and returning home, it's a mathematical expression.
Convergence and divergence has concise technical definitions using the likes of $\forall$ and $\exists$.
I challenge you find and understand them. ;) At this point you might be in a position to launch critique.
By the way, you should know that this stuff has practical applications used every day by engineers, physicists and others.

Wow. This goes on forever, doesn't it? — John Gill

... when people aren't even trying. (Wait, I see what you did there.) :)
I suppose we might show the definition of $\lim$, and that it doesn't rely on $\infty$ other than implicitly by way of the neverending numbers.
Probably won't matter to the deniers, though, I sort of doubt it'd be worthwhile.

Say, ∑n∈Nf(n)∑n∈Nf(n) is not a process like going shopping and returning home, it's a mathematical expression.
Convergence and divergence has concise technical definitions using the likes of ∀∀ and ∃∃.
I challenge you find and understand them. ;) At this point you might be in a position to launch critique.
By the way, you should know that this stuff has practical applications used every day by engineers, physicists and others. — jorndoe

A process is a process. If your intent is to create ambiguity in the definition of "process", such that it is possible to have a completed process, which by definition has no end, then be my guest. Do not expect me to follow along with such contradiction though.

And if you back up, justify, such contradiction with the report that it has practical applications, I would reply that such applications are nothing more than sophistry, deception.