Language is one of the things we do. Didn't Habermas reflect on this in his use of unavoidability and irreducibility? That it is action that has import? — Banno

I don’t recall what Habermas says about unavoidability and irreducibility (of language, I assume), but I have only read sections of the 1,000-page Theory of Communicative Action, along with a lot of secondary criticism, so I may have missed it entirely. I think it’s fair to say that Habermas sees rationality as procedural, and the procedure necessarily involves language. Anthony Giddens has a good overview of TCA in which he says that for Habermas, “rationality presumes communication, because something is rational only if it meets the conditions necessary to forge an understanding with at least one other person.” (“Reason Without Revolution?” in Habermas and Modernity, Richard J. Bernstein, ed.). In general, Habermas sees reason and communication as activities that we do together, which fits the picture of language (perhaps including logical and mathematical languages) as already given in the life-world we find ourselves born into. You can't fly solo.

I think it's very clear that "infinitesimals" do not qualify as Platonic objects, because they do not have the "well-defined", or even "definable" nature which is required of a Platonic object. — Metaphysician Undercover

Do not qualify yet. Once infinity and it's opposite are well defined (and infinity isn't just taken as an axiom), they likely would be Platonic objects. At least I have enough belief in the "logicism" of mathematics that it is so.

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.

A very shallow analysis
— Wayfarer
:grin: If you like. You insist on telling us, at great length, about the ineffable. Fair enough. I'll continue to point out that you haven't, thereby, said anything. — Banno

That's it. Nothing has been said, although plenty has been shown. :wink:

He wasn't there again today. Oh, how I wish he'd go away. — Banno

:lol: Although it's really more like: He wasn't there again today. Oh, how I wish he'd come to stay.

Electrons have only one dimension. Do they exist? :chin:

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.

they only exist as particles under specific conditions of measurement. Otherwise, their existence is uncertain or indeterminate.

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.

I've sometimes observed that the last aphorism in Wittgenstein's Tractatus ('that of which we cannot speak') is often used as a firewall against metaphysics. That is certainly how the Vienna Circle understood and deployed it. But if something can't be said, it might be important to say why and surely philosophy has a role to play there.

, , 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.

...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.

Another thing. Earlier in this thread, I linked to the Smithsonian Mag article on this topic, 'What is Math?' There was a statement made in that essay:

Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?

Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)

Notice that question from Pigliucci - 'what kind of existence does it have?' That's the underlying question in this whole topic. That, and the reflexive association with intelligible objects and religious belief. It's because it's a metaphysical issue, and the metaphysics is hard to reconcile with naturalism. Which is why I keep going back to Platonism (not that I'm any kind of expert in it) - because it allows for levels of knowing and being, and hierarchical ontology. (And this also is being brought up to date e.g. Vervaeke's reconstitution of neoplatonism.)

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.

'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?"

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.

You're talking epistemology here, not ontology or metaphysics.

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

Of course, but philosophy has no atmosphere to breathe in when it comes to the ineffable—it's the realm that myth and poetry and religion attempt to fill, and we do well to remember that our imaginative creations are just that, and refrain from participating in the hubris that leads to fundamentalism.

Notice that question from Pigliucci - 'what kind of existence does it have?' That's the underlying question in this whole topic. — Wayfarer

You will say it doesn't have any kind of existence but is nonetheless real. And that just begs the question: 'Real in what way?' which is just a repeat of the question 'What kind of existence does it have?'.

Without any application, it's all just playing with words.

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.

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. — Banno

I am interested in much of what @Wayfarer is interested in too. but not in the fundamentalist way. In other words, I don't believe in the possibility of the direct knowing of transcendent truths, which he does, but I do believe in the possibility of personal transformation, as in altered states of consciousness; I just don't make any assumptions as to what is the significance of altered states in any ontological or metaphysical connection.

Because of that I am labeled a "positivist" and summarily dismissed.

:up:

but I do believe in the possibility of personal transformation, as in altered states of consciousness — Janus

Perhaps an aside, but I am curious. To what end? What is the point of the personal transformation you are thinking of - where does it lead?

