Can you state just why you think that incompatibility obtains? — Janus

That's not my claim. It is in the article that I referred to, The Indispensability Argument in the Philosophy of Mathematics. That article says that mathematical knowledge and rationalist philosophy is incompatible with 'our best epistemic theories' which as @Banno pointed out is a reference to naturalism. Make of that what you will.

As for the unfathomable subtlety of living organisms, I'm all for it. I think many things we describe as 'instinct' are impossible to fathom, but that's a completely separate issue.

There is a longstanding dispute over the univocity of being (and predication) between the Thomists and the Scotists beginning in the Medieval period. The Scotists held to univocity (and Heidegger's first dissertation was on this topic, on a text then believed to be Duns Scotus'). — Leontiskos

I've learned that the dissident theological movement, Radical Orthodoxy, sees Duns Scotus' univocity (in combination with Ockham's nominalism) as the source of the decline of modern culture.

Like Macbeth, Western man made an evil decision, which has become the efficient and final cause of other evil decisions. Have we forgotten our encounter with the witches on the heath? It occurred in the late fourteenth century, and what the witches said to the protagonist of this drama was that man could realize himself more fully if he would only abandon his belief in the reality of transcendentals. The powers of darkness were working subtly, as always, and they couched this proposition in the seemingly innocent form of an attack upon universals. The defeat of logical realism in the great medieval debate was the crucial event in the history of Western culture; from this flowed those acts which issue now in modern decadence. — Richard Weaver, Ideas have Consequences

Many thanks for your perspective.

The conjecture of 'Fine Tuning" raises the spectre of Intelligent Design — Gnomon

What I see is only that the causal sequence that gave rise to life and mind didn’t commence with the formation of earth, or the formation of stars. But the point I was making was simply that the very idea of a ‘vast univere’ in which we are a ‘mere blip’ is something that only rational sentient beings understand. Again the objective view relegates us to blip-hood in our own minds.

I've been reading The Huxleys, Alison Bashford. There are many ideas like this in their writings. I think I quoted Julian Huxley already in this thread, along the lines that in h. sapiens, evolution has become self-aware. His brother Alduous, with whom he had a life-long intellectual companionship, was the more spiritually-inclined of the two. Interestingly, the book shows that throughout the generations beginning with T.H.H. ('Darwin's Bulldog') they all pursued themes of the intersection of religion, philosophy and science. It also shows that T H H was scrupulously agnostic, as distinct from atheist, and that he disdained the Dawkin's style of scorched-earth scientific atheism.

Only in your misreading of them. A perceptive reader would notice a much closer resemblance to form-matter dualism which is quite a different thing to Decartes’.

//there’s nothing in the post you’re responding to which suggests mind-matter duality. The contrast, rather than the dualism, was between sensable and intelligible. But in reality, mind synthesises both elements in arriving at judgement.//

Nothing whatever to do with Cartesian dualism, which never made any such distinction.

//but thank you all the same//

Does life have any potential to be anything beyond suffering, or is that too much of a pessimistic stance? — Arnie

It is well known that the 'four noble truths' of Buddhism begin with the observation or axiom that life is suffering - the Buddhist term is 'dukkha' which is difficult to translate, but which is usually represented as suffering or sorrowful (the word actually comes from a badly-fitted axle hole). In a forum post such as this it would probably be unwise to try and spell out all the remaining 'noble truths' in detail, save that they say that suffering has a cause and so also a solution or ending, which is the purpose of the Buddhist eightfold path.

The value of a single human life? — Gingethinkerrr

You might be pleased to learn that the inestimable value of each human life was a foundational principle of the great philosopher Immanuel Kant. He argued that human beings possess intrinsic worth, which he calls "dignity," because of their capacity for rationality and autonomy. This dignity makes each person deserving of respect and moral consideration. Kant insists that we must never treat others merely as means to an end but always as ends in themselves. This means recognizing and valuing the inherent worth of every individual, thus ensuring that our actions support their ability to make autonomous decisions and pursue their own goals. This principle is intended to guide all human interactions and forms a foundational aspect of Kant's ethics, emphasizing duty and the universality of moral law.

And welcome to thephilosophyforum.

