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.

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.

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.

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.

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

How is sex an external representation of a mind disassociating with itself? — Bob Ross

From that comment, I think you have an incorrect picture of what Kastrup means by 'dissociated alter'. From a glossary entry on Bernardo Kastrup's terminology:

In Bernardo Kastrup’s framework, dissociated alters are conceptualized as individual living organisms, including humans, which are distinct expressions or manifestations of a single, overarching cosmic consciousness. According to this idealist ontology, there exists only one cosmic consciousness, and all living beings are dissociated alters of this consciousness. These alters are surrounded by the thoughts of cosmic consciousness, and the inanimate world we perceive is the extrinsic appearance of these thoughts. Living organisms, including humans, are the extrinsic appearances of other dissociated alters. This framework suggests that our subjective experiences and perceptions are localized within these dissociated alters, which are essentially segments of the broader cosmic consciousness.

This plainly bears comparision with the Plotinus' philosophy of 'the One' as well as with Advaita Vedanta. For a detailed account, see The Universe in Consciousness.

On a more serious note, there's an excellent current text available online which provides a succint and accurate account of the Platonic forms - Eric S Perl, Thinking Being - Introduction to Metaphysics in the Classical Tradition (.pdf). The chapter on Reading Plato. As it is directly relevant to the OP, I'll quote one passage at length.

Is there such a thing as health? Of course there is. Can you see it? Of course not. This does not mean that the forms are occult entities floating ‘somewhere else’ in ‘another world,’ a ‘Platonic heaven.’ It simply says that the intelligible identities which are the reality, the whatness, of things (such as "health") are not themselves physical things to be perceived by the senses, but must be grasped by thought.

It is in this sense, too, that Plato’s references to the forms as ‘patterns’ or ‘paradigms’, of which instances are ‘images,’ must be understood. All too often, ‘paradigm’ is taken to mean ‘model to be copied.’ The following has been offered as an example of this meaning of παράδειγμα (parádeigma) in classical Greek: “[T]he architect of a temple requiring, say, twenty-four Corinthian capitals would have one made to his own specifications, then instruct his masons to produce twenty-three more just like it.” Such a model is itself one of the instances: when we have the original and the twenty-three copies, we have twenty-four capitals of the same kind. It is the interpretation of forms as paradigms in this sense that leads to the ‘third man argument’ by regarding the form as another instance and the remaining instances as ‘copies’ of the form. This interpretation of Plato’s ‘paradigmatism’ reflects a pictorial imagination of the forms as, so to speak, higher-order sensibles located in ‘another world,’ rather than as the very intelligible identities, the whatnesses, of sensible things.

But forms cannot be paradigms in this sense. Just as the intelligible ‘look’ that is common to many things of the same kind, a form, as we have seen, is not an additional thing of that kind. Likewise, it makes no sense to say that a body, a physical, sensible thing, is a copy, in the sense of a replica or duplicate, of an intelligible idea. Indeed, Plato expressly distinguishes between a copy and an image: “Would there be two things, that is, Cratylus and an image of Cratylus, if some God copied not only your color and shape, as painters do, but also … all the things you have? — Eric D Perl Thinking Being, p31 ff

I say that 'forms' are much more like 'intelligible principles' than what they are often confused for, which is a kind of ethereal shape. I think much of the dismissal of them is based on centuries of poor schoolroom teaching by those who really hadn't grasped that fact. But there are contemporary sources, such as Rebecca Goldstein's Plato at the Googleplex, and Iris Murdoch's Sovereignty of the Good, which provide a much more nuanced account of their continuing relevance.

Oh, so you don't know what it is, but you do know it's a fantasy.

a platonic realm — Banno

What do you think that might comprise? An ethereal palace, replete with ideal dogs and cats?