It's likes the arts— leads nowhere except to novel and perhaps inspiring experiences.

Ok. It's just that the words 'personal transformation' sound a bit more serious than just amusing oneself.

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.

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. — Banno

I agree. It just occurred to me that physics posits items that are just as far fetched as abstract objects, point particles being one of them.

Ok. It's just that the words 'personal transformation' sound a bit more serious than just amusing oneself. — Tom Storm

I think we are being transformed all the time by our experiences. We can also take a more conscious hand in that transformation via various practices—the arts, meditation, spending time in the wilderness, running and even sports. There are many kinds of cultivation.

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... — Banno

I think when it comes to matters of faith personal experiences may serve as justifications for one's own (but certainly not anyone else's) beliefs (although personally I prefer to draw no conclusions).

Justification of direct knowledge is not intersubjectively possible, in any sense analogous to the way empirical and logical justification is. So, it looks like I'm an empiricist and a logical positivist, which I'm not really in the fullest sense. I'm with Wittgenstein in thinking that the most important aspects of life cannot be definitively spoken about—they can only be alluded to. It's handwaving all the way down, but for me there is a great deal of beauty to be found in some hand-waving.

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

How, pray tell, did we get from a brief comment about Bohr and Heisenberg, first to 'the ineffable', and then 'religious fundamentalism'?

It reinforces the earlier point I keep making - mere mention of the Platonic intuition that 'ideas are real' just automatically pushes these buttons. Everyone involved in this conversation ought to be aware of that. it's cultural conditioning, pure and simple.

The way it must be constructed to satisfy physicalism, is to say that ideas are the product of brains, which are the product of evolution, which is the product of the interaction of physical forces. To question that, is to be called a fundamentalist, because it is itself a kind of disguised fundamentalism.

Physics is the question of what matter is. Metaphysics is the question of what is real. People of a rational, scientific bent tend to think that the two are coextensive—that everything is physical. Many who think differently are inspired by religion to posit the existence of God and souls; Nagel affirms that he’s an atheist, but he also asserts that there’s an entirely different realm of non-physical stuff that exists—namely, mental stuff. The vast flow of perceptions, ideas, and emotions that arise in each human mind is something that, in his view, actually exists as something other than merely the electrical firings in the brain that gives rise to them—and exists as surely as a brain, a chair, an atom, or a gamma ray.

In other words, even if it were possible to map out the exact pattern of brain waves that give rise to a person’s momentary complex of awareness, that mapping would only explain the physical correlate of these experiences, but it wouldn’t be them. A person doesn’t experience patterns, and her experiences are as irreducibly real as her brain waves are, and different from them.

Nagel offers mental activity as a special realm of being and life as a special condition—in the same way that biology is a special realm of science, distinct from physics. His argument is that, if the mental things arising from the minds of living things are a distinct realm of existence, then strictly physical theories about the origins of life, such as Darwinian theory, cannot be entirely correct. Life cannot have arisen solely from a primordial chemical reaction, and the process of natural selection cannot account for the creation of the realm of mind. — Thoughts are Real (Review of Nagel, Mind and Cosmos)

Back to the Third Realm.

...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".

I think it’s fair to say that Habermas sees rationality as procedural, and the procedure necessarily involves language.

A pretty common position. I think Robert Sokolowski does a good job explaining the intuitions that support this position, and demonstrating the centrality of the role of language in the "Human Conversation" (our collective pursuit of knowledge), without running into the problem of reducing reason to language alone (or worse something like computation, symbol manipulation, etc.).

At any rate, such a conception of reason would seem to risk loading the dice against mathematical platonism (and Platonism for that matter) to some degree. All the problems of the "linguistic turn" come up. One cannot "get outside language" or "outside the linguistic community." If these are problems that are assumed as essentially axiomatic, then it hardly seems that one can step outside the sensible world and attain a noetic apprehension of mathematical objects when "notetic apprehension" just is something to do with language (which is grounded in communities, etc.).