I've tried to have you fill this out explicitly. If what you say here were so we would have a neat case of quantification variance to work with - the difference between real and existent. But i do nto think you have been able to proved a coherent account. — Banno

Thanks for that, we have some agreement. Maybe to you the issue I'm commenting on 'goes without saying' but I think there's something that needs to be spelled out:

I don't agree with the premise of the argument - that naturalism is our "best" epistemic theory. — Banno

So you wouldn't endorse

Quine’s belief that we should defer all questions about what exists to natural science is really an expression of what he calls, and has come to be known as, naturalism

//

I will try and re-state what I see the difference between real and existent at a high level.

By 'existent' I refer to manifest or phenomenal existence. Broadly speaking, this refers to sensable objects (I prefer that spelling as it avoids the equivocation with the other meaning of 'sensible') - tables and chairs, stars and planets, oceans and continents. They're phenomenal in the sense of appearing to subjects as sensable objects or conglomerates.

I am differentiating this from what used to be called 'intelligible objects' - logical principles, numbers, conventions, qualifiers and so on. For example, if I were to say to you, 'show me the law of the excluded middle', you would have to explain it to me. It's not really an 'object' at all in the same sense as the proverbial chair or apple. You might point to a glossary entry, but that too comprises the explanation of a concept. The same with all kinds of arithmetical proofs and principles. Even natural laws - the laws of motion, for example. All of these can only be grasped by a rational intelligence. I could not demonstrate or explain them to a cow or a dog. They are what could be described as 'noumenal' in the general (not Kantian) sense, being 'objects of intellect' (nous) - only graspable by a rational mind. (Significant that the Collins Dictionary definition of ‘noumenal’ is ‘real’ as distinct from ‘phenomenal appearance’, echoing scholastic realism.)

As I said at the outset, in regular speech it is quite clear to say 'the number 7 exists'. But when you ask what it is, then you are not pointing to a sensable object - that is the symbol - but a rational act. (That's the sense in which I mean that 'counting is an act', but it doesn't mean that the demonstrations of rudimentary reasoning in higher animals amounts to reason per se.)

In Plato these levels or kinds of knowledge were distinguished per the Analogy of the Divided Line . Those distinctions are what have been forgotten, abandoned or lost in the intervening millenia due to the dominance of nominalism and empiricism. But In reality, thought itself, the rational mind, operates through a process of synthesis which blends and binds the phenomenal and noumenal into synthetic judgements (per Kant).

That is the back-story of why the need is felt for 'the indispensability argument for mathematics', and the difficulties of accomodating mathematical knowledge into the procrustean bed of empiricist naturalism.

I believe in evolution but the Theory Of Evolution is woefully incomplete. I don't believe mutations can create a person, the works of Shakespeare, a Mozart symphony... — EnPassant

It was never intended to, but it’s occupied the space left by the collapse of creation mythologies.

you want to deploy the indispensability argument, no? — Banno

I’m not arguing in favor of it. I’m asking why it’s even necessary. I’m questioning the claim that ‘according to our best epistemic theories, mathematical knowledge ought not to be possible.’ It obviously is possible, so what does that say about the shortcomings of ‘our best epistemic theories’?

I quoted the paragraph which says 'rationalists claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.' And I think the 'rational insight arising from pure thought' is, in fact, reason. But according to this article, this appears to contradict naturalism. So I'm saying, so much the worse for naturalism.

@Srap Tasmaner may recall the debate over the 'argument from reason'.

Here it is again, but as I'm yelling into a gale, I'll desist.

SO I must be misunderstanding what you are saying. — Banno

Indeed you are. I will reply later, dealing with domestic duties today.

A crow that collects three sticks or whatever is acting, — Banno

Take it aside and explain to it the meaning of ‘prime number’.

Quine’s belief that we should defer all questions about what exists to natural science is really an expression of what he calls, and has come to be known as, naturalism. — Banno

And Lloyd Gerson lays out the thesis, in Platonism and Naturalism: The Possibility of Philosophy, that (1) philosophy proper is Platonist, and (2) is incompatible with naturalism. This is what I believe is the underlying issue but I’ll fill that out later

Seems an odd position for you to be defending. — Banno

How am I defending it? I’ll come back to this later.

It's not complete, but it is better than looking for platonic realms. — Banno

Do you agree or disagree that mathematical knowledge is incompatible with ‘our best epidemic theories?’ That is the point on which the argument hinges.

