Snails do not have access to a platonic reality. It's not some mystical or divine intervention, but a simple result of a snail adding calcium to the edge of it's shell. — Banno

Sure. I agree. Josh and I were talking about "constructive interaction" with the environment and how that might be the genesis of universals like numbers. He said:

It’s not that the world isn’t involved, it’s just that the world only reaches us through our constructive interactions with it. We are an intrinsic part of the world, and the Real is the effect of a two-way interaction. — Joshs

That led to a little discussion of how that actually works in individual cases. We didn't get very far tho.

We bring one and two into existence, by an intentional act - it's something we do. Some important aspects of this. First, its we who bring this about, collectively; this is not a private act nor something that is just going on in the mind of one individual. Hence there are right and wrong ways to count. — Banno

I agree with the emphasis on the collective creation of counting (if the non-Fregean story is correct). I'm not sure I'd go so far as to say that intersubjective agreement results in the idea of "being right about counting." One can imagine mistakes in math that are widely accepted, but then corrected by reference to some Popperian discovery in the 3rd world. Wouldn't we say that it was that discovery that now made us "right," rather than the fact that everyone now agrees? After all, we agreed before, too.

Next, the existence had here is that of being the subject of a quantification, as in "Two is an even number". — Banno

Extremely important. @Michael and I are having a related conversation about what role "existence" plays in descriptions of platonism, and it hinges on a similar point. In the case you describe later in your post, the moral of the story would be: "P" is brought into existence depending upon an interpretation of (ideal logical) language; there are no facts in the world that change as a result of that interpretation. If -- as I do -- I lean toward the quantificational interpretation that allows P to be a "new thing," and if you dispute it, we aren't offering arguments pro and con about the object of the concept "to exist". We are specifying that very concept, rather than assuming it, in our differing interpretations. Or so it seems to me. And I think it's the gist of your comment here:

the account I gave above indicates how stuff like numbers and property and so on are constructed, by modelling that construction in a higher order logic. — Banno

I'm not a best-friend of formalism, but this is the kind of case where formal models really excel.

Yes, these parallels with Augustine are good. Anyone who favors the idea of an intelligible realm is going to have to say whether there's anything populating it before we humans arrive; Plato, Augustine, and Frege opt for saying that it's full, and we encounter it as such.

Which leads to the passages from Peirce and Nagel. History of philosophy isn't my forte, and I defer to Nagel on this, though it does seem a little oversimplified? I suppose there is a generalized "fear of religion," especially in analytic phil., but anti-Platonists seem to be offering genuine justifications for their position, that have to be taken seriously in their own right. And though my own sympathies are with religious modes of life, I don't doubt for a minute that one can be an anti-Platonist and a non-believer without also subscribing to what you're calling "the relativization of reason." Nagel himself is a good example. So is Habermas. And really, so is (most of) analytic phil., which questions various points concerning reason but rarely abandons it to relativism; the questioning is itself usually done using entirely standard assumptions about reason and its grounding.

Which is maybe just to say that evolutionary explanations aren't the only game in town, if one is dubious about platonism.

History of philosophy isn't my forte, and I defer to Nagel on this, though it does seem a little oversimplified? — J

Not at all. History of ideas is very much my interest - more so that what is taught as philosophy nowadays - and I see the issue in terms of the cultural dialectics sorrounding philosophy, religion and science. The major point I take from it, aside from the often-quoted passage about the fear of religion, which really is a major underlying factor in my view, the bulk of the essay is a defense of reason against attempts to explain it as a product of evolution. The main argument being, to say that it is, is to undermine the sovereignty of reason:

The only form that genuine reasoning can take consists in seeing the validity of the arguments, in virtue of what they say. As soon as one tries to step outside of such thoughts, one loses contact with their true content. And one cannot be outside and inside them at the same time: If one thinks in logic, one cannot simultaneously regard those thoughts as mere psychological dispositions, however caused or however biologically grounded. If one decides that some of one's psychological dispositions are, as a contingent matter of fact, reliable methods of reaching the truth (as one may with perception, for example), then in doing so one must rely on other thoughts that one actually thinks, without regarding them as mere dispositions. One cannot embed all one's reasoning in a psychological theory, including the reasonings that have led to that psychological theory. The epistemological buck must stop somewhere. By this I mean not that there must be some premises that are forever unrevisable but, rather, that in any process of reasoning or argument there must be some thoughts that one simply thinks from the inside--rather than thinking of them as biologically programmed dispositions. — Thomas Nagel op cit

Whereas I'm pretty confident the majority opinion is that reason can only be understood in terms of evolutionary development, because what else is there?

