ok.

You said your constructivism was compatible with realism — frank

This?

This view preserves mathematical realism (mathematical statements have objective truth values) while avoiding the metaphysical commitments of Platonism (no need for causally inert, spatiotemporally transcendent entities). — Banno

Tell me what you think realism is - how you are here using it... Ontological realism (Platonism), Semantic realism, Quantificational or something else/combined? I've been pretty explicit that the 'reality' of numbers is little more than our ability to quantify over them.

...?

I don't think you understand what math realism is. — frank

Do you?

Well then, tell me. Say something. Commit.

@frank

π is not 3.1415926... but it is the ratio of the circumference of a circle to its diameter. — Banno

Compare your interpretation of quus. There are multiple ways for us to continue the sequence 3.1415926... but only one is π. This is were Kripke starts to slip.

Quus: scepticism arises if meaning is tied to finite behaviour alone.
π: determinacy is secured by publicly available rules and standards.

Here:

“Direct realism” is not a position that emerged from philosophers asking how perception is best understood, so much as a reaction to dialectical pressure created by a certain picture of perception, roughly: the idea that what we are immediately aware of are internal intermediaries, be they sense-data, representations, appearances, mental images, from which the external world is inferred.

Once that picture is in place, a binary seems forced: either we perceive the world indirectly, via inner objects; or we perceive it directly, without intermediaries. “Direct realism” is then coined as the negation of the first horn. It is not so much a positive theory as a reactive label: not that. This already suggests the diagnosis: the term exists because something has gone wrong earlier in the framing.

What those who reject indirect realism are actually rejecting may not be indirectness as such, but the reification of something “given” — an object of awareness that is prior to, or independent of, our conceptual, practical, and normative engagement with the world. Once you posit sense-data, qualia as objects, appearances as inner items, you generate the “veil of perception” problem automatically. “Direct realism” then looks like the heroic attempt to tear down the veil. But if you never put the veil there in the first place, there is nothing to tear down.

You see the cat. Perhaps you see it in the mirror, or turn to see it directly. And here the word "directly" has a use. You see the ship indirectly through the screen of your camera, but directly when you look over the top; and here the word "directly" has a use. The philosophical use of ‘indirect’ is parasitic on ordinary contrasts that do not support the theory. “Directly” is contrastive and context-bound, it does not name a metaphysical relation of mind to object, it does not imply the absence of causal mediation.

What you do not see is a sense datum, a representation, an appearance, or a mental image. You might well see by constructing such a representation, and all the physics and physiology that involves. But to claim that what you see is that construct and not the cat is a mistake.

One can admit that neural representations exist and denying that such things are the objects of perception. These neural representations are our seeing, not what we see. — Banno

He says we can talk about what goes in in the first 10,000 decimal places of pi, but it makes no sense to talk about the full extension. — frank

Where?

“The decimal expansion of π is not a completed object. It is an instruction for producing digits.” RFM I §32

“It is not as if all the digits were already there and we merely hadn’t yet discovered them.” RFM I §35

Wittgenstein is certainly not saying that talk of the value of π does not make sense. It does make sense to talk of the value of π. We do so all over mathematics. Consider: which digit are we not able in principle to determine? There is no digit that is in principle undeterminable; but there is also no completed totality of digits waiting to be surveyed.

The response is not to reify the procedure that produces each digit; yet π is a quantified value within mathematics. It figures under quantifiers, enters inequalities, is bounded, approximated, compared, integrated over, etc. None of that is in dispute, and none of it commits us to Platonism. π is quantified intensionally, via its defining rules and inferential role — not extensionally, as a completed set of digits.

π is not 3.1415926... but it is the ratio of the circumference of a circle to its diameter.

How is your catch of the day? Indispensability not such good bait?

So in what specific ways are you different from a platonist? — frank

:brow:

...platonism is the view that mathematical stuff, numbers and triangles and so on, exist independently of human minds, language, and thought, and are located outside of space and time. — Banno

Platonism is not just "numbers exist", as Meta supposes.

Why are you changing the topic back away from indispensability...?

I'm familiar with the article. What I am not sure of is how you see it as problematic for the account I gave.

Just to be clear, the indispensability argument gives us reason to commit to the existence of mathematical entities. The proffered account does just that.

So, where's the issue?

And this somehow shows my proposal is problematic?

If you were willing to set this out as an argument, rather than just wave at it, we might have an interesting discussion.

Failure to...

The straw man to which I referred is the one proffered as the only alternative to indirect realism, is the contentious "direct realism" of their imaginings.

So can you show, or even suggest, a problem with it? Something more than mere disparagement ?

