@Srap Tasmaner - reflections on 'clickiness'...

Moreover, the deployment of our concepts is not governed by truth, but by their ranged of illumination. This is not on account of their being arbitrary ('subjective'), but absolutely necessary. — StreetlightX

Math-games, eh? But is there one math-game to rule them all?

Otherwise, we are free to keep it. What I want to add to this is that philosophical concepts are just like this. The concepts we employ are a function of what we aim to capture with them; to employ one concept rather than another is to bring out one aspect of the world rather than another. Moreover, the deployment of our concepts is not governed by truth, but by their ranged of illumination. — StreetlightX

I would say there is a difference here between math and philosophy then based on what you present. Math dictates (by discovery) that a different method be used to solve a problem. In philosophy, there are no clear-cut discoverable rules that can only be used to answer a particular problem. Rather, any number of worldviews/heuristics/logical constructs can be employed to answer the philosophical problem at hand. The math "dictates" that a particular rule must be used, philosophical problems have no demand. If the mathematician has a hang-up on a particular math that "works" for that problem, that is their decision, but they will probably fail to answer the question at hand based on a bias.

The math "dictates" that a particular rule must be used — schopenhauer1

The literal point of the thread is that the math dictates nothing about the choices that must be made. The problems dictate the directions into which we take math; and we are motivated by problems (there are of course intra-mathematical problems, but these too are no different (cf. Rosen on Modelling).

Yes, then change what I said to the "problems of math", I don't see how this counters the argument. If a problem requires a new method in math to solve a problem, it cannot be disputed that the math solves the problem if it "works". Pythagoras for example, didn't like irrationals, but it worked to solve problems that when worked out can clearly be seen by all. Philosophy has no clear-cut agreement whether the problem "works". If another method follows the same problem, then both can be said to "work", then it is preference for which is easier for that person to use. What cannot be contested is that a method(s) is agreed upon to work in a mathematical community. Philosophy can never really have this definitive satisfaction that their solutions "work". So one illumination is agreed upon, where in the other camp the illumination is a continuous dialogue. The discovery aspect to math is the difference. You cannot stretch this to philosophy no matter how you want it to be so.

Yes, then change what I said to the "problems of math", — schopenhauer1

But I'm not talking about the problems of math. At least, not exclusively. So there's no good reason to make any such change.

Further, among the points that B&C stress is that it is not at all 'discovery' that is at stake, but what they call - following Wittgenstein - concept-determination: "what is going on here is best described neither as ‘discovery’ nor as ‘invention’ of something entirely new. There are facts to be revealed, and creativity to be exhibited, but what is crucial is the opening up of different aspects of something ... which prompts a choice that sooner or later ‘catches on’... and proves fruitful."

There are times when you just gotta hack things together then see if it works after. Even when someone has blazed the trail in front of you they don't leave all the signposts. Sometimes you have to make your own, revealing the location you marked by putting a sign along the way.

This indeterminacy actually shows up as a methodological problem in stats, called researcher degrees of freedom.

But I'm not talking about the problems of math. At least, not exclusively. So there's no good reason to make any such change. — StreetlightX

But you compared the problems of math to the problems of philosophy, so I am just taking your cue.

Further, among the points that B&C stress is that it is not at all 'discovery' that is at stake, but what they call - following Wittgenstein - concept-determination: "what is going on here is best described neither as ‘discovery’ nor as ‘invention’ of something entirely new. There are facts to be revealed, and creativity to be exhibited, but what is crucial is the opening up of different aspects of something ... which prompts a choice that sooner or later ‘catches on’... and proves fruitful." — StreetlightX

And the point I am trying to make contra your comparison is that while both might have an "opening up of different aspects of something..which catches on", the "proves fruitful" part is what is different between the two. In philosophy, there can never be a set agreement that "this works" to solve a particular philosophical problem (at least perhaps outside of logic...something akin to math). However, in the math community "this works" can be agreed upon. So "proves fruitful" here is not just a difference of degree but quality. Fruitful in math really cannot be disputed- everytime one does the problem it works out. If this method is overtaken by a better one, then that one will be agreed upon to work out, in a step-wise fashion. If both methods work for the same problem, then it is preference perhaps. However, in philosophical problems, it is always open-ended. Fruitful here can be continuous, ongoing, etc. It is more invented, and not dictated by the rules of the game as much as math which is very much driven by the results that are calculated- what are the constricted givens of the math world.