There's also been discussion of another book from time to time, The Eclipse of Reason, Max Horkheimer, which makes the case that the sovereignty of reason as understood in classical philosophy has been progressively subsumed by instrumentalism and pragamatism - the utilitarian ends to which reason can be directed. In fact the whole conception of reason changed with the scientific revolution (per Alexander Koyré). It is no longer understood as a cosmic animating principle, but as a human invention (numbers are invented not discovered). That's what I mean by the relativising of reason (reference).

So - they're the themes I'm exploring. But I agree that it is a different to the subject matter to philosophy per se.

Links of interest:

Does Reason Know what it is Missing? - on Habermas' dialogue with Catholicism.

Join the Ur-Platonist Alliance! - Edward Feser on Lloyd Gerson

The sequence of natural numbers is a human construction. But although we create this sequence, it creates its own autonomous problems in its turn. The distinction between odd and even numbers is not created by us; it is an unintended and unavoidable consequence of our creation. — Objective Knowledge, 118

I don't entirely agree with this. The characteristics of the natural numbers—oddness, evenness, divisibility and primeness—are clearly shown in the ways groups of actual things can be divided up. I do agree, though, that once these characteristics are formulated as rules then the characteristics of extremely large numbers—numbers too large to be worked with by arranging actual objects in order to discover such characteristics—follow logically.

I see the issue in terms of the cultural dialectics sorrounding philosophy, religion and science — Wayfarer

Mathematical platonism originated with Frege. He lived at a point when a scientific outlook was coming to dominate the intellectual scene, but panpsychism and the idea of a world-mind mixed easily with mechanistic thinking at that point. There was no conflict until the 20th Century when eliminatism became fashionable.

I lean toward the quantificational interpretation that allows P to be a "new thing" — J

It's not as if we must choose and stick to only the quantificational interpretation, or alternately we must only ever use the substitutional interpretation. Which we use depends on what we are doing, on the task in hand.

...intersubjective agreement... — J

A misleading phrase, since it implies a background of subjectivity prior to, say, counting; the incoherence of the solipsistic homunculus talking to other homunculi. What is salient is that arithmetic is an interaction between people, and this is so even if one occasionally counts to oneself.

The only form that genuine reasoning can take consists in seeing the validity of the arguments, in virtue of what they say. — Thomas Nagel op cit

A pity then that did not address the argument of my post directly, but instead could only see it as reactionary 'fear of religion'. But yes, it is circular to reason that evolution is needed in order to explain reason. The relevance of that remains obtuse.

We can take Quine's joke seriously: to be is to be the subject of some quantification; and us that to reply to the OP.

Well, Frege is a modern representative of it, but it really does go back to the ancients:

Neoplatonic mathematics is governed by a fundamental distinction which is indeed inherent in Greek science in general, but is here most strongly formulated. According to this distinction, one branch of mathematics participates in the contemplation of that which is in no way subject to change, or to becoming and passing away. This branch contemplates that which is always such as it is and which alone is capable of being known: for that which is known in the act of knowing, being a communicable and teachable possession, must be something that is once and for all fixed — Jacob Klein, Greek Mathematical Thought and the Origin of Algebra

That is a common thread throughout practically all pre-modern philosophy.

We can take Quine's joke seriously: to be is to be the subject of some quantification — Banno

Oh, I think he meant it! But more to follow.

Well, Frege is a modern representative of it, but it really does go back to the ancients — Wayfarer

This isn't true. It's unfortunate that this wasn't made clear at the outset.. Mathematical platonism, otherwise known as realism, is just the view that mathematical objects are neither mental nor physical. We call them abstract objects. That's it. There's no accompanying doctrine.

So - they're the themes I'm exploring. But I agree that it is a different to the subject matter to philosophy per se. — Wayfarer

Sure. And Nagel is one of my favorites. I was raising a brow at the idea that fear of religion, specifically, accounts for the current interest in naturalized Explanations of Everything. The passage you cited -- and just about all of The Last Word -- articulates a position that I think is broadly correct, but you can hold it and still be an atheist to the core. Likewise, you can find it unconvincing on the merits, not because you're afraid you'll "cross the Tiber" (as they used to say) if you start believing that reason provides a privileged access to the world.

And yes, the Habermas discussion is very interesting. He's still alive, you know -- we could ask him what he thinks now (at 95!).