Mathematical platonism is the view that mathematical stuff, numbers and triangles and so on, exist independently of human minds, language, and thought, and are located outside of space and time.

The proffered alternative is that mathematical statements are true, and we can talk about mathematical objects existing, but this doesn't require positing some separate realm outside space and time where numbers "live." Instead, mathematical language works the way it does - we can truly say "there is a prime number between 7 and 11" - without needing to tell some grand metaphysical story about what makes this true. The truth of mathematical statements is connected to their role in our practices, proofs, and language games rather than correspondence to abstract objects in a Platonic heaven.
This view preserves mathematical realism (mathematical statements have objective truth values) while avoiding the metaphysical commitments of Platonism (no need for causally inert, spatiotemporally transcendent entities).

Stay cryptic. It's your only defence.

If the rules of a single system contradict each other — Metaphysician Undercover

Which system? What contradiction?

That's education, learning the rules. — Metaphysician Undercover

Better, education is learning to use the rules. And the issue is, what can we do with the rules.

Opening up, instead of closing off.

I'm actually Socrates. — frank

Everyone here uses that excuse.

Baloney — frank

...as bait? Maybe. Oily, so it'll attract something...

You always are fishing. It's what you do. What I so rudely call "failure to commit".

Well, yes - conceptual clarification. It's the elimination of the muddle of "first-person phenomenal experience", a philosophical fiction, bringing with it metaphysical baggage that isn’t doing real explanatory work; it presumes a private, ineffable inner object that is unnecessary to account for perception. What remains, after discarding the fiction, is simple: we see apples (or tennis balls), not mental phantasms. The causal/physiological machinery can be acknowledged without turning it into a metaphysical object.

No, I'll leave the baiting to you.

It kind of looks like your alternative involves people walking into a fictional world and pretending it's all real, drawing conclusions based on it's reality, when they know good and well it's all a lie. — frank

That's were you live.

Well, finitism doesn't automatically reject set theory. Arguing in terms of 'isms' will not get us as far as setting out the detail. some might see ZFC or other set theories from a finitist perspective, treating infinite sets as symbolic devices or potentialities, without committing to their actual existence. Finitism rejects the Platonist reading of infinite sets, but I think I've shown that there is at least one alternative.

I'm happy to call him a finitist, for what that's worth - the interesting thing is how that plays out.

My contention - and I haven't put it together into a PhD yet, so it is incomplete - is that he lacked, or missed, the mechanism that allows us to move from a rule to a quantification, the "counts as" of the constitutive definition.

That's the direction taken by Austin, and then Searle, and a large part of why their work is worth considering alongside that finitism. We bring things into existence by with we do with words, in a way that Wittgenstein might not have recognised.

According to the SEP he was a finitist. — frank

Well, again, that needs some finesse:

Though commentators and critics do not agree as to whether the later Wittgenstein is still a finitist and whether, if he is, his finitism is as radical as his intermediate rejection of unbounded mathematical quantification (Maddy 1986: 300–301, 310), the overwhelming evidence indicates that the later Wittgenstein still rejects the actual infinite (RFM V, §21; Zettel §274, 1947) and infinite mathematical extensions. — Stanford

This is well worth working through, as well as was he right?

My contention - and I haven't put it together into a PhD yet, so it is incomplete - is that he lacked, or missed, the mechanism that allows us to move from a definition to a quantification, the "counts as" of the constitutive definition.

That's the direction taken by Austin, and then Searle, and a large part of why their work is worth considering alongside that finitism. We bring things into existence by with we do with words, in a way that Wittgenstein might not have recognised.

Wittgenstein would agree with this view, and it's why he rejected set theory. — frank

Platonism treats numbers as independently existing; psychologism treats them as things in the mind; Wittgenstein showed how they are a public, social practice.

W. didn't reject set theory What he rejected was both the platonic and psychological interpretations of set theory, together with the false antipathy that thinks we must choose one or the other.

Might be the encoding. a nine with a dot over it, marking repetition.

$\dot{9}$ = 999999...

I'll go back and edit.


Edit: Ah - it's an English/USA thing? - do you use $\overline{9}$ ? Interesting. Down another rabbit hole. This is the sort of of thing I was taught, and taught to others: Recurring Decimals.

Depends on whether the first symbolism is time dependent. Does counting actually require temporal steps. Can you think of 1,2,3 as instantaneous? Just speculating. — jgill

"1, 2, 3..." isn't rigorous, of course - there are many different ways to continue the sequence. ℕ is rigorous - well, at least more rigourous. So there is a sense in which they are the same only if the sequence is continued in a certain way... the rule is shown in the doing, as Wittgenstein put it.

