Comments

  • Mathematical platonism
    ↪Arcane Sandwich

    Try

    https://www.umsu.de/trees/#~6x~3((Gx~5Ax)~1(Ax~5Gx))
  • Australian politics
    ↪Arcane Sandwich
    Of course.
  • Mathematical platonism
    ∀x((Gx) ≢ (Ax)) — Arcane Sandwich
    is just
    U(x)~((Gx⊃Ax) & (Ax⊃Gx))

    Looks fine.

    Cheers. Good chat.
  • Mathematical platonism
    ↪Arcane Sandwich


    "Pegasus exists (is an item of) the domain of Greek mythology" looks to be a round about way to say that Pegasus is a myth. But we can do that without using any notion of existence. I just did. We can add that if something is a Greek Myth, then it is not an Aztec myth, and conclude that Pegasus is not an Aztec myth. All well and good and done without introducing two-placed existential predicates.

    Greek myth(Pegasus)
    For all x, Greek Myth(x) ≢ Aztec myth(x)
    Hence
    ~ Aztec myth(Pegasus)

    So I'm still not seeing the need for Bung's approach.

    I could not follow that last paragraph, my apologies.
  • The Blind Spot of Science and the Neglect of Lived Experience
    But when Husserl points out that the intersubjectively produced empirical objects are entities that no one actually sees... — Joshs
    Husserl can't see the butterflies?
  • Mathematical platonism
    ↪Arcane Sandwich
    Oh, no rush; indeed no obligation. Respond, or not, at your leisure. Go to bed!

    Thanks for clarifying re Quine.
  • Mathematical platonism
    ↪Arcane Sandwich
    Do you think that the analysis you offered here is much different from what I offered
    ↪here
    ?

    Becasue I don't think it is.

    I must have misunderstood you. You appeared to be saying that Quine had a problem with quantification. He didn't, he had a problem with individual constants, replacing them entirely with quantified variables.

    The solution I offered makes use of them, contra Quine, and in accord with Kripke's answer to the sort of problem you presented. Pegasus does exist, which is to say no more than that he is an individual in the domain of discourse.
  • Mathematical platonism
    The existential quantifier, ∃, does not have ontological import. Quine is averse to it because he thinks that it does have ontological import. But he's just plain wrong. — Arcane Sandwich
    That doesn't chime with my understanding. Did you mean ∃! ? But that's not a quantifier.

    Quine certainly used quantification, to the extent that questions of existence and reality are for Quine to be answered using quantification. While I think this a but too tight, he's not wrong.
  • Mathematical platonism
    ↪Arcane Sandwich
    Thanks.

    By way of continuing the example, here's how I might parse the same sort of thing.

    Pegasus is an individual in the domain we are discussing. So not a predicate. We can write ∃(x)(x=a) were "a" is a constant that refers to Pegasus. It says very little.

    Since it is true that Bellerophon rode Pegasus to Mount Helicon, there is something that was ridden to Mount Helicon, by existential introduction.

    Something like "Pegasus exists in the context of Greek mythology, but it does not exist in the actual world" says little more than that Pegasus is an individual in the domain of Greek Myth, but perhaps not in the domain of chairs and rocks. Do you see a problem with such a simple and direct approach?

    Note the dropping of the words "conceptually" and "really". They do not appear to be doing anything.

    If needed, we could well put Pegasus and Mount Helicon into the same domain, and add a predicate something like "real", and say that Mount Helicon is real, but Pegasus is not real. But that has no implications for Pegasus' existence, as set out. It remains that Pegasus exists, but this amounts to little more than that Pegasus is one of the things about which we can talk - it is an item in the domain.

    What I've said here will be misunderstood and augmented by others, but to my eye it dissolves the issue of the OP. Infinitesimals exist, since they can be the subject of a quantification. Pegasus exists, since it can be the subject of a quantification. But neither are the sort of thing you might run into in the street.