The key difference between Frege and Popper here is...whether the 3rd realm exists independently of human thought, or is created by our thought. If Burge is right, then there's no doubt what Frege believed: complete independence. Popper stakes out a middle ground — J

Compare:

Frege believed that number is real in the sense that it is quite independent of thought: 'thought content exists independently of thinking "in the same way", he 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." ' Furthermore in The Basic Laws of Arithmetic he says that 'the laws of truth are authoritative because of their timelessness: they "are boundary stones set in an eternal foundation, which our thought can overflow, but never displace. It is because of this, that they authority for our thought if it would attain to truth." — Tyler Burge

Intelligible objects must be independent of particular minds because they are common to all who think. In coming to grasp them, an individual mind does not alter them in any way, it cannot convert them into its exclusive possessions or transform them into parts of itself. Moreover, the mind discovers them rather than forming or constructing them, and its grasp of them can be more or less adequate. Augustine concludes from these observations that intelligible objects must exist independently of individual human minds. — Cambridge Companion to Augustine

Plainly Augustine has theological commitments that Frege lacks, but nevertheless the Platonist elements they have in common are significant. Augustine adds that reason is: “a kind of head or eye of our soul ... which does not belong to the nature of animals” (lib. arb. 2.6.13).11", clearly a reference to the tripartite soul of Plato, in which reason is a governing faculty, responsible for wisdom and seeking truth. Frege's notion that logical laws are "boundary stones set in an eternal foundation" parallels Plato's Forms and Augustine's intelligible objects as timeless, immutable realities. They are not dependent on human minds, cultures, or contingent physical realities but are 'discernable by reason', where 'reason' represents the faculty that is capable of grasping incorporeal truths.

We bring one and two into existence, by and intentional act - it's something we do. — Banno

Hence, these MUST be understood as constructions, hence contingent facts, our own creations, in fact, not immutable truths, which still retain a theological undertone that does not sit well with our secular age. Thomas Nagel quotes C S Peirce:

The only end of science, as such, is to learn the lesson that the universe has to teach it. In Induction it simply surrenders itself to the force of facts. But it finds . . . that this is not enough. It is driven in desperation to call upon its inward sympathy with nature, its instinct for aid, just as we find Galileo at the dawn of modern science making his appeal to il lume naturale. . . . The value of Facts to it, lies only in this, that they belong to Nature; and nature is something great, and beautiful, and sacred, and eternal, and real - the object of its worship and its aspiration.

The soul's deeper parts can only be reached through its surface. In this way the eternal forms, that mathematics and philosophy and the other sciences make us acquainted with will, by slow percolation, gradually reach the very core of one's being, and will come to influence our lives; and this they will do, not because they involve truths of merely vital importance, but because they [are] ideal and eternal verities. — Evolutionary Naturalism and the Fear of Religion

This is part of the preamble in which Nagel then describes the 'fear of religion' as one of the main motivations for the rejection of Platonism and the adoption of evolutionary naturalism:

Even without God, the idea of a natural sympathy between the deepest truths of nature and the deepest layers of the human mind, which can be exploited to allow gradual development of a truer and truer conception of reality, makes us more at home in the universe than is secularly comfortable.

That's the cultural dynamic that I think is behind the rejection of platonism in mathematics and the subsequent relativisation of reason.

If anyone is spending their holiday on TPF, poor devils, then Merry Christmas! — J

Beats crossword puzzles! And, same to you. :party:

Perhaps the attempt to understand God in terms of rational principles is a misguided attempt to understand a God who is understood, to the extent he is understood, as willful. — Fooloso4

By 'theological voluntarism', associated with Protestant conceptions of Divinity, and very different from the philosophical rationalism of scholastic theology.

But the PSR says that everything has an explanation. — Clearbury

I don't know if it does. It says that everything that exists has a reason for its existence. But everything that exists is the domain of phenomena, 'what appears'. The 'first cause', whether conceived of as a personalistic God or not, is not something that exists, but the condition of the possibility of the existence of everything that exists. It's on a different ontological level to what exists - that's what 'transcendence' means. (See God Does Not Exist.)

I am going to call that a ticketyboo. — Clearbury

Hardly does justice to the topic.

Existence isn't a property; that would imply there are objects in the world that lack it - which is absurd. All objects in the world exist. — Relativist

What about the mathematical and analytical tools that are used to determine what in the world exists, especially on the scales of the atomic or cosmological. Are they themselves also things that exist? (I seem to recall that atomic physics relies heavily on the imaginary number the square root of minus one in normalisation procedures, which would suggest not. ) For that matter, there's Terrence Deacon's absentials which are also defined as not materially existent but often amongst the definining properties of entential activities. From the glossary entry:

• a state of things not yet realized
• a specific separate object of a representation,
• a general type of property that may or may not exist,
• an abstract quality,
• an experience, and so forth-just not that which is actually present.
• something missing, separate, and possibly nonexistent
• irrelevant when it comes to inanimate things, but a defining property of life and mind
• what is absent matters.
• a purpose not yet actualized,
• a quality of feeling, a functional value just discovered
• not just superimposed probable physical relationships
• each an intrinsically absent aspect of something present

Absentials do not exist, but play a defining role in the existence of what he calls ententional agents.