Yet it is also stated in the definition of ℕ

However, this is not time-dependent. We understand what it would be to continue the sequence correctly or incorrectly, without doing so.

Really? 0.999... = 1 ?

Ask ChatGPT about the popularity of NSA. It is on target. — jgill

:smile:

And so maths is a game that never ends...?

So far as I can see, $0.\dot{9}$ is not an infinitesimal, but a real.

And again we must avoid mixing up a completed infinite definition with a process imagined as still going on.

I'm tempted to use a constitutive definition here, that 0.999... counts as 1.

I don't think that leads to any contradictions, and cleans things up nicely. But...?

No, the claim is that we do not directly see the tennis. We still indirectly see the tennis, much like when watching it on TV. — Michael

If you like. then it is the indirect realist who introduces "direct" and "indirect", and who is going to haver to explain their use.

The point about the "direct realist" being a straw man.

And back to Austin's account of seeing things directly and indirectly.

And again, when you sit in the front row of the Rod Laver Arena, you are not seeing some mental phantasm of the tennis.

Wittgenstein understood set theory is platonism — Metaphysician Undercover

What nonsense. Platonism treats mathematical propositions as descriptions of independently existing objects; psychologism treats them as reports of mental acts. Both misunderstand mathematics, which consists in public techniques governed by rules.

Your potted history is inaccurate; but any so brief account will be. Leibniz, Euler, and even Newton routinely identified infinite sums with finite values. They did not treat limits as approximations, indeed they did not systematically distinguish approximation from equality in the modern sense. For them, infinitesimals existed as actual quantities—smaller than any finite number but not zero, and hence dy/dx was literally a fraction. Limits reconceptualised these by showing that for every ε > 0 (no matter how small), there exists some δ > 0 such that whenever 0 < |x - a| < δ, we have |f(x) - L| < ε.

Now some students see the ε-δ definition and think: "For any ε > 0, we can get within ε of L. So we can get arbitrarily close to L. But we never actually reach L. Therefore the limit only approximates L"

Here the student misses the quantification: For All ε > 0, the value is within ε of L.
This isn't saying we can get close. It's saying we can get closer than any specified distance. The universal quantifier ∀ε is doing the heavy lifting. And bear in mind that ∀x F(x) is ~∃x ~f(x). This is an existential statement: there is a limit; it has a specific value, and is not an approximation.

This is a pedagogic point, not a mathematical one - and one that I learn is brushed over in the schools of engineering, perhaps because their use of rounding is so routine.

And to that history we can add the return of infinitesimals, this time with a firm foundation, in the development of non-standard analysis and hyperreal numbers, *ℝ, whcih includes positive numbers smaller than any standard real, but not zero.

For two centuries, students were told: "Leibniz and Euler were being sloppy. Infinitesimals don't exist. Here's the rigorous way (ε-δ)." Then Robinson showed "Actually, infinitesimals exist just fine. The old guys were on to something."



$0.\dot{9}$ really does equal 1.

An interesting read, bringing us back to the Stoics in the context of disability.

The Post Paralysis Peace Paradox

I don't think it's fire-walled... let me know.

Neither has it been shown that something goes wrong in practice if we treat a convergent series as unequal to the limit by an infinitesimal amount. — frank

Can you set this out clearly, so we can see what you are claiming?

This argument that "we see tennis, even if on TV; therefore direct realism is true" is ridiculous. — Michael

Quite so. And it's not what was argued. The indirect realist makes the ridiculous claim that even when you are at the Rod Laver Arena, you do not see the tennis, but an image of the tennis. That we do not have "direct" perception of the tennis when watching it on the screen, or in the front row.

First a small point. If mathematics is a practice, as I have argued here, then it's not a surprise that one might changing from a recursive approach that is able to treat infinite sequences as a whole, to sequences of choice that do not.

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.

This in contrast to the Wittgensteinian approach, ontological questions dissolve into grammar and use.

What has not been shown is that something goes wrong, concretely, in classical practice if sequences are treated as completed totalities.

Yep. Yet the limit is not something the sequence is chasing, but a property of the sequence as a whole...?

...you should cut me some slack... — frank

Meh. You seem more interested in the drama than the maths.

You're mostly just playing sillybuggers as it stands.

I enjoy these chances to exercise my math muscles a bit more directly than usual, — Srap Tasmaner

Yes! What I'm finding interesting here are the links to set theory and first order logic, but it's a strain to recall the little undergrad calculus I did study.

I've tried to present my working as explicitly as possible - and ChatGPT is invaluable here, for both checking arguments and formatting Mathjax. I'd have hoped that if there were real objections, the objector would at least take the trouble to set them out formally.

Ok. We can drop it. — frank

Failure to commit. Again.