    And what is going on here is a clarification of what we mean by saying that something exists, made by looking at how a formal language can deal coherently with the problem.
  • The Blind Spot of Science and the Neglect of Lived Experience
    This argument is interminable. — Wayfarer
    Only because you won't shut up... :wink:

    ↪Arcane Sandwich
    See https://thephilosophyforum.com/discussion/15574/is-the-distinction-between-metaphysical-realism-anti-realism-useless-andor-wrong/p1
    There is an extended, somewhat absurd, argument about whether there is gold in those hills, form about page 10. I won't blame you for not reading it, but thought I'd at least let you know some of the back story.
  • Mathematical platonism
    ↪J
    I would have been happier with an example that was a long way form philosophy - to look at how these terms are used in their natural environment, as it were, rather then in the lab. The sort of method advocated more by Austin than Wittgenstein, but captured in "Don't think, look" at PI §66. A way to check that our language hasn't taken a vacation.

    I played with your idea of a high C for a while. The supposition was that the high C could be understood - perhaps C6, two ledger lines above the treble clef; but unachievable by a certain musician. We have a clear idea of what that is, even if it cannot be sung by our soprano. Is there a note that no soprano might sing? Apparently the highest achieved is F#8, truly awesome. But again, we know what it would mean for someone to say, sing C9, even if this is never achieved. These are tied to our use of this sort of language.

    But is there a highest C? Here we start to leave firm ground. Is there a C higher than any other C? A first approach might be to claim that since for any given C there is a higher C, one octave above, that there can not be a highest "high C". In theses terms the term "Highest high C" doesn't have any referent, and very little sense.

    But taking a step further, a high C must travel through a medium, and at frequencies above around 10 Ghz, the separation of air molecules is such that sound fails ton be transmitted. There will presumably then be a particular high C, far beyond anything we might hear, that is the highest C that can be transmitted via air. Higher frequencies might be achieved in other mediums.

    Hopefully this digression shows that the context sets limits on the terms being used. Until we have a clear idea of what we mean by "high C" we don't have an answer to questions about what is the highest high C. Similarly, perhaps until we have a clear idea of what sorts of things are ineffable, we don't have a clear answer to the issues being discussed around ineffability. Trouble is, we don't have a way of saying what it is that is ineffable without the danger of thereby contradicting ourselves.

    Put in more Wittgensteinian terms, we don't have a clear language game around ineffability. So we end up making one up. And that is fraught with the potential for contradiction.

    Anyway, a bit off topic, I supose. Thanks for the reply.
  • The Blind Spot of Science and the Neglect of Lived Experience
    Actually I think you’ve conflated this thread with the other one, — Wayfarer
    They are on much the same.
    ...it is meaningless to say that of anything that it exists outside of or independently of any perspective — Wayfarer
    Yes, to say. And yet there is gold in those hills, even if no one says it.
  • Mathematical platonism
    Free Logic is not the only option. You can keep classical logic while tracing a distinction (as Bunge does) between real existence and conceptual existence. — Arcane Sandwich
    I haven't followed Bung, and you provide no reference, so I've no clear idea what he might be saying, but that sounds like a variation on free logic.

    That mathematics is "pleasing" looks to be besides the point.
  • The Blind Spot of Science and the Neglect of Lived Experience
    I don't see how you've shown this at all. In your example, perspective absolutely is an attribute of the world. "How we say things" is a consequence of how we experience them, and how we experience them says something about how the world is (else we need to write off empiricism). "How we say things" isn't something that is arbitrarily related to how the world is, nor do our practices of speech just happen to be what they are. Terms for perspective are universal across all languages because perspective is universal. — Count Timothy von Icarus

    Thanks for replying to my old post.

    Just briefly, here is the argument I presented. We start with the butterfly moving from left to right for @Wayfarer, but right to left for me. There's an apparent contradiction here, in that I describe the movement of the butterfly as being the opposite of the way Wayf sees it. We resolve this by understanding that although we are both seeing the same thing, we describe it differently; and we develop a way of achieving agreement, we agree that the butterfly is flying towards the mountain. What we have done here is agree that we see things differently, and then to find a way of setting out what is the case in such a way that we are in agreement. In effect we phrase what is going on so that the individual perspective does not imply a contradiction.