Rather than the problem of an infinite regress, the problem is one of the limits of human reason. — Fooloso4

While I can see your point, natural theology will suggest that the regularities and rationally-intelligible principles that constitute what we describe as natural laws suggest a prior cause. And indeed that the whole idea of apriori truths implicitly suggests it. The fact that science itself can't explain scientific laws is no fault of science, but it does legitimately imply a deeper level of explanation than the scientific. One could argue among the aims of philosophy is to discern the boundary of what can be explained in terms of natural laws, and to intuit what may lie beyond it, even if it can't be stated in scientific terms.

It's not in dispute that a necessarily existing thing exists and can't not. But if the PSR is true, then there will be an explanation of that. You haven't provided one, I think. — Clearbury

As the OP is on Christmas break (which strictly speaking I also am, but never mind), I'll volunteer a response. The point about necessary being is that it needs no explanation. It is the terminus of explanation for all question about 'why is that the case?' A trivial example is the case of a simple arithmetical equation, what is the sum of two plus two? The answer of course is 'four' and there is no point in asking why it is. Asking "why is 2 + 2 = 4?" misconstrues the nature of necessity. The explanation for such truths lies in their self-evidence within the system within which they're true, and no further "why" can be meaningfully posed.

Similarly, in metaphysics, the idea of a necessary being functions as the ultimate 'terminus of explanation' under the principle of sufficient reason. The PSR asserts that everything must have an explanation, either in terms of an external cause or in terms of its own nature. For contingent beings, the PSR demands a cause or reason external to themselves. But for a necessary being, its necessity is its explanation.

One has to disabuse oneself of modernity. — unenlightened

I've been reading Hans Jonas: The Phenomenon of Life (1966) which is a highly-regarded work in phenomenology and existentialism. He points out that for pre-moderns, life was the norm, what with the Universe being so obviously alive, whilst death was anomalous, something that had to be explained, in terms of the classical myths of immortality. He explains that this flips with the Renaissance so that dead matter becomes the norm, and life itself an anomaly, which now has to be explained in terms of physical laws, so called. Fascinating read.

It’s a perfectly meaningless expression. But Happy Christmas, regardless. :party:

How is anything? :chin: Anyway it’s Christmas Day, I’ll reply later (and Happy Christmas :party: )

Every sentient creature is surrounded by objects but only rational sentient beings know arithmetic. Anyway if you read the quote in context it makes a point which is clearly salient to the OP (although I’m not going to try and explain it all over again.)

I think that is to greatly underestimate the intelligence and intellectual honesty of those you disagree with — Janus

I’m not criticizing individuals but ideas. In this case, empiricist philosophy which can’t admit the reality of number because of it being ‘outside time and space’. If you take that as any kind of ad hom, it’s on you.

I’m not totally on board with Kastrup but I don’t know if it is implausible. Human infants possess an un-formed intelligence which will normally come to maturity as instances or instantiations of human consciousness. What differentiates one individual from another is the contents of consciousness but underlying that is a kind of generic ‘mind’ or ‘mindedness’. Works for me.

number is real and materially instantiated in the diversity of forms given to our perceptions. — Janus

The nature of the particular contents therefore makes no difference at all. This fact, as rudimentary as it is incontestable, already rules out a certain class of views concerning the origination of the number concepts: namely, the ones which restrict those concepts to special content domains, e.g., that of physical contents.

Yet if decisions were made in the direction of these ideals, might not they be tending towards the ethical? — ENOAH

Of course :ok:

since you are always arguing that reality is entirely constructed by consciousness — Janus

I never have used that expression nor would I put it like that //although on reflection I suppose it is fair//.

What I do say is that material objects are perceived by the senses and so can’t be truly mind-independent, because sense data must be interpreted by the mind for any object to be cognised. What interests me about the passage I quoted, is that mathematical functions and the like are not the product of your or my mind, but can only be grasped by a mind. That’s the sense in which they’re what Augustine describes as ‘intelligible objects’ in the earlier post about that.