By comparison:

Intellectus is the higher, so that if we call it ' understanding', the Coleridgean distinction which puts 'reason' above ' understanding' inverts the traditional order. Boethius, it will be remembered, distinguishes intelligentia from ratio; the former being enjoyed in its perfection by angels. Intellectus is that in man which approximates most nearly to angelic intelligentia; it is infact obumbrata intelligentia, clouded intelligence, or a shadow of intelligence. Its relation to reason is thus described by Aquinas: 'intellect (intelligere) is the simple (i.e. indivisible, uncompounded) grasp of an intelligible truth, whereas reasoning (ratiocinari) is the progression towards an intelligible truth by going from one understood (intellecto) point to another. The difference between them is thus like the difference between rest and motion or between possession and acquisition.

We are enjoying intellectus when we 'just see' a self-evident truth; we are exercising ratio when we proceed step by step to prove a truth which is not self-evident. A cognitive life in which all truth can be simply ' seen' would be the life of an intelligentia, an angel. A life of unmitigated ratio where nothing was simply ' seen' and all had to be proved, would presumably be impossible; for nothing can be proved if nothing is self-evident. Man's mental life is spent in laboriously connecting those frequent, but momentary, flashes of intelligentia which constitute intellectus.

When ratio is used with this precision and distinguished from intellectus, it is, I take it, very much what we mean by 'reason' today; that is, as Johnson defines it, 'The power by which man deduces one proposition from another, or proceeds from premises to consequences'.

C.S. Lewis - The Discarded Image

That is, reason has become merely ratio. And if intellectus survives, it survives as a grasp of simple axioms, the principle of non-contradiction, etc. In fact though, this is still extremely different because the idea was that all wholes could be "grasped as wholes" and that there might be gradations to a sort of noetic understandings (not the understanding had through discursive demonstrations). The sequestering of intellectus to merely the realm of axioms and "hinge propositions" is sort of the flip side of the reduction of reason to ratio.

Anyhow, the following paragraphs are less relevant but I find the application to literature interesting:

But, having so defined it, he gives as his first example, from Hooker, 'Reason is the director of man's will, discovering in action what is good'. There would seem to be a startling discrepancy between the example and the definition. No doubt, if A is good for its own sake, we may discover by reasoning that, since B is the means to A, therefore B would be a good thing to do. But by what sort of deduction, and from what sort of premises, could we reach the proposition 'A is good for its own sake' ?

This must be accepted from some other source before the reasoning can begin; a source which has been variously identified-with 'conscience' (conceived as the Voice of God), with some moral 'sense' or 'taste', with an emotion ('a good heart'), with the standards of one's social group, with the super-ego.Yet nearly all moralists before the eighteenth century regarded Reason as the organ of morality. The moral conflict was depicted as one between Passion and Reason, not between Passion and 'conscience', or ' duty', or ' goodness'. Prospero, in forgiving his enemies, declares that he is siding, not with his charity or mercy, but with'his nobler reason' (Tempest, v, i, 26). The explanation is that nearly all of them believed the fundamental moral maxims were intellectually grasped. If they had been using the strict medieval distinction, they would have made morality an affair not of ratio but of intellectus.

...The belief that to recognise a duty was to perceive a truth-not because you had a good heart but because you were an intellectual being-had roots in antiquity. Plato preserved the Socratic idea that morality was an affair of knowledge; bad men were bad because they did not know what was good. Aristotle, while attacking this view and giving an important place to upbringing and habituation, still made 'right reason' ( 6p6os Myos) essential to good conduct. The Stoics believed in a Natural Law which all rational men, in virtue of their rationality, saw to be binding on them. St Paul has a curious function in this story. His statement in Romans (ii. 14 sq.) that there is a law ' written in the hearts' even of Gentiles who do not know 'the law', is in full conformity with the Stoic conception, and would for centuries be so understood.

Nor, during those centuries, would the word hearts have had merely emotional associations. The Hebrew word which St Paul represents by Kap5ia would be more nearly translated ' Mind' ; and in Latin, one who is cordatus is not a man of feeling but a man of sense. But later, when fewer people thought in Latin, and the new ethics of feeling were corning into fashion, this Pauline use of hearts may well have seemed to support the novelty.