    Later we find ourselves on the other side of the mountain, and see the butterfly moving in the same direction, but now away from the mountain. In order to capture this we can change the description again, to say that they are moving towards the East.

    The direction in which the butterfly is moving, in each case, stays the same. But we have three different descriptions, left to right, towards the mountains, and towards the East. Now the butterfly was always heading East, even when heading towards the mountain or from left to right. What has changes is not the movement of the butterfly, but the description used. We developed a way of setting out that movement that did not depend on the position of the observer. True, the observer still has a perspective, but that perspective is removed from the utterance.

    There are three aspects to this account that I think are salient.

    First, it is an application of the Principle of Relativity, the general form of which is to present scientific principles in such a way that they apply equally to all observers. Saying that the butterfly is moving towards the East will be true in all three case, while saying that they are moving towards the mountain or from left to right will be false for some observes.

    This leads to considering interpretations of each observation in such a way as to achieve agreement. The butterfly moves from left to right for Wayfarer ≡ The butterfly moves towards the mountain for someone on it's inland side ≡ the butterfly is moving towards the East. We can apply the Principle of Charity to reach agreement on all these observations.

    And this speaks to the communality of language, that what we say about how things are is part and parcel of our role as members of a community. This in firm opposition to the view that some individuals observations are somehow paramount, or must form the foundation of knowledge. Knowledge is not built from solipsism.

    This is in contrast to Wayfarer's thesis that science neglects lived experience. A better way to think of this is that science combines multiple lived experiences in order to achieve agreement and verity. So sure, "our entire perceptual and cognitive apparatus biases our understanding of the world", and yet we can work to minimise that bias by paying attention to contexts and wording our utterances with care, so that they work in the widest available context. Not the view form nowhere but the view from anywhere.
  • Mathematical platonism
    But physical objects existed before logic (propositional, first order, second order, etc.) was invented. — Arcane Sandwich
    :meh:
    How is that salient?

    I would say that the existential quantifier, symbolized by ∃, should be distinguished from a first-order existence predicate. — Arcane Sandwich
    I agree. Free logic. Not used here.

    I can also explain why interpreting the existential quantifier as if it had ontological import necessarily leads to a contradiction. — Arcane Sandwich
    Sure. Quine's point being that treating that π exists is just to say it is the value of a bound variable has no ontological import. That was kinda the joke.

    Later.
  • Mathematical platonism
    ↪Arcane Sandwich
    I just don't see any of that as helpful. π in't imagined, it's r/c. To say that π exists is just to say it is the value of a bound variable.

    ∃(x)(x=r/c)
  • Mathematical platonism
    ↪Arcane Sandwich
    Ok.

    I can't see that saying π is a fiction is any better than saying it is the subject of a quantification. Indeed, that π is the ratio of the diameter of a circle to it's circumference isn't in any useful way like saying Frodo walked into Mordor.
  • Mathematical platonism
    "a certain, je ne sais quoi — Count Timothy von Icarus
    Nice. That catches something of the drift.
  • Mathematical platonism
    I don't agree with Austin's diagnosis. — Arcane Sandwich
    Ok. I'll leave you to that.
  • Mathematical platonism
    it refers to the quality of being real. — Arcane Sandwich
    What's that, then? See
    ↪javi2541997
  • Mathematical platonism
    ↪Arcane Sandwich
    Sure, one doesn't need to use imaginary numbers to count apples. Why should that make them more or less real than integers? Moreover, what does "real" do here.
  • Mathematical platonism
    "What physical, ordinary object in the world, can be accurately referenced by the number that stands for the square root of minus one."? There is no such object. — Arcane Sandwich
    https://www.geeksforgeeks.org/applications-of-imaginary-numbers-in-real-life/
  • Mathematical platonism
    ↪Count Timothy von Icarus
    I have a friend who's legs were removed curtesy of the US military using an old cluster bomb. He sometimes finds himself, on waking, checking that they are not there. Not so jejune,
    ↪Wayfarer
    .
  • Mathematical platonism
    ↪J
    :wink:

    What
    ↪Count Timothy von Icarus
    and
    ↪Wayfarer
    are proposing is important - but it is also just froth.
  • Mathematical platonism
    Is the only answer 'Shuddup already' — Wayfarer
    Yep.