Your article says "We learn about ordinary objects, at least in part, by using our senses." — Banno

Fair enough - but it’s not ‘my article’, it’s an encyclopedia article on a genuine controversy. Why it’s a controversy, and what the implications are, are what I’m interested in. Quite agree that rational abilities underpin our world, but not that they can be reduced to ‘social intentionality’ (which sounds like the kind of thing a Josh’s would say :yikes:)

This is where "reality" becomes a tempting term to introduce. — J

Phenomena - apparent, appearing
Noumenal - object of nous/intellect
Imaginary - fictional and literary

Forum content on a database can be worth a lot. — Shawn

Really? Where do I sell?

In any case, once it's posted, it's published, to all intents and purposes, in that anyone can read the posts here, even if only registered members can enter them.

I wonder what you mean when you say that numbers are real. — Janus

That they have a common reference, that the value of a number is not a matter of opinion or choice. I would like to say 'objective' but I don't think 'objective' is quite the right term - we refer to numbers to ascertain what is objective, but they are not really known as objects except for in the figurative sense of 'object of thought' (which actually is near to the original, as distinct from Kantian, view of the meaning of ’noumenal')

I'd love to agree. But where? Where can you show me innate good without walking me through constructs? — ENOAH

The part of you that would love to agree is an indication.

Republicans, led by former president Donald Trump, are refusing to commit to accept November’s election results with six months until voters head to the polls, raising concerns that the country could see a repeat of the violent aftermath of Trump’s loss four years ago.

The question has become something of a litmus test, particularly among the long list of possible running mates for Trump, whose relationship with his first vice president, Mike Pence, ruptured because Pence resisted Trump’s pressure to overturn the 2020 election.

In a vivid recent example, Sen. Tim Scott (R-S.C.) was pressed at least six times in a TV interview Sunday on whether he would accept this November’s results. He repeatedly declined to do so, only saying he was looking forward to Trump being president again.

He continued to evade the question even as the interviewer, NBC News’s Kristen Welker, reminded him that a “hallmark of our democracy is that both candidates agree to a peaceful transfer of power.”


“This is why so many Americans believe that NBC is an extension of the Democrat party at the end of the day,” Scott said at one point. “… I believe that President Trump will be our next president. It’s that simple.” — Top Republicans, led by Trump, refuse to commit to accept 2024 election results, Washington Post

**THE COUP ATTEMPT IS ONGOING**

Of course, numbers are the currency of quantification. But I don't think my previous reply to you was 'a cop-out.' I think in all likelihood you skimmed over it, and that in reality it's a live issue and also related to the OP (although I will admit that I'm not well-trained in the lexicon (dare I say jargon) of analytic philosophy.)

Certainly counting may be an act, indeed I believe it is, but that doesn't address the issue of the realness or otherwise of numbers - which is the ontological issue. And why that is significant is because of the centrality of mathematics to science, mathematical physics in particular, and because of the challenge that poses to physicalism. (I'm baffled that this is regarded as trivial.) Again I'll refer to an IEP article The Indispensability Argument in the Philosophy of Mathematics which I'm sure will be amenable to you, as it is based around an argument by Putnam and Quine, whom you probably know better than I do. So I would like to get your view of the matter:

In his seminal 1973 paper, “Mathematical Truth,” Paul Benacerraf presented a problem facing all accounts of mathematical truth and knowledge. Standard readings of mathematical claims entail the existence of mathematical objects. But, our best epistemic theories seem to deny that knowledge of mathematical objects is possible.

What are these 'best epistemic theories', and why are they irreconciliable with knowledge of mathematical objects? Well, the article goes on to explain:

Mathematical objects are in many ways unlike ordinary physical objects such as trees and cars. We learn about ordinary objects, at least in part, by using our senses. It is not obvious that we learn about mathematical objects this way. Indeed, it is difficult to see how we could use our senses to learn about mathematical objects. We do not see integers, or hold sets. Even geometric figures are not the kinds of things that we can sense. Consider any point in space; call it P. P is only a point, too small for us to see, or otherwise sense. Now imagine a precise fixed distance away from P, say an inch and a half. The collection of all points that are exactly an inch and a half away from P is a sphere. The points on the sphere are, like P, too small to sense. We have no sense experience of the geometric sphere.

And further along:

(Rationalist) philosophers claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.

I think this is the nub of the issue. Our 'best epistemic theories' are, of course, naturalist - as I'm sure Quine would affirm - and are underpinned by empiricism - that knowledge originates from sensory experience. And also, there is the suggestion that mathematical ability is innate - another strike against rationalism, from the empiricist point of view.

