I think you would approve of Wittgenstein's view. He was a finitist, and a math anti-realist. He didn't believe in set theory. He thought it was bullshit. — frank

But this does not invalidate ZFC nor the axiom of choice, nor need we conclude that a limit is something the sequence approaches dynamically rather than a property of the sequence as a completed object.

And the larger point: At issue is whether there is one basic ontology for mathematics. Sime is seeking to replace one ontology with another, to insist that we should think of infinite sequences as processes or algorithms, not completed totalities. — Banno

Yes, I attach value to mathematics, but that's like saying I attach value to logic or to language or, you know, to thinking. The basis of mathematics is woven into the way we think, and mathematics itself is primarily a matter of doing that more systematically, more self-consciously, more carefully, more reflectively. The way many on this forum say you can't escape philosophy or metaphysics, I believe you can't escape mathematics, or at least that primordial mathematics of apprehending structure and relation. — Srap Tasmaner

When you say you are critiquing mathematical principles, here's what I imagine: you open your math book to page 1; there's a definition there, maybe it strikes you as questionable in some way; you announce that mathematics is built on a faulty foundation and close the book. "It's all rubbish!" You never make it past what you describe as the "principles" which you reject. — Srap Tasmaner

So I enjoy these chances to exercise my math muscles a bit more directly than usual, and I take deep offense at Metaphysician Undercover's repeated dismissal of mathematics as a tissue of lies, half-truths, and obfuscations. — Srap Tasmaner

Such potentially infinite sequences do not possess a limit unless the choices are made in accordance with an epsilon-delta strategy that obeys the definition of "limit". So in this case, we can speak of approaching a limit, because Eloise and Abelard are endlessly cooperating to produce a strategy for continuing a live sequence that literally approaches their desired limit, as opposed to the previous case of Eloise having a one-move winning-strategy when competing against Abelard for proving a convergence property of a dead algorithm. — sime

This is exactly arse about. The limit is a result of the sequence. Those who care to look can see exactly that in the proofs offered earlier. — Banno

he key is that an infinite sequence may have a finite sum: ½ + ¼ + ⅛ ... = 1 — Banno

Added: the pedagogic problem - it's not a mathematical problem - is how to dissipate the notion that the limit is "a little bit more" than the sequence? — Banno

Notice that the limit is set out in terms of the sequence - the limit is provided by the sequence alone! so the limit results form the sequence. But it need not be one of the elements of the sequence. It's not something the sequence reaches toward — it is a property of the sequence itself. — Banno

The limit isn't something the sequence is trying to get to; it's a concise description of how the sequence behaves. The sequence doesn't "know about" or "aim for" its limit - the limit is simply our label for a pattern in the sequence's terms. — Banno

If ∣x∣<ε for every ε>0, then x=0 is not a stipulation about limits; it is a theorem about the real numbers, derived from the order structure of ℝ. — Banno

I'm not concerned about credibility or showing that I'm working. — frank

The electron is, in fact, conceived by scientists as a point. It's startling, but true. — frank

For (2) to be possible, I must be offering you the actual value. — Srap Tasmaner

But an electron is conceived as a point. — frank

Isn't that the same as the idea of an infinitesimal in math? — frank

According to Zvi Rosen, the sum and the limit are not equal (according to Cauchy). They're just as close as we "want" them to be. — frank

The salient bit today is that a limit is not a rounding off. — Banno

It's not that the adjacent members of a sequence become "infinitely close": they become "arbitrarily close", and so the series (in this case, the sum of the members of the sequence) becomes arbitrarily close to — well, that's the thing, to what? And that's your limit. — Srap Tasmaner

The difference between the limit and the sum is an infinitely small number. — frank

We could say that this solves Zeno's paradox as along as space and time actually conform to the calculus framework. I think the average scientist would agree that they do conform, but there is still room to reject the calculus angle. — frank

The key is that an infinite sequence may have a finite sum: ½ + ¼ + ⅛ ... = 1 — Banno

Replayed songs are physical — Corvus

But you cannot access the other folks mind, hence you wouldn't know what song is being played in his/her mind. — Corvus

No. — Corvus

Does that explain it? — frank

Ok. All I know is that it's common sense that if you're driving from Washington DC to Alaska, you will, at some point, be in British Columbia. Those who claim this view is wrong should at least acknowledge that what they're saying sounds bizarre. — frank

Trump is being fetishized as the personification of pure evil... — Tzeentch

Maybe there's no joy there. Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke. — Srap Tasmaner

Some people reject talking about infinite collections, I think, or reject talking about performing operations on them. — Srap Tasmaner

Who would say no to that? How could you get from A to B without arriving at a point that's halfway between? — frank

DHS notes a more than 1000% increase in assaults on ICE agents. — AmadeusD

So you think that "to be is to be the value of a variable" is a platonist principle? — Ludwig V

Except that ordinal numbers don't assign a value; that assigns a place in an order. — Ludwig V

No, it isn't. It is about whatever I am assigning a value to. — Ludwig V

Not all words refer to anything. That's why there's such a fuss about dragons and the present king of France. — Ludwig V

I think many people believe that if something is referred to, it counts as an object. — Ludwig V

So you are right to foreground what we do with numbers - or numerals if you prefer. But I think you slip up when you say that the numeral refers to an idea. That just resuscitates that argument you gave about numbers as ideas. The assignation of value in this context is public and shared, so it cannot be about ideas in our individual minds. — Ludwig V