Edit: So I guess, there is more "imposed" then is indicated from the OP about math. One can find a new math ugly, not to one's liking, but it cannot be argued that the results work. Eventually, this forces one's hand. Not so much in philosophy.

Nonsense. People forget the simplest ideas when it comes to math. It's often recognized that once a mathematical object is defined that it can only be manipulated logically. Yet, if we ever logically manipulated an object into a statement, such as A is not equal to A, then we would know A could not exist. People overlook the fact that a mathematical object can only exist if its existence is consistent with logic. It works both ways.

By way of a footnote, the discovery and acceptance of the concept of zero was resisted by Western mathematicians for a long period of time, for religious and philosophical reasons. As is well-known, the incorporation of zero into maths - indispensable for decimal notation - was first accomplished by Indian (or possibly Chinese) mathematicians, who had no such inhibitions.

indispensable for decimal notation - was first accomplished by Indian (or possibly Chinese) mathematicians, who had no such inhibitions. — Wayfarer

My point earlier to StreetlightX was that a difference between math and philosophy on new insights or "illuminations" is that math must eventually acquiesce to what works to get the results. Philosophy can keep going on open-ended infinitum. 0 was eventually incorporated because it worked and everyone could see the results when it was used. The results demands or "dictates" its eventual use- even if it is at first ignored or scorned. No one for example will say, "Hey we now all acknowledge the Age of Schopenhauerean Ethics" when solving ethical problems. As much as I'd like to see that, it ain't gonna happen.

I agree with the pragmatic angle, but the conclusion seems more Platonic. Each of these particular turns in the history of mathematical thought were "forced" by the need to move from the particular to the general. A constraint that was breaking a symmetry needed eventually to be unbroken so as to move up to the next level of abstraction or generality.

Irrational numbers - following the Aristotelian argument supplied in the paper - can be seen as having to relax the constraint that a number is either odd or even. The argument shows that an irrational number is not definitely one or the other. So a higher degree of symmetry is obtained by removing this constraint as an "obvious necessity".

The same goes for complex numbers - removing the constraint that number lines be one dimensional. And non-Euclidean geometry - removing the constraint that worlds be flat thus parallel lines apply.

Well, for B&C, the important point to note is that nothing in the math itself forced this choice, rather than the other. — StreetlightX

No. The tension here was between the numerical and geometric view - the geometric one representing the presumed continuity of physical nature, the numerical one representing the desire to talk about that in a system of discrete signs.

So you have the continuity of symmetry and the discreteness or individuation that results from symmetry-breaking. And then the metaphysical tension regarding whether to understand acts of individuation as actual discrete ruptures - stand alone existences - or merely contextualised developments, breakings that have only relatively definite existence.

The disturbing thing was that geometry and algebra did seem to be two commensurate or complementary ways of talking about the same world. But eventually the cracks got revealed.

Numbers work as zero-dimensional and non-geometric signs when their physical dimensionality is suppressed or constrained. But to maintain a complementarity between numbers and geometry as maths seeks to advance, gradually the constraints on that dimensionality have to be relaxed in a systematic fashion to keep the two worlds connected as we move deeper towards maths capable of higher symmetry.

So in summary, it does all start in pragmatic acts of measurement - the semiotic trick of giving names to things. We break the continuity of physical experience by imposing a system of discrete marks upon it.

This is the useful trick. The semiotic ability to construct constraints that break the symmetry of the experienced world. And the Ancient Greeks were dazzled by this new Platonic reality that rational geometry opened up. A fundamental connection between discrete number and continuous dimension appeared to be forged.