And yet, I can't help but think that it's obvious that humans do indeed have a 'non-sensory capacity for understanding mathematical truths' and that if that does throw shade on the view of humans as 'physical creatures', then so much the worse for it.

What say you?

But whereas you seem to be saying that these individuals are "real", I'm pointing out that they remain shorthand for an activity we can perform. — Banno

Excellent point and I will respond soon.

Rather, they are something we do. — Banno

And I don't think you understand the argument, nor want to understand it.

Actually I'm relieved that there's at least one other contributor more voluble than I :yikes: .

The historical theme that I refer to is the long aftermath of the dispute between realism and nominalism amongst the Medievals. I say that nominalism won the argument, and that, as history is written by the victors, it is now so thoroughly embedded in the philosophical lexicon that we no longer notice it, it has become axiomatic.

Due to some long-ago epiphany, I became interested in Platonic and Aristotelian realism (as I've mentioned many times). The fact that numbers and the like exist as intelligible objects strikes me as having profound philosophical importance, because, while they're indispensable to natural science, they are transcendental in the sense of being 'true in all possible worlds'. I don't believe they have a naturalistic explanation, as they are epistemologically prior to any coherent naturalism (which must assume the soundness of logic and math to establish any of its claims.)

It seems to me much modern philosophy wants to ignore this or explain it away (hence the convoluted Indispensability of Mathematics in the Natural Sciences argument by Putnam and Quine.) This is subject of one of the stock articles I now refer to What is Math (Smithsonian Magazine). James Brown, a maths emeritus, argues for the Platonist view, while the representatives of empiricism pour scorn:

Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

I think that speaks volumes! It goes on....

"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?

Ain't that the truth.

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?)

See, they're all actually getting close to the issue I'm talking about, but they flee screaming, because of the metaphysical implications. And nobody likes metaphysics. It's like Basil Fawlty's, 'don't mention the war'. :wink:

↪Wayfarer seems to want two sorts of quantifiers, real and exist. He's immediately committed at least to some sort of free logic. He is this giving us permission to talk of things that do not exist, but are real - like numbers. — Banno

I say it's a real philosophical distinction which has become lost due to specifics of intellectual history. I can make the case for it, but it would be a very long one. I'm not 'content with mystisicm', a term generally spat out as a pejorative, especially by analytical philosophy. But it is a real and crucial issue which overflows the bounds of propositional language (as I believe Wittgenstein hints at in the mystical aphorisms at the end of his book.)

science presents us with a cosmos that seems to be almost infinitely older and larger than humanity, — Janus

Yes, science does that, and science is a human enterprise. Please try to understand this point, I am not trying to be confrontational or arguing for its own sake.

All the problems come up when someone then asks you, Why make that choice? I don't mean just you, I mean anyone who wants to say something using words like "real" and "exist". What sort of case are philosophers supposed to make for their choices here? — J

I say there is a crucial but neglected distinction between 'what is real' and 'what exists'. It is found in apophatic theology - the stance of Paul Tillich and others that 'God does not exist' (see this brief OP.) Ultimately this goes back to the distinguishing of reality from appearance - which is at the root of the Western philosophical tradition, although generally neglected in current philosophy. Philosophically, the root can be found in Parmenides and the subsequent Platonic tradition, in which 'the One' is understood to be 'beyond existence and non-existence'. But as all of that is now gone and forgotten to anyone other than a few specialist academics, I don't expect it to be understood.

Existence refers to what is finite and fallen and cut of from its true being. Within the finite realm issues of conflict between abound between autonomy and heteronomy. Resolution of these conflicts lies in the essential realm (the Ground of Meaning/the Ground of Being) which humans are both estranged and yet also dependent on. In existence man is that finite being who is aware both of his belonging to and separation from the infinite. Therefore existence is estrangement.

Do humans prefer altruism over selfishness? Is one ethically better than the other? — Joshs

Indeed. I think Richard Polt's point is perfectly clear, which is why I often refer to it, although Heidegger's obscurantism can be used to muddy any waters one chooses.

The point about appealing to evolutionary biology in support of an ethic is exactly an instance of the naturalistic fallacy. This fallacy, identified by philosopher G. E. Moore in his work Principia Ethica, occurs when one tries to define what is "good" in terms of natural properties or states of affairs, such as what is "natural" or what has evolved biologically. For example, just because a behavior like altruism has evolved in certain species as beneficial for survival does not necessarily mean such behavior can be considered morally good. Ethical norms typically involve evaluative judgments that cannot be directly derived from facts about the natural world.

