It beats me why people want to deny that there is a physical environment that we share with others, as well as with other kinds of physical entities, who perceive pretty much the same persistent features of the environment as we do. — Janus

One can agree that in very general terms higher animals perceive pretty much the same persistent features of the environment as we do without having to then conclude that there is such a thing as a ‘physical’, organism -independent basis for this commonality. Analytic philosophers fou sit necessary to jettison the ‘myth of the given’ , the idea that we directly perceive the stuff of the world unmediated by our own schemes . Phenomology didn’t deny that we perceive an ‘out there’. They only denied that the ‘out there’ come packages as physical stuff. Enactivists say that each organism co-creates its environment in relation to its needs , goals and aims as an ongoing environmental
process. So each specie’s world is in some sens u inquest to its own functional goals.

counting is something we do, not something we discover.

Its a way of talking about stuff. The way of talking is made up. The stuff isn't. — Banno

But there must first be a peculiar way of thinking about stuff before it makes sense to calculate and measure. We must assume stuff is objectively self -persistent in some fashion. It would make no sense to ‘count’ some variable aspect unless that aspect belonged to something that did not vary during the counting. So even the ‘stuff’ rendered as identically self-persisting is made up (an idealization)

What?

If what you are saying is that there must be something to count before one counts, then... well, sure, but I don't see the relevance.

The point I’m labouring is that there are real intelligible objects. — Wayfarer

Are we talking about spoons here, or that there are five spoons?

I don't see what work "real" is doing in your post.

One can agree that in very general terms higher animals perceive pretty much the same persistent features of the environment as we do without having to then conclude that there is such a thing as a ‘physical’, organism -independent basis for this commonality. Analytic philosophers fou sit necessary to jettison the ‘myth of the given’ , the idea that we directly perceive the stuff of the world unmediated by our own schemes . Phenomology didn’t deny that we perceive an ‘out there’. They only denied that the ‘out there’ come packages as physical stuff. Enactivists say that each organism co-creates its environment in relation to its needs , goals and aims as an ongoing environmental
process. So each specie’s world is in some sens u inquest to its own functional goals. — Joshs

If there were no "physical organism-independent basis for this commonality" then what would explain the commonality? A universal mind? The rejection of the myth of the given is not a rejection of perception independent invariances, but a rejection of the idea that the way different percipients apprehend those invariances is completely independent of their various cognitive faculties.

"What is the Myth of the Given?
Wilfrid Sellars, who is responsible for the label, notoriously neglects to explain in
general terms what he means by it. As he remarks, the idea of givenness for knowledge,
givenness to a knowing subject, can be innocuous.
1 So how does it become pernicious?
Here is a suggestion: Givenness in the sense of the Myth would be an availability for
cognition to subjects whose getting what is supposedly Given to them does not draw on
capacities required for the sort of cognition in question.
If that is what Givenness would be, it is straightforward that it must be mythical.
Having something Given to one would be being given something for knowledge without
needing to have capacities that would be necessary for one to be able to get to know it.
And that is incoherent".

From here: file:///C:/Users/dynam/AppData/Local/Temp/mcdowell-Avoiding-the-Myth-of-the-Given1.pdf

So even the ‘stuff’ rendered as self-persisting is made up (an idealization) — Joshs

1. Babies show recognition of object constancy at 8 months.

2. They may or may not be able to speak at that age, but their rationality is definitely limited.

3. So maybe what we're seeing in their behavior isn't ideas, but instinct.

4. The problem is that the concept of instinct requires a backdrop of object constancy.

5. That means I'm actually asserting that my own assertions arise from instinct rather than knowledge.

C: There's an impending collapse of sense here.

And yet that could happen if they thought that the max compressive resistance of their concrete is say A, but also 2*A, and also 329*A. If we allow contradictions free reign in mathematics, everything follows.

I am wondering now whether I should have said I accept the following:

The consistency of certain systems (PA and the like) cannot be constructively proved by any means. — sime

The consistency of, for example, PA cannot be proven by finitistic means, but I don't know whether it can't be proven by constructive though non-finitistic means. First, to pin the question down exactly, we would need an exact mathematical definition of 'constructive'. Second, even if we say, for sake of argument, we mean a particular constructive set theory or other constructive theory, then we would have to prove that such theories do not prove the consistency of classical PA.

As to Gentzen's proof, if I am not incorrect, we can describe the situation in two ways:

Let 'T' stand for transfinite induction on e_0.

(1) PRA+T |- Con(PA)

In other words, by finitistic means plus transfinite induction on e_0 we prove that PA is consistent.

vs

(2) PRA |- Con(PRA+T) -> Con(PA)

In other words, by finitistic means we prove that Con(PRA+T) implies Con(PA).

