But they don't involve any incommensurability, right? I'm trying to understand the connection. — Snakes Alive

I'm not entirely sure what you mean by 'involve any incommensurability'. The idea is that if one assumes commensuribility, then the Zeno paradoxes are one of the results of that assumption when one tries to construct a continuum from discontinuities.

The 'cut' issue Wittgenstein is highlighting is a feature of the Dedekind cut construction, with that formalism it isn't immediately clear that the irrational numbers are a dense set in the number line (which means there's an irrational arbitrarily close to every other number), whereas the sequential construction presents the density of the irrationals in the reals in a natural way; it piggybacks on top of the density of the rationals in the real numbers; which is clear from how the decimal representation of numbers works.

...Something under-appreciated about the mathematics of limits, which shows itself in the enduring confusion that 0.99... isn't equal to 1, is that when you're evaluating the limit of something; all the steps to reach that limit have already happened. So 'every finite step' misses the mark, as the limit is characterised as after all steps have been done. This means when you characterise irrationals as convergent sequences of rationals, the infinity of the sequence iterates has been 'done in advance'. If you truncate the series at some point you obtain an approximation to the limit, but really the limit has 'already been evaluated' as soon as you write it down. Similarly, you can conjure up the reals by augmenting the rationals with all such sequences; since the sequences have already terminated in advance. — fdrake

One of the things I like about Wittgenstein's approach is that he 'accepts' the number line on the condition that it isn't leveraged to ground a 'general theory of real numbers' (W: "The picture of the number line is an absolutely natural one up to a certain point; that is to say so long as it is not used for a general theory of real numbers"). I think one of the things he's getting at is the conditional nature of the (dense) number line: if you want to construct irrationals from the rationals, then and only then will your number line be dense. What he worries over is the 'transcendental illusion' of confusing effect for cause: of saying that it is because the number line is dense that one can make cuts and thus construct the Reals (and with them, the irrationals).

This is why he makes a distinction between 'geometric illustration' and 'geometric application': the number line is a geometric illustration that 'falls out' of an arithmetic procedure (the Dedekin cut), but it is not the case that the number line 'really is there' in any Platonic-realist sense which is then 'discovered' in the course of employing the cuts (Witty again for reference: "The geometrical illustration of Analysis is indeed inessential; not, however, the geometrical application. Originally the geometrical illustrations were applications of Analysis. Where they cease to be this they can be wholly misleading"). Instead, the cuts retroactively 'makes' the Real number line the dense continuum that is seems to be.

(I'm again at my limit of mathematical understanding here, but I wonder if there is a similar issue when one moves 'one level up' from the Reals to the Complex numbers: is there an arithmetic procedure analogous to Dedekind cuts that can generate Complex numbers that, as it were, retroactively makes the Complex plane (?) ... dense(?) (again, I think my vocabulary is probably wrong and that we don't speak of the plane being 'dense', but y'know... filled with 'arbitrarily close' numbers... whatever it's called on the Complex plane). And then maybe reiterated for the quaternions and octonions and so on and so on? - idk if it works like that?).

Anyway, the point is that yeah, I totally get that it's not quite fair to speak of 'every finite step' because the evaluation of the limit posits the steps as 'always-already' completed, but I guess what I'm trying to say is that this is precisely the (possible) problem: it's insofar as limit procedures already 'presuppose', in advance, the number line, that one can misleadingly take it that all the numbers are just 'there' in some Platonic fashion, just waiting to be discovered (whereas Witty emphasises that it is only the application of the math which determines, reteroactively, as it were, what the math itself looks like. Again, sorry if I'm getting my terms mixed up, but I hope the sense of comments come through.

Not sure that does answer my point, which is about infinite divisibility. It's the divisibility of the line that creates the numbers - IOW it's the notionally zero-width cut that separates the continuum into parts that creates the numbers, it's not that you're building up a bunch of nothings into a something, as with Zeno.

And so long as you can do that, you can find a common measure. — gurugeorge

The above can serve as something of a reply to this as well, I hope. Again, it's not that there is a continuum 'already there', waiting to be cut: rather, the cut and the continuum are co-extensive insofar as the one presupposes the other; the cut extends math itself and does not just apply to 'existing' math. This is what I was trying to get at in my initial post to you when I said that the number line is something of a fiction.