It’s also a version of the naturalistic fallacy.

I was responding to your appeal to 'DNA and evolutionary psychology'. Here, you're appealing to science to account for the faculty of conscience. But it may not throw much light. Science is concerned with what can be measured, with quantitative evaluation (what is) whereas ethical judgement by its very nature concerns 'oughts'. it is not at all coincidental that Hume, one of the founding figures of the Scottish Enlightenment, recognised this dilemma.

This is from an OP by a philosopher about the shortcomings of such an approach.

I have no beef with entomology or evolution, but I refuse to admit that they teach me much about ethics. Consider the fact that human action ranges to the extremes. People can perform extraordinary acts of altruism, including kindness toward other species — or they can utterly fail to be altruistic, even toward their own children. So whatever tendencies we may have inherited leave ample room for variation; our choices will determine which end of the spectrum we approach. This is where ethical discourse comes in — not in explaining how we’re “built,” but in deliberating on our own future acts. Should I cheat on this test? Should I give this stranger a ride? Knowing how my selfish and altruistic feelings evolved doesn’t help me decide at all. Most, though not all, moral codes advise me to cultivate altruism. But since the human race has evolved to be capable of a wide range of both selfish and altruistic behavior, there is no reason to say that altruism is superior to selfishness in any biological sense. — Anything but Human, Richard Polt

Yes, well if you really drill this down to the very DNA or evolutionary psychology…. — Shawn

…you will run into Hume’s is/ought problem.

Hand-waving to you might be sign language to someone else. ;-)

Couldn’t you say that the innate in conscience is where the good is gleaned, where the good is constructed? — Fire Ologist

I can see some sense in which it's a 'construct' but I also believe there is an innate good, although not everyone will agree.

Do you agree with this, namely that the notion of good in inherent in the primacy of experience, and not something that can be learned by simply looking up a definition and analyzing it? — Shawn

I had rather thought that discerning the good was the role traditionally assigned to conscience, and that those who do not do good have a deficiency in that respect. And also that while this is something that might be shaped by experience, it is still essentially innate, rather than acquired - in that, someone who lacks all conscience, such as a sociopath, is not going to acquire one through experience.

The botched and imperfect world we live in, full of design flaws and disease also seems to indicate sloppy work. — Tom Storm

Always seemed to me that there was never an expectation in Christianity that 'the world' could be other than a 'vale of tears'. The point of the Christian faith is not to fix that, but to transcend it. Heaven, or the Life Eternal, is where there is no suffering or evil or corruption ('there's no sickness, toil or danger in the place to which I go.') Whereas because there's no conception of that in secular culture, we expect earthly existence to be as perfect as possible, and then blame the God we don't believe in for spoiling it.

In order to always have a secure compass in hand so as to find one's way in life, and to see life always in the correct light without going astray, nothing is more suitable than getting used to seeing the world as something like a penal colony. This view finds its...justification not only in my philosophy, but also in the wisdom of all times, namely, in Brahmanism, Buddhism, Empedocles, Pythagoras [...] Even in genuine and correctly understood Christianity, our existence is regarded as the result of a liability or a misstep. ... We will thus always keep our position in mind and regard every human, first and foremost, as a being that exists only on account of sinfulness, and who's life is an expiation of the offence committed through birth. Exactly this constitutes what Christianity calls the sinful nature of man. — Schopenhauer's Compass, Urs App

a distinction that I can't make sense of — Banno

So much the worse for the ‘linguistic turn’ in analytic philosophy. Ontology concerns bigger questions, although like metaphysics it’s often regarded as obsolete in the academy.

It's (potentially) a choice between grammars, between languages. — Banno

I prefer to think of it more as an ontological question. As the SEP article on Platonism in Philosophy of Maths says:

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. 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.

Or as Rebecca Goldstein says of Gödel:

Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason.

I think the resistance to this idea is because it suggests that the non-physical reality of number, hence as the SEP excerpt says challenges the prevailing naturalist dogma. But then, in the SEP entry on Idealism, which you have previously quoted, it is said that:

the idealist, rather than being anti-realist, is in fact … a realist concerning elements more usually dismissed from reality.

All of this is a case in point.

I say “numbers exist”; you say “numbers do not exist”. — J

As mentioned in the other thread, I have been very interested in this question since first posting on forums (even though I can't really get my head around the technicalities of the arguments presented in the OP so my comments here might be tangential to those.)