    But that's not to stop you from doing.

    Shut up and calculate. :wink:
  • Mathematical platonism
    ↪Arcane Sandwich
    That seems to be pretty much what I have been arguing.

    That's what I'm not sure about. I don't think I'm asking for the inexpressible itself (call it P) to be expressed; that would indeed be impossible. Rather, I want to know why P is inexpressible. Call that explanation Q. Does it really follow that, if P is inexpressible, Q must be as well? — J

    So we might proceed by looking at examples.

    But your high "C" will not do, becasue the singer being unable to reach it is not inexpressible; we know what note they are trying to reach, and that they cannot reach it.

    So give an example of something that is inexpressible...

    See the problem?
  • Mathematical platonism
    I just want to point out that these two views are not the same. You can indeed move on from inexpressibility to a demonstration or showing of what can't be expressed. But first (or conjointly) you can also say why, as Wayfarer suggests. Or would the claim be that inexpressibility itself can only be demonstrated, not justified? — J
    If something is inexpressible, then by that very fact one cannot say why... Doing so would be to give expression to the inexpressible.

    In the end, past all the justification and discussion, there is the act. Any justification becomes besides the point.

    And this applies to ritual, ethics and maths:
    What this shews is that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" and "going against it" in actual cases. — PI §201

    This is the answer offered to the problems of rule following, and it's the only one that works - that in the end, it's just what we do.

    And in so far as it is just what we do, the rationalisations, arguments and justifications are almost irrelevant, mere superstructure or appendix.

    So while "it might be important to say why" (@Wayfarer) it remains that "what cannot be said may be shown or done" (Banno)

    So we still have, from the Tractatus, "The world is all that is the case" and "What we cannot speak about we must pass over in silence", both quite so. And then we go to the next step, which is that nevertheless, we must act, and be part of that world.

    When Moore holds up a hand and says "Here is a hand" he is performing an act, making a declaration; here he cannot be wrong.
  • The Blind Spot of Science and the Neglect of Lived Experience
    ↪Janus
    This is much the same point
    ↪that I tried to make, at some length
    , in continuing @Wayfarer's story
    ↪The Mind Created World
    . Science proceeds not by ignoring first-person accounts but by looking for common grounds, by finding agreement.
  • Mathematical platonism
    ...religious fundamentalism... — Wayfarer
    Not my words.

    it's cultural conditioning, pure and simple. — Wayfarer
    Probably, but so what. Any amount of social or psychological explanation for Banno's foibles will not change the veracity and validity of arguments Banno sets out. And by the same token your defence of spirituality beyond what is reasonable might be explained by your Catholic upbringing. All irrelevant, as you know.

    For what it's worth I will repeat that physicalism is perhaps as mistaken as spiritualism. That I reject spiritualism and mysticism does not mean that I accept scientism and physicalism.

    if the mental things arising from the minds of living things are a distinct realm of existence... — Thoughts are Real (Review of Nagel, Mind and Cosmos)
    If.

    Read what I said about existence, above. The bit about there not being "different kinds of existence".
  • Mathematical platonism
    I don't believe in the possibility of the direct knowing of transcendent truths — Janus
    What troubles me is the presumption to knowledge - justified true beliefs - in the absence of a coherent way of providing a justification.

    Which of course leads into the discussion of what is to count as a justification...

    And @Wayfarerand I apparently have quite divergent views on this.
  • Mathematical platonism
    ↪Janus
    Yep. I'll add that the existential generalisation described in my previous post is at the least a first point of agreement, something that holds for us all just in virtue of our acceptance of certain aspects of the way we talk. Hence it can work as a hinge on which we might build some agreement. Whereas saying that some things are real but don't exist or exist but are not real just builds befuddlement.