I'm getting the impression that your objection is simply to the concept of an abstract object, which you call platonism. Would that be fair? — Ludwig V

In the Roman number system "V" counts as five. The Chinese system has 五 (wǔ) for the same number. The ancient greeks used the letters of their alphabet as numerals, so five was the letter epsilon. If you just talk about numerals, you lose the equivalences across different systems. — Ludwig V

That's pretty cool. — frank

An abstract object is something that isn't physical, but it's not simply mental either. — frank

What of quantification? — Banno

So math is just language games, right? — frank

If each individual 1 is a token of the type <1>, you have to say what sort of thing the type is. That's not going to work out. — Srap Tasmaner

if they are only in the mind, he owes us a story about how we manage to do things with them in the world. — Banno

Notice that this odd position is blandly asserted, not supported by any argument. — Banno

He relies on presuming that all reference must be object-reference, — Banno

Numbers are not just ideas in the mind, but are rooted as objects in our shared practices. — Ludwig V

But we need another step - "1 counts as a number" - to get the procedure moving.

...

It's not platonic. — Banno

Do you mean the premiss that space can be infinitely divided, not merely conceptually, but also physically? — Ludwig V

But a physical limit to the process of division doesn't undermine the conceptual description. — Ludwig V

We've already left Meta behind, since he has claimed numbers are not ordered... — Banno

I was thinking some days ago that, though I'm not sure what the favored way to do this is, if pressed to define the natural numbers I would just construct them: 1 is a natural number, and if n is a natural number then so is n+1. I would define them in exactly the same way we set up mathematical induction. (Which is why I commented to Metaphysician Undercover that the natural numbers "being infinite" is not part of their definition, as I see it, but a dead easy theorem.) — Srap Tasmaner

But we need another step - "1 counts as a number" - to get the procedure moving. — Banno

It's not platonic. — Banno

So we get "One counts as a number" and "every number has a subsequent number" and discover that the pattern does not end, and then learn to talk of the whole as being unbounded and that infinite counts as being unbounded... iterating the "...counts as..." to invoke more language games. — Banno

Because next can mean two different things.

1) Next in the definition (logical next).
In mathematics, next often just means “the item with the next label in the sequence.” It’s part of how the rule is set up, so if you tell me where you are, the rule tells you what counts as the next one. That doesn’t require anything to be happening in time. — Sam26

"Next" here implies a relation, and mathematics is the study of the relations between its "objects," which it is happy to treat as effectively undefined. — Srap Tasmaner

Empirically, that may be true - especially if you regard a field (gravity, magnetism) as a medium. But setting up a set of co-ordinates does not require a medium in addition, so far as I can see. — Ludwig V

In math, process doesn’t have to mean a thing happening in time. It may just mean a rule, a precise recipe that tells you how to get the next step, or how to compute the nth term. Infinity shows up because the rule has no final step. — Sam26

For me, empty space is not a mediium. — Ludwig V

Space is a co-ordinate system, which defines the possibilities where certain kinds of object may be. Objects are distinct from mediums because the latter are found everywhere, but objects have a locating within space. — Ludwig V

so the advice is to remain calm, don't open the door unless they show you a warrant,. — frank

It seems to me that the question of a medium in space is secondary. The first move is to set up a co-ordinates and rules for plotting the position of objects on those. (In other words, the concept is defined by the practice.) Once we have co-ordinate and objects, the question of a medium makes some sense. How non-mathematicians develop the concept is another question. But we can be pretty sure it is by interacting with the ordinary world. Mathematics, in my book, is a development of that. — Ludwig V

The paradox of Zeno's paradox, for me, is that Achilles is precluded from reaching a point that defines the system - the limit. The first step is to divided the distance from the start to the goal, limit, by 2, and so on. The limit is not an optional add-on, (as it seems to be in the case the natural numbers). — Ludwig V

In particular, these laws are always aimed at suppressing small businesses, because small businessmen are less dependent on the power and can overthrow it. — Linkey

The question is, at what level of explanation should this incompatibility be situated? at the physical level, as physics usually assumes, or at the level of the rules of mathematics? — sime

I think we should consider the fact that Newton and Leibniz didn't invent calculus for the purpose of solving Zeno's paradox, but for describing trajectories under gravity. Hence the mathematical definition of differentiation that we inherited from them and use today, isn't defined as a resource-transforming operation that takes a mutable function and mutates it into its derivative; rather our classical differentiation is merely defined as a mapping between two stateless and immutable functions. — sime

But if Zeno's paradox is to be exorcised from calculus, such that calculus has a dynamical model, then I can't see an alternative than to treat abstract functions like pieces of plasticine, that can be sliced into bits or rolled into a smooth curve, but not at the same time. — sime

We can be pretty confident that space is not infinitely divisible and yet still use calculus to plot satellite orbits. — Banno

As the Trump administration oversees the sale of Venezuela's petroleum worldwide, Senate Democrats are questioning who is benefitting from the contracts.

In one of the first transactions, the U.S. granted Vitol, the world's largest independent oil broker, a license worth roughly $250 million. A senior partner at Vitol, John Addison, gave roughly $6 million to Trump-aligned political action committees during the presidential election, according to donation records compiled by OpenSecrets. — Stephen Groves, The Associated Press