And gradually the very nature of that trick - even for the physical world itself, the pan-semiotic Cosmos - got revealed. The whole damn world exists as a constraint on dimensionality, a grand tale of global symmetry-breaking and localised individuation.

So maths only ever had one route by which to recover its physical origins. At every turn, it needed to work out what further localising constraint had been added to the deal that could be successfully generalised away. Mathematical advance had to see that a symmetry had been broken, which could then be unbroken to reveal the higher, more abstracted, less particularised, realm that lay beyond.

Go up a level and you could still give names to things. Topology abandoned geometry's definite measurements of length, but still deals in individuated abstracta. Semiotics could still work as the trick.

But the route was always Platonically predestined and necessary. If existence takes definite shape due to constraints, due to symmetry-breakings, then the only way to understand that is by following the path backwards that abstracts away those constraints, unbreaks those symmetries, to reveal how the how show works.

the Ancient Greeks were dazzled by this new Platonic reality that rational geometry opened up — apokrisis

Justly so. I think we ought to recall the sense of awe with which the ancients treated mathematics and reason. After all, at that point in civilisation, life truly was 'nasty, brutish and short' for the great mass of people. But then early civilisations also witnessed the construction of the pyramids, the discovery of astronomical tables, and many other such principles. (The story of Archimedes' achievements is quite amazing. Not to mention the Antikythera device.)

So, wasn't it only natural that the power of reason, and the uncanny efficacy of geometry and arithmetic for the construction of such apparent miracles, would be regarded with a kind of awe? It was what separated humans from beasts, and also philosophers from the hoi polloi; the 'rational animal' alone could fathom such principles. Hence the later identification of 'the rational intellect' and the soul.

the 'choice' to allow imaginary numbers (the square root of negative numbers, like √-1 — StreetlightX

By way of a footnote, the discovery and acceptance of the concept of zero was resisted by Western mathematicians for a long period of time, for religious and philosophical reasons. As is well-known, the incorporation of zero into maths - indispensable for decimal notation - was first accomplished by Indian (or possibly Chinese) mathematicians, who had no such inhibitions. — Wayfarer

We have numerous choices as to what zero actually represents, and this is evident if we start to look at the difference between the different representations of zero, like the number line, in which zero represents a count of one unit, or in some cases zero represents the point of separation between positive and negative. Then there is the representation of imaginary numbers, in which I don't quite know what zero represents.

But the route was always Platonically predestined and necessary. If existence takes definite shape due to constraints, due to symmetry-breakings, then the only way to understand that is by following the path backwards that abstracts away those constraints, unbreaks those symmetries, to reveal how the how show works. — apokrisis

I disagree. While I think the whole symmetry-breaking story is a useful framing and pedagogic tool - I turn to it too, occasionally - I think it is a mistake to reify it into a metaphysical picture. It ends up treating the pragmatics as mere accidents on the way to some eternal Platonic story which was there from the beginning - which is nothing more than theology through and through. But, to borrow a bit from Rosen - I don't believe there is any 'largest model' here: there is no 'most general generality' - what I suppose you call 'vagueness'.

Vagueness is for me the ultimate transcendental illusion: it takes a perfectly valid move - the step from particular to general, always motivated by a particular problem (B&C's 'decision points') - and then illegitimately extrapolates that step into what one might call an 'unmotivated generality'. It tries to think an abstract generality shorn of any reference to the particular, cutting off it's roots to any particular problem that would motivate it - other than a nice, just-so story. Abstraction without a (necessary) foot in the real.

So basically I can agree with you up right up until the point where you invoke unmotivated generality as a Platonic bow to tie the whole developmental story together. It's this very last step that shifts a perfectly rigorous and valid methodology into a procrustean metaphysics that tries to retroactively fit concrete developments into a pre-ordained story. It's just a theological-Platonic hangover/residue that needs to be rejected.