    So there are primes.
  • Mathematical platonism
    As far as quantum physics is concerned, one simple point is that made by both Bohr and Heisenberg - physics reveals nature as exposed to our method of question, not as she is in herself. That leaves ample breathing-room for philosophy. — Wayfarer
    And they both might have continued by saying that the method of questioning that is appropriate is that of physics, not that of philosophy. And I'd agree - much of what is called "philosophy" in this forum is just attempting to do physics, badly, and without the numbers.
  • Mathematical platonism
    'what kind of existence does it have?' — Wayfarer
    I'd suggest that this is not a good question to ask, becasue it presumes that there are different kinds of existence. But if we take Quine as a guide, then the issue is quite a bit simpler. Prime numbers and electrons can both be subject to existential introduction, a quantification. That is, from "The electron was deflected to the right" we can conclude "There exists an electron"; from "11 is the first prime number greater than 10" we can conclude "There exist prime numbers". And that's where we might pause to ask "what more is there to existence?"
  • Mathematical platonism
    ...a firewall against metaphysics. — Wayfarer
    I quite agree, and take that to be one of the main themes of the Investigations - that what cannot be said may be shown or done.
  • Mathematical platonism
    ↪frank
    ,
    ↪Wayfarer
    , you are about to go Quantum, aren't you.

    The point being made in my post is that there is a difference in method between finding the value of π and the value of the mass of an electron.
  • Mathematical platonism
    ↪Janus
    I'm pleased you got the joke. My general opinion of @Wayfarer is that we agree about most things, but that he adds more than is needed; where silence is appropriate he keeps talking. But this is becasue he wants to show us something more, presumable thinking that we (I?) don't already see it. Maybe I don't.
  • Mathematical platonism
    ↪J
    You've read more than I, then, since my knowledge is from the commentaries rather than the original, and mostly as an example of the application of the thinking of Austin and Wittgenstein, as an application of the linguistic approach to wider areas. We do things with words; indeed so overwhelmingly are we embedded in a social structure built with language that like the ubiquitous fish in water we fail to notice our surrounds.

    As I understand it, what is unavoidable is our mutual agreement concerning the way the world is, and the language we use to discuss it; and what is irreducible are certain activities we perform, including illocutionary acts and normative assumptions. So "property" unavoidably takes as granted that there is land, that the land can be subdivided into sections, and that we can talk about the land; and it takes as irreducible the idea that I can dispose of this piece of land as I see fit, while you cannot.

    While a dumb animal might defend a territory, it does not own it in the way a person owns a piece of land in virtue of deeds and purchases and so on. In this way "property" is dependent on our being embedded in an irreducible social structure that is unavoidable.

    Much to the confusion of libertarians everywhere.

    Mathematics is presumable also unavoidable and irreducible, in ways that might well be worth setting out.
  • Mathematical platonism
    I'm asking if infinitesimals exist in the sense that would satisfy mathematical platonism. — Michael

    And that is the question I answered. I gather I need to be more explicit. You gave the following as yout definition...

    Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

    There are mathematical objects in the sense that we can quantify over mathematical things such as numbers and triangles. There are even numbers, therefore there are numbers. There are isosceles triangles, therefore there are triangles.

    These are not independent of "us and our language", since if we were not here there would be no mathematics. Mathematical entities are however independent of any single individual, but built by a community int he way that a language is built. Hence the "us".

    We decide the truth or falsity of statements about planets and electrons by experimentation and inference. We use telescopes and potentiometers. We do not use such devices to decide the truth or falsehood of mathematical entities. We do use inference. The truth of mathematical suppositions is agreed in much the same way that the truth of Ohm's Law or Kepler's Law is agreed. If that is what is meant by "objective", then they are objective. We discover things about mathematical entities, in that we find unexpected results in our construction. This is not the same as discovering that the orbit of a planet is elliptical or that electrons have a specific mass.

    I take all of this to be saying that infinitesimals exist, but not in the way set out in your quote.
  • An Analysis of "On Certainty"
    ↪Sam26
    Well done!
  • Mathematical platonism
    What one believes, desires, feels or cares about will still be there, even if one is honest.
Home » Banno