It is not, at least to me, apparent that those are not constructive (I don't know whether they are or are not constructive). However, I did find a paper that mentions that T is constructively challenged by some writers (actually, it's not T that's discussed but an alternative assumption used in place of T).

But you also say:

Gentzen proved that the inconsistency of PA implies the inconsistency of PRA + transfinite induction on the ordinals. — sime

That is of the form ~Con(PA) -> ~Con(PRA+T). But, at least prima facie, that does not intuitionistically imply Con(PRA+T) -> Con(PA). Yet, the latter version is the one we more often see. So I don't know why you stated Gentzen as ~Con(PA) -> ~Con(PRA+T).

Are we talking about spoons here, or that there are five spoons? — Banno

Five of anything is 'five'. We're talking about (among other things) numbers - whether they have any reality outside individual minds, outside of the individual act of counting. That's what 'real' means. Anti-realists will say no. Realists will say yes.

I don't see what work "real" is doing in your post. — Banno

It's a discussion about mathematical realism, isn't it?

We're talking about (among other things) numbers - whether they have any reality outside individual minds, outside of the individual act of counting. — Wayfarer

It's that formulation that is at issue, that very division that presupposes things and minds, the notion that there is an inside and an outside.

I'd like to do an analysis of "realism" to see when it came into use. I suspect it is a reaction to idealism; that realism wasn't needed until the nonsense of idealism came about.

https://www.etymonline.com/search?q=realism

https://www.wolframalpha.com/input/?i=realism

https://www.inphoproject.org/idea/480.html

And yet that could happen if they thought that the max compressive resistance of their concrete is say A, but also 2*A, and also 329*A. If we allow contradictions free reign in mathematics, everything follows. — Olivier5

And yet I can't see that there could be, on account of there being some inconsistencies or paradoxes in certain areas of math, any reason for them to say such kinds of things.

it is said that 0.9999999999999... equals 1, because .33333333333333... equals one third and three thirds equal one. — Janus

That is not a rigorous mathematical proof. However, there is a rigorous mathematical proof that .999... = 1.

'.999..' is an informal way of describing a certain infinite summation. And infinite summation is defined by convergence. And we prove that the sequence converges to 1.

No offence, but I think you pay too much attention to naïve materialists. — Olivier5

I've had many a discussion with contributors here who are convinced that ideas are essentially 'brain structures', and that the brain is shaped by evolutionary adaptation according to Darwinian principles to operate in a way that is advantageous for survival. Do you think that is a view held by a minority of academics?

Yes, I saw that myself when I realized that 0.333333333 is no more equal to one third than 0.9999999 is equal to 1, so I deleted it, probably while you were responding to my unedited post.

Your proof from convergence seems no different though, since 0.999999 can converge infinitely with 1 out ever reaching it, just as 0.33333 converges infinitely on one third, which seems to be exactly the point against the argument I deleted.

I'll leave my post as it is so that your following self-correction makes sense in context.

:up:

I'd like to do an analysis of "realism" to see when it came into use. — Banno

From your #2




Notice the difference between 3 and 5. 3 is scientific realism, today's realism, what people mean by realism in ordinary speech. 5 is scholastic realism, realism concerning universals. Mathematical realism is nearer to (5) and (3) and tends to contradict (3), because it is assumed by (3) that numbers can't exist outside of minds, and because another axiom of (3) is that reality is what exists independently of any mind.

0.999999 can converge infinitely with 1 out ever reaching it — Janus

That's not the point.

'.999...'
is informal for

SUM[n = 1 to inf] 9/10^n

And SUM[n = 1 to inf] 9/10^n is the limit of a sequence. And that limit is 1.

/

By the way, when you wrote, "0.333333333 is no more equal to one third than 0.9999999 is equal to 1", I glanced over it too quickly and took you to mean that we can't assume .333... = 1/3 any more than we can assume .999... = 1". That is correct. But we can go on to prove both that .999... = 1 and that .333... = 1/3. We don't assume them, but we do prove them.

OK, what you said you thought I meant is correct, but as for the proof, my math is not very sophisticated, so I'll have to take your word for it.

So, if the proof is a proof, then it seems that we do have an inconsistency, and my argument that it has no bearing on structural engineering would stand.

(Note that when I say we have an inconsistency I just mean that the intuition that .9999 does not equal 1 is inconsistent with any proof that the two are equal).

Inconsistency, in a formal sense, is not a clash between a theorem and the fact that certain people have a different intuition.