Yeah, it's a question of ratios: the ratio of the sides to the hypotenuse can never be made commensurate (which means that the 'size' of the unit isn't actually relevant: only the proportion between them - and that stays invariant).

Just briefly - about to sleep - one of the things Rosen develops is that - to continue the balloon analogy - while it's true that Weirstrass developed a method to, as it were, contain the Zeno paradoxes, mathematical artefacts end up showing up 'one level up' again ('the air gets displaced in the balloon'). It's at this point I stop being able to follow the nitty gritty of the math, but here's where he ends up going:

"In the context I have developed here, the transition from rationals to reals gets around the original Zeno paradoxes, mainly because the reals are uncountable, shocking as this word might be to Pythagoras. In particular, a line segment is no longer a countable union of its constituent points, and hence, as long as we demand only countable additivity of our measure [lambda], we seem to be safe. We are certainly so as long as we do not stray too far from line segments (or intervals); i.e., we limit ourselves to a universe populated only by things built from intervals by at most countable applications of set-theoretic operations (unions, intersections, complementations). This is the universe of the Borel sets.

But if R is now the real numbers ... we are now playing with 2^R. Thus we would recapture the Pythagorean ideals, and fully embody the primacy of arithmetic... if it were the case that everything in 2^R were a Borel set. We could then say that we had enlarged arithmetic “enough” to be able to stop. Unfortunately, that is not the way it is. Not only is there a set in 2^R that is not a Borel set, but it turns out that there are many of them; it turns out further that there are so many that a Borel set is of the greatest rarity among them. It is in fact nongeneric for something in 2^R to be Borel". I'll send you the chapter, and, perhaps, if you have the time you might see where this goes (basically you get a progression form irrationals -> Zeno -> Borel sets as nongeneric -> ... -> Godel; Rosen also mentions Banach-Tarski as another instance of an artefact produced by the assumption of commensuribility).

So I guess what I wonder - genuine question - is whether the paradoxes of intuition are only seen to be paradoxes of intuition in the light of now-established mathematical development, so that what once seemed to be a paradox of formalism becomes a paradox of intuition after we come up with a way to develop the math a little (usually by 'enlarging the arithmetic'). I'm wondering because if it's possible to view the development of math (so far) as the development of ever more ingenious ways of burying incommensurability through higher-order abstractions for the sake of upholding commensuribility, then at the limit, would it be possible to say that the distinction between paradoxes of intuition and paradoxes of formalism are just the recto and verso of the same (ultimately) unsurpressable undertow of incommensuribility, just seen from different temporal angles? (as in - PoI 'after' mathematical development, PoF 'before' development). Hope this question makes sense.

They're all variations on the same theme of constructing a continuum out of the discontinuous.

Just to round off my references, compare also Deleuze:

"In the history of number, we see that every systematic type is constructed on the basis of an essential inequality [read: incommensurability - SX], and retains that inequality in relation to the next-lowest type: thus, fractions involve the impossibility of reducing the relation between two quantities to a whole number; irrational numbers in turn express the impossibility of determining a common aliquot part for two quantities, and thus the impossibility of reducing their relation to even a fractional number, and so on. It is true that a given type of number does not retain an inequality in its essence without banishing or cancelling it within the new order that it installs. Thus, fractional numbers compensate for their characteristic inequality by the equality of an aliquot part; irrational numbers subordinate their inequality to an equality of purely geometric relations - or, better still, arithmetically speaking, to a limit-equality indicated by a convergent series of rational numbers". (Difference and Repetition).

Given the existence of irrationals, isn't the point made here already accepted? The existence of irrationals has been known since ancient times, as you say.