Moreover, the deployment of our concepts is not governed by truth, but by their range of illumination. This is not on account of their being arbitrary ('subjective'), but absolutely necessary. — StreetlightX

I think this is almost right in a few ways, but when you say "their range of illumination" I'm reminded of the old adage about looking by the lamppost for the lost keys at night.

IOW, in venturing beyond the known, we do in fact use truth as a guide - in the sense of it being an ideal to aim towards.

And the point I am trying to make contra your comparison is that while both might have an "opening up of different aspects of something..which catches on", the "proves fruitful" part is what is different between the two. — schopenhauer1

I think you're missing the point though - what 'proves fruitful' is the choice made between two possible 'paths'. We're not talking about 'solving problems': we're talking about determining concepts: should number be treated in this way or that? Should infinity be thought of like this or like that? The point is that the normative force of this 'should' is provided by a concrete problem (which may be intra-mathematical or not) which any choice that is made is responsive to. Mutatis mutandis the way in which we form concepts in philosophy are similarly responsive to the problems they address: in neither case is it a matter of solving problems, but determining concepts.

It is an interesting paper - thanks for the link. I have one perhaps stupid question:
The two criteria the authors mention need actually be unpacked as the following (as far as I can see):
1) There is a one-one correspondence between numbers and lengths and there is an arbitraily chosen unit length.
2) Numbers can be expressed as ratios of the varying amounts of the arbitrarily chosen unit length mention in (1).

Is there really a choice as to which one to give up? Giving up 1) would be to give up using the notion of a unit of length, which would entail nothing could be measured.

Giving up 1) would be to give up using the notion of a unit of length, which would entail nothing could be measured. — MetaphysicsNow

I believe the idea is that it is no longer part of the definition of number that it be a measure - which doesn't mean it can't be used for measuring.

I think you're missing the point though - what 'proves fruitful' is the choice made between two possible 'paths'. We're not talking about 'solving problems': we're talking about determining concepts: should number be treated in this way or that? Should infinity be thought of like this or like that? The point is that the normative force of this 'should' is provided by a concrete problem (with may be intra-mathematical or not) which any choice that is made is responsive to. Mutatis mutandis the way in which we form concepts in philosophy are similarly responsive to the problems they address: in neither case is it a matter of solving problems, but determining concepts. — StreetlightX

So from the B&C article itself it says:

We now take all this for granted, but if we go back to the origin of the
determination, we can see that it was by no means necessary. At the core of this
determination was a choice of conceptual aspect, and although we might find it hard
now to see things in any other way, it is important to recognize that the choice was
there and that our concept of number might have developed in another way. We
should also note that while the choice between the different ways of seeing – of
determining the concept – was, we might say, forced by mathematics itself (the proof
above), the outcome of the choice was not so determined. The choice between
criteria, whatever its motivation, does not answer uniquely to intra-mathematical
considerations; mathematics itself, we might say, allows either choice, while
eventually accepting the choice that is made. — B&C

Again, I don't discount that there is some sort of choice as to which criteria fits the picture better, but rather, I disagree with the move to equate it with philosophical creativity. In the math world (similar to the science world), eventually there will be a consensus (for a time being) of the criteria. This consensus is based on the fact that it can be used to solve a wide range of problems demonstrably. This demonstrable ability of math is lacking in philosophy. There is a sort of constraint or "dictate" going on here in math, moving it along to a more clear picture, that is not able to be had in philosophical problems. Philosophical problems are more like interesting flourishes of thought. Whether math has necessity or not, the problems are constrained enough to have its own dictates through demonstration. Philosophy does not. They are flights of fancy, if you will, that can be entertained or not entertained with no demonstrable constraints on the flights of fancy one chooses. It is too open-ended for any consensus. There is no "pairing off" of previous notions in a step-wise fashion.

The authors talk about criteria, not definitions (although, by giving criteria you might supply a definition I suppose). I don't think the Greeks would have defined numbers by those two criteria, for the very notion of number appears in the criteria, so the definition would be circular in a very unilluminating way.