The context in which .999... = 1 is classical mathematics as introduced ordinarily in Calculus 1 then more rigorously explicated in Real Analysis, especially made rigorously axiomatic by the set theoretic development of the real numbers. The infinite sum as the point of convergence of a sequence is what mathematicians MEAN by .999... Saying, in face of that context, that .999... does not equal 1 is not informed intuition but rather ignorance of the actual mathematics. Claiming, without mathematical basis, that .999... does not equal 1 is claptrap that needs to be remedied by study of the basics of the subject. (I am addressing here only what is ordinarily meant in mathematics, as I recognize that there are other approaches including finitistic views and computationalist views).

when I say we have an inconsistency I just mean that the intuition that .9999 does not equal 1 is inconsistent with any proof that the two are equal — Janus

That's not what mathematicians, including Turing, mean by 'inconsistency'. That's not (as far as I can tell) a sense of inconsistency at issue with the "bridge collapses" issue mentioned in connection with Turing, which is the sense of a system proving a formula and its negation, thus, by the principle of explosion, the system proving every formula whatsoever in the language of the theory.

What?

If what you are saying is that there must be something to count before one counts, then... well, sure, but I don't see the relevance. — Banno

One must make certain assumptions about this something: namely that it stays put as what it is, that it is res extensia, that it has duration. But nothing in the world stays put as what it is. Moment to moment it transforms itself ever so subtly. Self-identicality is an illusion of sorts. This is my point, as stated differently by Husserl and Heidegger:

For Husserl, extension, duration and magnitude are all implied by the idealizing thinking of self-identical objects. The ideal geometry of a line made possible the empirical intuitions pertaining to various characteristics of number.


“A true object in the sense of logic is an object which is absolutely identical "with itself," that is, which is, absolutely identically, what it is; or, to express it in another way: an object is through its determinations, its quiddities , its predicates, and it is identical if these quiddities are identical as belonging to it or when their belonging absolutely excludes their not belonging. Purely mathematical thinking is related to possible objects which are thought determinately through ideal-"exact" mathematical (limit-) concepts, e.g., spatial shapes of natural objects which, as experienced, stand in a vague way under shape-concepts and [thus] have their
shape-determinations; but it is of the nature of these experiential data that one can and by rights must posit, beneath the identical object which exhibits itself in harmonious experience as existing, an ideally identical object which is ideal in all its determinations; all [its]
determinations are exact —that is, whatever [instances] fall under their generality are equal—and this equality excludes inequality; or, what is the same thing, an exact determination, in belonging to an object, excludes the possibility that this determination not belong to the
same object.”

“ In this sphere of magnitudes, and initially of spatial magnitudes—first of all in classes of privileged cases (straight lines, limited plane figures, and the corresponding cases of spatial magnitudes), first of all in the empirical intuition that magnitudes divide into equal parts and are composed again of equal parts—or of aggregates of like elements which decompose into
partial aggregates and can be expanded into new aggregates through the addition of elements or
of aggregates of such elements—in this sphere, there arose the "exact" comparisons of magnitudes which led back to the comparison of numbers. Upon the vague "greater," "smaller," "more," 'less," and the vague "equal" one could determinately superimpose the exact "so much" greater or less, or "how many times" greater or less, and the exact "equal."
Every such exact consideration presupposed the possibility of stipulating an equality which excluded the greater and the smaller and of stipulating units of magnitude which were strictly substitutable for one another, were identical as magnitudes, i.e., which stood under an identical concept or essence of magnitude.”

“Thus it was possible to conceive of processes converging idealiter through which an absolute
equal could be constructed ideally as the limit of the constant approach to equality, provided that one member [of the system] was thought of as absolutely fixed, as absolutely identical with itself in magnitude. In this exact thinking with ideas one operated with ideal concepts of the unchanging, of rest, of lack of qualitative change, with ideal concepts of equality and of the
general (magnitude, shape) that gives absolute equalities in any number of ideally unchanged and thus qualitatively identical instances; every change was constructed out of phases which were looked upon as momentary, exact, and unchanging, having exact magnitudes, etc.”

Husserl and Heidegger share a focus on Galileo as originator of modern mathematical science based on an idealization of geometric spatio-temporality as objective bodies in causal interaction. Heidegger traces the origin of empirical science to the concept of enduring substance.

“Mathematical knowledge is regarded as the one way of apprehending beings which can always be certain of the secure possession of the being of the beings which it apprehends. Whatever has the kind of being adequate to the being accessible in mathematical knowledge is in the true sense. This being is what always is what it is. Thus what can be shown to have the character of constantly remaining, as remanens capax mutationem, constitutes the true being of beings which can be experienced in the world. What enduringly remains truly is. This is the sort of thing that mathematics knows. What mathematics makes accessible in beings constitutes their being. Thus the being of the "world" is, so to speak, dictated to it in terms of a definite idea of being which is embedded in the concept of substantiality and in terms of an idea of knowledge which cognizes beings in this way. Descartes does not allow the kind of being of innerworldly beings to present itself, but rather prescribes to the world, so to speak, its "true" being on the basis of an idea of being (being = constant objective presence) the source of which has not been revealed and the justification of which has not been demonstrated. Thus it is not primarily his dependence upon a science, mathematics, which just happens to be especially esteemed, that determines his
ontology of the world, rather his ontology is determined by a basic ontological orientation toward being as constant objective presence, which mathematical knowledge is exceptionally well suited to grasp.”