How does the Pythagorean doctrine of commensurability lead to Zeno's paradoxes? — Snakes Alive

Doesn't the principle just fall out as a corollary of the infinite divisibility of length (or any other measurable relation)? — gurugeorge

AFAIK the number line is complete (wholes, rationals, integers, rationals, and irrationals). We even have the real-imaginary number space. — TheMadFool

Just wanna come back and address these together as they all hit on similar points that I think deserve to be expanded upon. The idea as I understand it is this - there is in fact one way to 'save' the assumption of commensurability after the introduction of the irrationals, and it is this: to treat irrationals as the limit of a convergent series of rational numbers. In this way, we don't actually have to deal with incommensurate values per se, only rationals (Rosen: "At each finite step, only rationalities would be involved, and only at the end, in the limit, would we actually meet our new irrational. This was the method of exhaustion...")

In the last century, this was formalized with the procedure of 'Dedekind cuts', which enable the construction of the real numbers (irrational + rational numbers) from the rational numbers alone. One 'side-effect' of constructing the irrationals in this way was to definitively construe the number line as continuous (a 'gapless' number line). The idea is basically that simply by initiating a procedure of step-by-step counting, one can eventually arrive at an irrational at the limit of that process.

However - and here we get to Zeno - the attempt to 'save' commensuribility in this way simply pushes the problem back a step, rather than properly solving it. For what Zeno points out is that even if you add up the points on a number line to arrive at an irrational, no single point itself has any length, and that adding a bunch of lengthless points cannot itself yield any length (which in turn allows one to make wild (paradoxical) conclusions like √2 = 0).

So what the Zeno paradoxes essentially mark is the irreducibly of incommensuribility. Making the irrationals the limit of a converging series of rationals in order to save commensuribility is a bit like trying to suppress a half inflated balloon: short of breaking the balloon, all one can ever do is shift the air around. One of the take-aways from this is that the very idea of the (continuous) number-line is a kind of fiction, an attempt to glue together geometry and arithmetic in a way that isn't actually possible (every attempt to 'glue' them together produces artifices or problems, either in the form of irrationals, or later, in the form of Zeno's paradoxes - and, even further down the line, Godel's paradox).

[Incidentally this is something that Wittgenstein was well aware of: "The misleading thing about Dedekind’s conception is the idea that the real numbers are there spread out in the number line. They may be known or not; that does not matter. And in this way all that one has to do is to cut or divide into classes, and one has dealt with them all. ... [But] the idea of a ‘cut’ is one such dangerous illustration. ... The geometrical illustration of Analysis is indeed inessential; not, however, the geometrical application. Originally the geometrical illustrations were applications of Analysis. Where they cease to be this they can be wholly misleading." (Wittgenstein, Lectures on the Foundations of Mathematics)

Compare Rosen: "The entire Pythagorean program to maintain the primacy of arithmetic over geometry (i.e., the identification of effectiveness with computation) and the identification of measurement with construction is inherently flawed and must be abandoned. That is, there are procedures that are perfectly effective but that cannot be assigned a computational counterpart. In effect, [Zeno] argued that what we today call Church’s Thesis must be abandoned, and, accordingly, that the concepts of measurement and construction with which we began were inherently far too narrow and must be extended beyond any form of arithmetic or counting.]

@fdrake I wonder if this story sounds right to you. I've struggled somewhat to put it together, from a few different sources.

Necrosophy? Wisdom of... dead things?

"The physical world cannot be separated from our own efforts to probe it. — Snakes Alive

Yeah, this is rubbish. Or at least, it does not follow.

You understand the OP and Kant differently from I then.

That measurement changes the object, insofar as it's being measured, means that we can't ever see something except qua measured, and so can't know what it is independently of that measurement occurring. — Snakes Alive

But this seems to be an incoherency: what you want to ask is something like: how can we know something that, in principle, be known? But this is not a question that makes sense. The idea seems to be that there is some set of knowable-things that, in principle, could be known if only we could... know it otherwise than though how we come to know things; but that's not how knowledge - or really anything - works. It's a category error to expect knowledge to be 'operative' independently of the kinds of thing which make it knowledge. It's not 'you can't know'; it's 'you're using the word knowledge wrong'. It's the attempt to wring blood out of an (oxymoronic) stone.