Yeah, I didn't mean otherwise when I spoke of definitions. Regardless, I'm not sure it matters for the point at hand.

Philosophical problems are more like interesting flourishes of thought. Whether math has necessity or not, the problems are constrained enough to have its own dictates through demonstration. Philosophy does not. They are flights of fancy, if you will, that can be entertained or not entertained with no demonstrable constraints on the flights of fancy one chooses. It is too open-ended for any consensus. — schopenhauer1

I suppose then that we simply disagree on this point. I'm firmly of the belief that every philosophy worth its salt has the kind of internal consistency that characterizes mathematical concepts, and they derive that consistency from the particular problems that animate them. The difference, to the degree there is one, lies only in the fact that philosophy has a far wider range of inspiration than math: its problems are drawn from a more diverse array of sources. To dismiss philosophy as 'flights of fancy' is to not understand it.

It struck me that badly-formulated philosophy can be compared a little to p-hacking: it doesn't derive its consistency from the need to adress a problem but simply from unmotivated hacking together of correlations between series.

Well, if the two criteria are criteria, what are they criteria for? As I say, I don't think they can be criteria that provide us with the rules for telling us (or the Greeks) what a number is or was, since they are expressed using the very notion of "number". So they might be criteria for describing the use of numbers. If so, then (1) captures the use of numbers for measuring things such as geometrical lengths that are amenable to the application of the notion of a discrete unit. (2) becomes something like the criteria that ratios of whole numbers - i.e. the numbers used to count those discrete units - can be used to express all possible numbers. Does the proof that the square root of two cannot be expressed as the ratio of two whole numbers really have any bearing at all on criteria (1) thus understood? I think that the answer to that question is "No". If that is right, then the authors are incorrect about the Greeks facing a choice about which criterion needed to be dropped (or, rather, modified) - and it is mathematics itself that just hands them the fact that (2) is not adequate.

To dismiss philosophy as 'flights of fancy' is to not understand it. — StreetlightX

To this I'll admit, I should have used another word. Perhaps "thought-explorations" or "generalized theoretical-investigations".

I'm firmly of the belief that every philosophy worth its salt has the kind of internal consistency that characterizes mathematical concepts, and they derive that consistency from the particular problems that animate them. — StreetlightX

A particular line of reasoning can have internal consistency, but there are so many theories from so many avenues, that can aim at solving a certain question, there can be no consensus really except perhaps those who are specialized in that particular philosophical school of thought. So maybe Wittgensteinians agree on certain principles and can branch out from there, or Schopenhauerians can in their camp. But then, this just goes to my point that in math there is no "school of thought" in math-proper. It is consensus of concepts that are demonstrable in the results they produce and works in a step-wise fashion. I guess my two main terms to see the difference between math and philosophy are "demonstrable" (which leads to eventually) "consensus" (and onward it goes).

The difference, to the degree there is one, lies only in the fact that philosophy has a far wider range of inspiration than math: its problems are drawn from a more diverse array of sources. — StreetlightX

But this is a much larger difference than you seem to be implying. The implication with math is that its constrained world "dictates" its next move, where in philosophy the diverse array of resources makes it too open-ended.

But the question is whether or not we can keep both at the same time. As they put it, the irrationals force the criteria to 'come apart'. The question is whether you can take them as a set or not. I think I underplayed this in the presentation of the OP, but yeah.

The irrationals show that criteria (2) as I developed it (which could be an incorrect development, I grant you) is just false - there is simply no "decision to make about which of the two criteria is more important to us" to quote the authors.

Basically, the authors need to work on the example a little more and expand on exactly what they take their criteria to be criteria for.

The irrationals show that criteria (2) as I developed it (which could be an incorrect development, I grant you) is just false - there is simply no "decision to make about which of the two criteria is more important to us" to quote the authors. — MetaphysicsNow

Not at all. The other option is simply to reject that irrationals are numbers tout court. And for the longest time this is just what happened. For a good history of this, see Daniel Heller-Roazen's The Fifth Hammer.