The importance of all this for our own purpose is that nearly every reference to Reason in the old poets will be in some measure misread if we have in mind only ' the power by which man deduces one proposition from another'. One of the most moving passages in Guillaume de Lorris' part of the Romance of the Rose (5813 sq.) is that where Reason, Reason the beautiful, a gracious lady,a humbled goddess, deigns to plead with the lover as a celestial mistress, a rival to his earthly love. This is frigid if Reason were only what Johnson made her. You cannot turn a calculating machine into a goddess. But Raison la bele is 'no such cold thing'. She is not even Wordsworth's personified Duty; not even-though this brings us nearer-the personified virtue of Aristotle's ode, ' for whose virgin beauty men will die' (o-O:s TIEpt, 1rap6eve, J.!Opcpas) .She is intelligentia obumbrata, the shadow of angelic naturein man. So again in Shakespeare's Lucrece we need to know fully who the 'spotted princess' (719-28) is: Tarquin' s Reason, rightful sovereign of his soul, nowmaculate.

Many references to Reason in Paradise Lost need the same gloss. It is true that we still have in our modern use of ' reasonable' a survival of the old sense, for when we complain that a selfish man is unreasonable we do not mean that he is guilty of a non sequitur or an undistributed middle. But it is far too humdrum and jejune to recall much of the old association.

Anyhow, a question philosophy of language has generally ignored (often deciding it belongs to some other field) is: "Why do we feel compelled to speak? Why speak at all, and why of this and not that?"

This is the question Umberto Eco puts front in center in Kant and the Platypus. I think it's also the question that comes front and center when it comes to mathematical objects (or at least if one wants to attempt to explain them in terms of language).

There is the historical question of: "why did disparate cultures all come up with terms for magnitude and multitude?" but also the more immediate question of "why do people feel impelled to use these terms so often?" The platonists seems to have some sort of answer. The more compelling rebuttal would seem to require not only showing problems with the platonists' position, but showing how else these questions might be answered.

Do not qualify yet. Once infinity and it's opposite are well defined (and infinity isn't just taken as an axiom), they likely would be Platonic objects. At least I have enough belief in the "logicism" of mathematics that it is so. — ssu

The point though, is that "infinity" and "infinitesimal" refer to completely different things. That "infinity" refers to a Platonic object does not imply that "infinitesimal" does.

”Intellectus is the higher, so that if we call it ' understanding', the Coleridgean distinction which puts 'reason' above ' understanding' inverts the traditional order. Boethius, it will be remembered, distinguishes intelligentia from ratio; the former being enjoyed in its perfection by angels” — Count Timothy von Icarus

Intellectus is the Latin term adopted by Roman philosophers like Cicero and later by medieval Scholastics to translate nous from Greek philosophical texts. It similarly denotes the capacity for intellectual intuition or understanding of universal principles. Nous (and therefore Intellectus) is a key term for the higher faculty of the soul, distinct from reason (ratio), which operates discursively. … In the Aristotelian scheme, nous is the faculty that underwrites the capacity of reason. For Aristotle, nous was distinct from the processing of sensory perception, including the use of imagination and memory, which animals can do. For Aristotle, discussion of nous is connected to discussion of how the human mind sets definitions in a consistent and communicable way (through the grasp of universals) and whether people must be born with some innate potential to understand the same universal categories in the same logical ways. Derived from this it was also sometimes argued, in classical and medieval philosophy, that the individual nous must require help of a spiritual and divine type. By this type of account, it also came to be argued that the human understanding (nous) somehow stems from this cosmic nous, which is however not just a recipient of order, but a creator of it. — various sources including Wikipedia

— various sources including Wikipedia — Wayfarer

"Various sources"? What does that mean, that it's an AI generated piece of crap, compiled from cherry picking sites most often visited? Isn't that just internet mob mentality?

do they? I'm sure it's heresy and utterly crazy for many and people will refer me to take basic math lessons, but what if you have ∞ for infinity and 1/∞ for infinitesimal as Platonic objects in Math?

And we do have problems in understanding infinity, so how Platonic that is, is a question. Or otherwise I guess the Continuum Hypothesis has been solved.