Reworded: Measurement changes the object, insofar as it's being measured, means that we can't ever see something except qua measured, and so... it makes no sense to speak of knowledge otherwise than this.

what does follow is that the only thing measurement ever allows us to partake in are those things that are ontologically dependent on it — Snakes Alive

But how is this not just a fancy way of phrasing a tautology (and thus a triviality)? 'The only things we can measure are those things that we can measure.' Once you deny the 'constitution' route, that's all one is really left with.

Ah right, so Nietzsche was like the Nazis because they had criteria of 'political exclusion'. But of course, all politics is nothing other than the negotiation of political criteria of exclusion. Mundane crap parading as insight.

When we seek knowledge of things, we don't generally think of ourselves as just reporting how the measured thing affects the measurement apparatus — Snakes Alive

This 'just' here is doing a lot of work, and it hides over an equivocation: a measurement on its own doesn't really yield any knowledge whatsoever; a measurement is nothing other than a motivated physical interaction - and it's the motivation that matters. One observes particle decay in the ATLAS detector: what is its significance? Well, you need to place that observation within a theoretical framework in which it has a place (even if that place is, at it were, 'out of place' - perhaps this means you need a new theory, etc). But measurement qua measurement is just setting things up to bump into each other in a certain way. Ideally this bumping tells you something about the phenomena, if you know what you're looking for: if it bumps in this way and not that, it means...; otherwise, it means...; significance is never given in the mere observation itself.

the universe is, you say, outside of our measuring it, but all we ascertain in measurement are our interactions with it! — Snakes Alive

But this is a false dichotomy through and through. All we ascertain in measurement are our interactions with it, yes, but who gives a fuck about us? Why are we the 'measure or measure', as it were? It's only by elevating measurement itself into a transcendental principle of 'universe-constitution' that you can get the kind of idealist peddling that Wayfrer would like. But that's unwarrented, unscientific hogwash.

But if you say all properties of the thing we're interested in are relative to the measurement in this way (what it is to have that property just is to look a certain way under certain conditions), then this collapses into Kantianism — Snakes Alive

Consider modifying this bold bit to something like: 'what it is to have that property is just to be interacted with in a certain way under certain conditions". Is this a Kantianism? But there are no noumena here: the idea is that all properties are relational in this way: any interaction whatsoever will yield a 'result' appropriate to that interaction. There's only a Kantianism if one tries to substaintialize an 'object' apart from these interactions (like a 'red' without the conditions of 'red': a nonsense). There's nothing special about measurement. If there were no measurement, the universe would still be there, quite independent of it, insofar as measurement is just a subclass of physical interactions, which take place all the time, everywhere. The charge of Kantianism only holds if measurement is not understood to belong to the larger class of physical interactions - that is, if you exceptionalize measurement. But this is just what the QM shows to be false: we are no different to anything else in the universe.

How am I doing this? — Snakes Alive

By assuming this entails Kantianism.

"Us" would be whoever's measuring. — Snakes Alive

And what is measuring? A: A physical interaction.

Very well – but if you say all properties of the thing we're interested in are relative to the measurement in this way (what it is to have that property just is to look a certain way under certain conditions), then this collapses into Kantianism. We deal with phenomena, and we learn about things only that they look certain ways to us. — Snakes Alive

But what is the status of 'us'? You're treating 'us' as an exception that is somehow different from 'everything else'; but this is just what is unwarranted.

we are not recording what the thing is independent of that measuremen — Snakes Alive

What is this 'the thing' 'independent of that measurement'? It's like asking 'what does red look like in the absence of light?': it's not that it looks other than it does in light, it's that the question is wrong. It misunderstands what it means to be red, which is just to look like that in the presence of light. That doesn't make it an 'artefact': good measurements really do capture something of what is so measured: red 'really does' look like that. But: 'what does it look like in the absence of a look?' - this is just a badly formed question, leading to false puzzles.

In other words, if you take seriously the idea that measurements only reveal things qua measured, and there's no reason to think that it's possible to measure things qua things as they are independent of or prior to measurement, then you immediately end up in Kantianism. — Snakes Alive

?

Measurement = physical interaction = not exclusive to humans = no Kantianism.