It might be interesting to check again. I'd just read this:

‘Esse is percipi,’ wrote the empiricist metaphysician George Berkeley around 1710: ‘To be is to be perceived.’ For something to exist or be real, for Berkeley and for many others (Immanuel Kant, for example), was for it to play certain roles in human perception or to correspond to our mental imagery. In a tribute to that style of metaphysics and a parody of it, in 1939 Quine said that ‘to be is to be the value of a variable.’ Now, Quine took himself to be ridiculing the grand pronouncements of metaphysics. But it was hard not to hear that ‘bound variable’ stuff as itself an ontological theory according to which existence is dependent on language: to be was to be picked out by the ‘something’ in sentences like ‘there is something that’s tall and green’ (or, in the language of logic, (∃x)(Fx&Gx), in which the existential quantifier binds the variable ‘x’). — Sartwell, The post-linguistic turn

The theme, one that may be becoming prevalent, is that post modernism has noticed that not just any narrative will do. Global warming does not care what narrative you adopt, and relativism works for oligarchs as well as anarchists. The truth doesn't care what you believe. That's for @Joshs.

articulates a position that I think is broadly correct, but you can hold it and still be an atheist to the core. — J

Well, Nagel says he is. But he's philosophically open to a somewhat religious perspective, the idea expressed in Mind and Cosmos of rational sentient beings as the universe coming to self-awareness. That is a theme that animates many kind-of religious philosophies, like Hermeticism. Besides, I think the missing dimension is not the idea of God, but to the entire category of the sacred.

I did register that Habermas was still with us. He has a massive corpus which again I've barely touched, but I came across his dialogue with then Cardinal Ratzinger. Whilst I am not Catholic, and find much that is disagreeable about that institution, Catholicism is still arguably one of the conduits through which a form of the philosophia perennis has been preserved and transmitted. In my recent (2022) trip to Florence, I was impressed by the frescos showing Aquinas laying down the lore to the assembled gathering of philosophers.


Spanish Chapel of the Church of Santa Maria Novella, Florence.

Mathematical platonism, otherwise known as realism, is just the view that mathematical objects are neither mental nor physical. We call them abstract objects. That's it. There's no accompanying doctrine. — frank

However it presents an obvious ontological question. As SEP puts it, and as I'm sure I've previously quoted:

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects that aren’t part of the causal and spatiotemporal order studied by the physical sciences.[1] Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.

Although these philosophical consequences are not unique to mathematical platonism, this particular form of platonism is unusually well suited to support such consequences. For mathematics is a remarkably successful discipline, both in its own right and as a tool for other sciences.[2] Few contemporary analytic philosophers are willing to contradict any of the core claims of a discipline whose scientific credentials are as strong as those of mathematics (Lewis 1991, pp. 57–9). So if philosophical analysis revealed mathematics to have some strange and surprising consequences, it would be unattractive simply to reject mathematics.[3] A form of Platonism based on a discipline whose scientific credentials are less impressive than those of mathematics would not be in this fortunate situation. For instance, when theology turns out to have some strange and surprising philosophical consequences, many philosophers do not hesitate to reject the relevant parts of theology.

Yes, it's a puzzle. It's not a conflict between religion and science tho, that's all I meant.

Perhaps not - but it's a metaphysical question, and insofar as metaphysics is usually associated with religion, rightly or not, it ends up being tarred with the same brush.

Everything is a conflict between science and religion, for . Here we are talking about Mathematics, and he must introduce god, but not in so many words. Moreover, he sees any objection to this unneeded insertion as further evidence of a supposed scientistic fear of religion.

God explains everything, and so might be introduced into any topic. But of course in explaining everything he explains nothing.

And of course Wayf is entitled to introduce god, just as others are entitled to point out that he is not necessary.

:grin: The number of experts in phil of math is tiny. You have to be an expert in two fields. Maybe one day such a person will pass our way! That would be cool.

Now, Quine took himself to be ridiculing the grand pronouncements of metaphysics. But it was hard not to hear that ‘bound variable’ stuff as itself an ontological theory according to which existence is dependent on language: — Sartwell, The post-linguistic turn

You can sense the parodic aspect of the Quinean formula, but I always took him to be saying, essentially, "There is no way to usefully define 'existence' such that all customers will be satisfied, so let's just limit existence-talk to what we do with quantification." And I don't think the resulting ontological "theory" says that existence is dependent on language -- it's dependent on a certain understanding of logical thought, which we're free to maintain is independent of language if we want to. See Frege the platonist.

Here we are talking about Mathematics, and he must introduce god, but not in so many words. Moreover, he sees any objection to this unneeded insertion as further evidence of a supposed scientistic fear of religion. — Banno

A very shallow analysis, Banno, although easy to stereotype, which is what you're doing. There's an excellent book mentioned by me and others from time to time, The Theological Origins of Modernity, Michael Allen Gillespie, 2009, which I read when first joining forums, and which gives the deep background to these disputes.