If there were no "physical organism-independent basis for this commonality" then what would explain the commonality? A universal mind? — Janus

The basis for this commonality is a reciprocal causality operating between organism and environment . Since other organisms belong to each organism’s environment , there is a complex web of interaction taking place at every level, between the mind and body , between body and environment , and between organisms in a wider environment. All of these levels are nested within one other. The result is that each individual has their own perspective on a world that they share with others. The difference between this enactivist model and physicalism is that the latter creates commonality by correspondence with a presumed already existent reality. The former, however , see the real world not as pre-existing and independent of sense-making organisms , but as a co-produced continual development pragmatically inseparable from an organism’s goals and needs. To ask if a thing exists in the world is to ask how it is pragmatically relevant to my ongoing direction of functioning, how it is useful to me. These are not separate questions but the same question.

Odd, how folk use Husserl and Heidegger as if they made things clear.

I need a translation.

Imagine that every moment of time things changed. Yeah, you reply, I already get that. So what? What does this have to do with mathematics? No, I mean, not that some aspect of a thing changes, but that everything about the world shifts in some subtle fashion every moment, so that there is no remainder, nothing left behind to compare to what changes , nothing that allows us to say, this change happens against such and such a backdrop of stability. What if we all are dropped into a subtly new world every moment? This is the basis of Husserl’s and Heidegger’s thinking. Theirs is a radically temporal , radically pragmatic orientation. Every event is a carrying forward and a transformation of a prior world of referential relations. if you start with such a premise , and take a look at the modern empirical notion of objects as presently occurring entities with duration it should strike you that at some point someone decided to ‘pretend’ that this constantly flowing, changing pragmatic unfolding of world froze itself into ‘objects’ with duration and extension.

But that's not how things are.

It wasn't "decided"; it's what we do. And we are able to do so because of how things are.

having done so , what can we conclude about the status of ‘truth’? Can we save some sense of it that doesn’t get sucked down into the relativity of use? Is ‘true’ just another thing we say in certain contexts for certain purposes? — Joshs

Just as objects are 'fictions'/inventions (eddies in the stream), so perhaps are meanings?

materiality is already ‘conceptual’ through and through in that the very notion of an empirical object is a complex perceptual construction , an idealization. Furthermore , it is this idealizing abstraction at the heart of our ideas of the spatial object that makes the mathematical
possible. They are parasitic on and presuppose each other. — Joshs

That sounds correct, but would you agree that the conceptual is social and 'material'? So that the game of reducing one to the other is perhaps futile? Or that the game is at least fundamentally impractical inasmuch as the distinction between conceptual and the material is not likely to lose its intense utility?

Math is too serious a matter to be left to philosophers. — Olivier5

:up:
But what we make of that math does seem to be a matter for philosophers (and everyone else, really.)

OK, thanks; I guess I misunderstood.

Every event is a carrying forward and a transformation of a prior world of referential relations. if you start with such a premise , and take a look at the modern empirical notion of objects as presently occurring entities with duration it should strike you that at some point someone decided to ‘pretend’ that this constantly flowing, changing pragmatic unfolding of world froze itself into ‘objects’ with duration and extension. — Joshs

That seems right on some level, but this game of pretend, presumably evolved, is no so easily shaken off. What's an event? Does it involve objects? 'Transformation' implies some thing that is transformed, maintaining its identity in some sense. As another poster has mentioned, this kind of point threatens to 'deconstruct itself,' which is not necessarily a bad thing.

The difference between this enactivist model and physicalism is that the latter creates commonality by correspondence with a presumed already existent reality. — Joshs

I get that the world of objects perceived in common is a relational, interactive world. But I don't think the commonality of the different organisms cognitive :machinery' is enough to explain the fact that we all, animals and people see the same objects in the same locations. Beyond all the perceptions of the world I think we have every reason to believe there is a world that is perceived; that is what it is regardless of how it is perceived and would be there just as it is (as it is, not as it is perceived, mind) in the absence of any percipients. We have every reason to believe that because it is the best explanation for a shared world; in fact it is the only explanation apart from some form of idealism; some notion that all minds are somehow conjoined or that there is a universal mind we all partake in..