In other words, if you take seriously the idea that measurements only reveal things qua measured, and there's no reason to think that it's possible to measure things qua things as they are independent of or prior to measurement, then you immediately end up in Kantianism. — Snakes Alive

But this would only be the case if you ignore the specificity of the experiments involved. The cool thing that QM teaches us is that measurement is just another kind of physical interaction. It places us on the same footing as everything else in the universe. That's just what it means to say that 'we are part of what we are trying to measure'. The whole import of QM is that it demolishes any pretence to human exclusivity - we don't stand apart from the world, we are already of it. There's no need to try and attempt to 'get outside' because one is 'already inside', if I may put it that way. This in turn bars any Kantian interpretation of QM, insofar as it denies any attempt to make of us exceptions to the rule.

’. What it seems to do is undermine the presumption of the mind- independent nature of the objects of physics — Wayfarer

Thanks for not addressing my point, charlatan. Now is about the time you post a coat-of-arms picture I believe, or whatever other detritus you usually spew in lieu of argument.

The problem with these kinds of claims is that they take a tautology - the fact that the results of measurement cannot occur with without acts of measurement - and try and wring from it a substantive claim that does not follow: that as a result, the universe itself is existentially dependent upon those acts of measurement.

Or put otherwise, it is simply trivially true that the experimenter cannot be separated from the experiment: and this speaks to the nature of experiments (as conducted by human/other experimenters) and not, as it were, the universe itself. So one needs to be very careful about how to treat these kinds of claims. It is undeniably true that the observer is always part of the picture - insofar as it is only ever a picture.

Or put yet otherwise, because this is a hard point to grasp: the relation between measurement and universe is absolutely asymmetrical. Of course any measurement of the universe will depend on the fact of it being so measured: that is what it means to be part of the universe. But it does not follow that the universe depends, 'existentially', on measurement. But this attempt to wring a substantive from a tautology is just the kind straightforward logical fallacy committed by quantum-woo charlatans day after day.

It is true that this does require a reevaluation of what 'objectivity' means: measurement isn't just about attaining a POV 'from nowhere'. It is always a kind of POV 'from somewhere'. But this speaks to the enterprise of measurement itself, and not anything else.

Pretty sure Heidi is just a foil for the larger point, but I would prefer if the foil weren't so shittily fabricated. Not that Cic can help himself from his trolling in any case.

Yeah, the equation of Heidegger's 'nothing' with some fuzzy fear of death or anxiety over non-existence has nothing to do with Heidegger's point. As Frank said, the word 'death' (or 'non-existence', for that matter) doesn't even appear in the essay this is from. So the OP is just making things up.

https://drive.google.com/file/d/1qlEcNiK8CBkJOU1YMyz_OSWIIAq6sMXl/view

Response by Zimbardo to his critics.

"State is the name of the coldest of all cold monsters. Coldly it tells lies too; and this lie crawls out of its mouth: “I, the state, am the people.” That is a lie! It was creators who created peoples and hung a faith and a love over them: thus they served life.It is annihilators who set traps for the many and call them “state”: they hang a sword and a hundred appetites over them.

Where there is still a people, it does not understand the state and hates it as the evil eye and the sin against customs and rights. ...Only where the state ends, there begins the human being who is not superfluous: there begins the song of necessity, the unique and inimitable tune."

“The history of the state is the history of the egoism of the masses and of the blind desire to exist”

"As little State as possible!"

"Socialism ― or the tyranny of the meanest and the most brainless, ―that is to say, the superficial, the envious, and the mummers, brought to its zenith, ―is, as a matter of fact, the logical conclusion of “modern ideas” and their latent anarchy: but in the genial atmosphere of democratic well-being the capacity for forming resolutions or even for coming to an end at all, is paralysed. Men follow―but no longer their reason. That is why socialism is on the whole a hopelessly bitter affair."

Nietzsche, sic passim. So peddle your fucking ignorance elsewhere.

No the whole post is trash from top to bottom.

^ At some point, even saying 'every word of that post was wrong and steeped in the stink of intellectual sewerage' just doesn't cut it.

Does current science have commensuribility as a principle? — TheMadFool

I don't know that one can speak of 'current science' as a reified whole. Just (individual) scientists and their views, organizations and their views, institutions and their views and so on. In any case, it seems obvious that the quest for a TOE - on the premise on commensuribility - still seems to many, if not most, as a legitimate and desirable endeavour.