… the apparent rejection or disappearance of religion and theology in fact conceals the continuing relevance of theological issues and commitments for the modern age. Viewed from this perspective, the process of secularization or disenchantment that has come to be seen as identical with modernity was in fact something different than it seemed, not the crushing victory of reason over infamy, to use Voltaire’s famous term, not the long drawn out death of God that Nietzsche proclaimed, and not the evermore distant withdrawal of the deus absconditus Heidegger points to, but the gradual transference of divine attributes to human beings (an infinite human will), the natural world (universal mechanical causality), social forces (the general will, the hidden hand), and history (the idea of progress, dialectical development, the cunning of reason). … — Reader Review

A background which is transparently clear in many of your comments.

@Count Timothy von Icarus already referred to the Analogy of the Divided Line upthread, in that, there is an hierarchical ontology, meaning different levels of being or existence. Which has what has been 'flattened out' by modern ontology, and why the ontology of abstract objects is so difficult to account for.

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.

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

That's a lovely display, thanks.

Lots to be said about Nagel and religion. Is he really open to religious belief? We know that he doesn't want religion to be true, and that he's provoked (in a good way) by the fact that so many philosophers he respects are deists of one stripe or another. Your distinction between belief in God and belief in a sacred dimension may be relevant here.

I've read a bunch of Habermas but as you say, there's a mega-bunch to read! You'd probably like Between Naturalism and Religion (2008), which addresses a lot of our topics here. His concept of a "detranscendentalized use of reason" is a real contribution. He is absolutely unwilling to give up the Nagelian position that reason is the "last word," and equally unwilling to accept the traditional foundationalist explanations for why this is so. In addition, several of the essays in the book are extremely sympathetic discussions of the role of religious belief -- and religious adherents -- in secular, liberal society.

You'd probably like Between Naturalism and Religion (2008) — J

I bet, looks right up my street, thanks for it.

Lots to be said about Nagel and religion. Is he really open to religious belief? — J

I sometimes wonder if he's being dragged kicking and screaming......

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

Fine! I have realised the link between Terrence Deacon's absentials and the via negativa. Anyway, as you say, enough for today, thanks all for the comments :pray:

there is an hierarchical ontology, meaning different levels of being or existence. — Wayfarer

Well, here we are again. That is absolutely one way to employ the terms "being" and "existence," a way with a distinguished history. If you were willing to say that the hierarchical organization may be an actual metaphysical structure but not necessarily described by the terms you've used (being, existence, ontology), I would be inclined to accept that. The map is not the labels.

Yes, indeed - and hereabouts so many of the discussions of existence, essence, and what is real would benefit from simply understanding quantification.

And I don't think the resulting ontological "theory" says that existence is dependent on language — J

Yep. And to this I would add that the relation between what exists and what we do is worth considering. 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?

Fine! I have realised the link between Terrence Deacon's absentials and the via negativa. Anyway, as you say, enough for today, thanks all for the comments :pray: — Wayfarer

:wink: My objections just show that you are right.

The theme, one that may be becoming prevalent, is that post modernism has noticed that not just any narrative will do. Global warming does not care what narrative you adopt, and relativism works for oligarchs as well as anarchists. The truth doesn't care what you believe. That's for Joshs. — Banno

Given your emphasis on language as use, I want to point out the implication of truth not caring about what one believes, desires, feels or cares about. That is , a notion of truth as pristinely separate from issues of affect, value, power and purpose, as though those factors were at best peripheral to, and at worst repressive of attainment of truth. Pragmatism and hermeneutics, which treat truth as a function of discursive practices, know better than that.

What one believes, desires, feels or cares about will still be there, even if one is honest.

Mathematical platonists distinguish themselves from non-platonists like nominalists. Each group seems to understand what the other means, hence their disagreement. I'm asking if infinitesimals exist in the sense that would satisfy mathematical platonism.

I'm asking if infinitesimals exist in the sense that would satisfy mathematical platonism. — Michael

Have you still not answered this question? 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.

This creates a schism in mathematics because calculus requires infinitesimals, while set theory assumes Platonism. So instead of employing infinitesimals, set theory views infinities as well-defined objects.

Of course mathematicians will not admit to an inconsistency between calculus and set theory, they would just claim that one is an extension of the other, just like many physicists would not admit to an inconsistency between Newtonian laws (governing objects) and Einsteinian laws (governing spacetime) . What they do instead, is veil the inconsistency behind a whole lot of extra axioms and principles, designed to smooth out the bumps, and hide the inconsistencies which exist between different applications which use different principles.

Simply put, "infinitesimal" refers to the continuity (like a "dimensional line", or space) which is assumed to lie between discrete objects (which may be infinite in number), as required to maintain separation between the assumed objects, making them discrete.

So the two, infinitesimal space, and infinite objects, require completely different accounting principles. The infinite objects are given by Platonism, but they require a "space" to be, in order to account for them being discrete objects, and since the objects are infinite, the "space' where they exist must be infinitesimal. Notice that "infinitesimal" refers to what is outside the Platonic objects.