My intuition about the matter is simply that numbers are real but that they don't exist.

In everyday speech, of course, it is fine to say 'the number 7 exists while the square root of 2 does not'. But if I ask you to point to the number 7, what you're pointing to is a symbol. It can as easily be symbolised 'seven', 'VII', 'seben', '0b111' and so on. But while the symbolic form exists, what it symbolises, a number, is an act, namely, the act of counting, which is grasped by the mind:

"in the same way", Frege says "that a pencil exists independently of grasping it. Thought contents are true and bear their relations to one another (and presumably to what they are about) independently of anyone's thinking these thought contents - "just as a planet, even before anyone saw it, was in interaction with other planets." — Frege on Knowing the Third Realm, Tyler Burge

And the act of counting does not exist in the sense that phenomenal objects exist. But I don't know if there's provision in the current philosophical lexicon to allow for phenomenal and intelligible objects to exist in different ways (although perhaps that's a matter for modal metaphysics.) At any rate, I'm of the view that the unspoken assumption about the matter is that existence is univocal, i.e. something either exists or it does not, whereas here I'm pointing out entities that don't exist as do sensible objects.

About the only discussion I'm aware of that elucidates this distinction (albeit in relation to universals rather than number per se) is in Russell's Problems of Philosophy>The World of Universals:

Consider such a proposition as 'Edinburgh is north of London'. Here we have a relation between two places, and it seems plain that the relation subsists independently of our knowledge of it. When we come to know that Edinburgh is north of London, we come to know something which has to do only with Edinburgh and London: we do not cause the truth of the proposition by coming to know it, on the contrary we merely apprehend a fact which was there before we knew it. The part of the earth's surface where Edinburgh stands would be north of the part where London stands, even if there were no human being to know about north and south, and even if there were no minds at all in the universe. ...We may therefore now assume it to be true that nothing mental is presupposed in the fact that Edinburgh is north of London. But this fact involves the relation 'north of', which is a universal; and it would be impossible for the whole fact to involve nothing mental if the relation 'north of', which is a constituent part of the fact, did involve anything mental. Hence we must admit that the relation, like the terms it relates, is not dependent upon thought, but belongs to the independent world which thought apprehends but does not create.

This conclusion, however, is met by the difficulty that the relation 'north of' does not seem to exist in the same sense in which Edinburgh and London exist. If we ask 'Where and when does this relation exist?' the answer must be 'Nowhere and nowhen'. There is no place or time where we can find the relation 'north of'. It does not exist in Edinburgh any more than in London, for it relates the two and is neutral as between them. Nor can we say that it exists at any particular time. Now everything that can be apprehended by the senses or by introspection exists at some particular time. Hence the relation 'north of' is radically different from such things. It is neither in space nor in time, neither material nor mental; yet it is something.

It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'....In the strict sense, it is not whiteness that is in our mind, but the act of thinking of whiteness. The connected ambiguity in the word 'idea', which we noted at the same time, also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an 'idea'. Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an 'idea' in the other sense, i.e. an act of thought; and thus we come to think that whiteness is mental. But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts.

We shall find it convenient only to speak of things existing when they are in time, that is to say, when we can point to some time at which they exist (not excluding the possibility of their existing at all times). Thus thoughts and feelings, minds and physical objects exist. But universals do not exist in this sense; we shall say that they subsist or have being, where 'being' is opposed to 'existence' as being timeless.

My bolds. My sense is that rational thought is shot through with these kinds of relations, that they hold reasoned argument together - something that we don't notice because we look through them, rather than at them. So in that sense, numbers and universals are real as the constituents of reason - athough not, as conceptualism says, as 'products of the mind', for the reason that Russell gives in the bolded phrase.

I believe that this supports what I am calling the philosophy of phenomenological idealism as outlined in The Mind-Created World op - that the so-called 'external world' is held together by such rational cognitive acts.

This is a way of looking at human consciousness and intelligence, but it doesn't mean much since we are such a tiny fraction of the cosmos, — Janus

We’re the only ‘tiny fraction of the cosmos’ who know what that means. It’s amusing in the extreme that objective science, which is a cognitive mode only available to h. Sapiens, then declares its authors insignificant in the ‘grand scheme’ - a grand scheme that is their own mental creation!

(I have read that that Tipler book is unbridled nonsense, but the Tipler and Barrow book The Cosmic Anthropic Principle seems reasonably well-regarded.)