AFAIK the number line is complete (wholes, rationals, integers, rationals, and irrationals). We even have the real-imaginary number space. — TheMadFool

The whole problem is precisely over the question of 'completion'. The assumption of commensuribility turns on the idea that, once we 'complete math', we could then use mathematical tools to create a one-to-one model of all reality (Rosen: "The idea behind seeking such a formalized universe is that, if it is big enough, then everything originally outside will, in some sense, have an exact image inside"). But if commensuribility does not hold - if not everything in the universe is in principle able to be subject to a single measure - then no such 'largest model' can exist.

Importantly this does not mean that modelling is a lost cause; instead, it means that modelling must be specific to the phenomenon so modelled: beyond certain bounds and threshold values, modelling simply ends up producing artifacts (at the limit, you get Godel's paradoxes!). You can have models of this and models of that but never THE model.

Seem to me, if the result of that disaster is mathematics, science and technology, we could use more such disasters. — Banno

But the assumption now stands in the way in of just those things.

Doo itt. It's among his most accessible and topical works.

I think trying to attribute all of that to Pythagoras [if that is what is being said] is at the very least drawing a long bow. — Wayfarer

Why? Pythagoras was the mathematical reductionist par excellence. It is not for nothing that the notion that 'everything is number' is chiefly associated with his name. There's barely a bow-string to pull, let alone draw long.

Consequently I'd argue that science has been reductionist because it has so for long been enmeshed in the 'broader philosophical tradition'. It's only now been able to begin to extricate itself from that muck and mire now that the Pythagorean inheritance is drawing its (hopefully) dying breaths. A cause for celebration.

It's not out of the question. That said, one of the ironies of your opposition to the OP is that the anti-Pythagorian thrust of the paper goes hand-in-hand with an anti-reductionist approach to the world. Among the consequences Rosen draws is that:

"Above all, we must give up reductionism as a universal strategy for studying the material world. But what can we do in a material world of complex systems if we must give up every landmark that has heretofore seemed to govern our relation to that world? I can give an optimistic answer to that question, beginning with the observation that we do not give up number theory simply because it is not formalizable. Godel’s Theorem pertains to the limitations of formalizability, not of mathematics. Likewise, complexity in the material world is not a limitation on science, but on certain ways of doing science."

It's only by opposing the Phythagorean disaster that one can, for instance, demonstrate the invalidity of the Church-Turing thesis as it pertains to the world. But, given that you're an arch-reductionist yourself, I suppose it's not surprising that you'd so willingly crawl into bed with the enemy.

But nevertheless the emphasis on pure reason, on things that could be known simply by virtue of the rational mind, is one of the major sources of Western philosophy. — Wayfarer

So much the worse for Western philosophy.

The conclusion to be drawn is not that mathematics is useless or somesuch; only that one should engage in mathematics without the (unscientific) assumption of commensuribility. Or put otherwise: don't assume the commensuribility of everything in the world (this is theology); instead, run tests, inject a good dose of empiricism and pay close attention to whatever phenomenon you aim to examine.

One of my great regrets in life is not picking up his The Degrees of Knowledge at a second hand bookshop once. I really want to read him one day. *sigh*.

Chiming in to agree with Luke here. The point of the beetle-box story is not that, if only the beetle really was in the box, that the word would make sense. It is that the bettle's being in the box is entirely irrelevant from the very beginning. The use of the word 'soul' is perhaps an exemplary case of the beetle-in-a-box: the fact that there are no such thing as 'souls' has no bearing on the fact that one can make perfect sense of the word 'soul'.

I always thought it was quite clear the Cic had a kind of pathological/sado-masochistic relationship to Heidegger, enjoying quite immensly his bad-faith engagements with any discussion of him.

But, I believe chapter XVI of The Republic — Posty McPostface

There are only 10 books (or 'chapters', if you prefer) in The Republic. It doesn't go to 16 in the copies I'm aware of. What are you referring to?

In any case Plato probably wouldn't have minded cause he was a wanker.