The underlying argument is very simple - it is that number is real but not materially existent. And reason Platonism is so strongly resisted is because it is incompatible with materialism naturalism on those grounds, as per the passage from the Smithsonian article upthread, ‘What is Math?’: 'The idea of something existing “outside of space and time” makes empiricists nervous.'

Hey that’s pretty good. One for the scrapbook. Although shedding the illusion is often rather more traumatic than a snake shedding its skin.

We can maintain that mathematical objects are mind-independent, self-subsistent and in every sense real, and we can also explain how we are cognitively related to them: they are invariants in our experience consciousness

Rather like objects as ‘permanent possibilities of sensation’ but here the objects are noumenal.

Mathemarical concepts for Husserl are no more ‘real’ than the spatial objects we interact with in the world. — Joshs

And no less.

Physicists, probably more than anyone else in science, are obsessed with simplicity, unification and "naturalness," and not without reason, because this attitude has accompanied spectacular advances in physics over the past two centuries. But how philosophically justified is it? And how sustainable? I suppose that goes to the question of the proverbial "unreasonable effectiveness of mathematics." — SophistiCat

Subject of a book by Sabine Hossenfelder, Lost in Math.

Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades.

The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these "too good to not be true" theories are actually untestable and they have left the field in a cul-de-sac. To escape, physicists must rethink their methods. Only by embracing reality as it is can science discover the truth.

(Although from my perspective, embracing reality 'as it is' will entail abandoning the axiom that it is only physical.)

I've discovered a Notre Dame Review about a book which I'll probably never get around to, but which finds some common ground between Platonism and Husserl, Phenomenology, Logic and the Philosophy of Mathematics, Richard Tieszen, from which:

In his later Ideas for a Pure Phenomenology and Phenomenological Philosophy (1913), Husserl develops the method of eidetic variation. Eidetic variation consists of a series of mental acts that aim to grasp an invariant, ideal, non-sensory object that serves as a substrate to a range of experiences. The same object is given across this range of experience and we experience its self-samenesss. Husserl suggested that this method would serve to sharpen our conceptual grasp of ideal objects, and Tieszen argues that this method is in fact close to the actual procedure employed in modern geometry. In abstract sciences, Tieszen writes, "objectivity and invariance go hand in hand" (p. 89), and invariance is best understood as givenness. An ontology of abstract objects, then, should rest on the elements of Husserlian epistemology.

Husserl called his position "transcendental" phenomenology, and Tieszen makes sense of this by claiming that it can be seen as an extension of Kant's transcendental idealism. The act of cognition constitutes its content as objective. Once we recognize the distinctive givenness of essences in our experience, we can extend Kant's realism about empirical objects grounded in sensible intuition to a broader realism that encompasses objects grounded in categorial intuition, including mathematical objects.

The view is very much like what Kant has to say about empirical objects and empirical realism, except that now it is also applied to mathematical experience. On the object side of his analysis Husserl can still claim to be a kind of realist about mathematical objects, for mathematical objects are not our own ideas (p. 57f.).

This view, Tieszen points out, can preserve all the advantages of Platonism with none of its pitfalls. We can maintain that mathematical objects are mind-independent, self-subsistent and in every sense real, and we can also explain how we are cognitively related to them: they are invariants in our experience, given fulfillments of mathematical intentions. The evidence that justifies our mathematical knowledge is of the same kind as the evidence available for empirical knowledge claims: we are given these objects. And, since they are given, not subjectively constructed, fictionalism, conventionalism, and similar compromise views turn out to be unnecessarily permissive. The only twist we add to a Platonic realism is that ideal objects are transcendentally constituted.

We can evidently say, for example, that mathematical objects are mind-independent and unchanging, but now we always add that they are constituted in consciousness in this manner, or that they are constituted by consciousness as having this sense … . They are constituted in consciousness, nonarbitrarily, in such a way that it is unnecessary to their existence that there be expressions for them or that there ever be awareness of them. (p. 13).

Bolds added. It is in accordance with my intuitive understanding.

I am of the view that inner as opposed to outer, objective aspects of 'reality' are important here in the tradition of human understanding. Science, similarly to religion may be embedded in mythic understanding. What do you think, especially in relation to the concept of myth? As far as I see it is a topic involving dialogue between ancient philosophy, as well as anthropological thinking and research. How may the development of ideas about 'gods' or one God be understood in the history of religion and philosophy? — Jack Cummins

They're good questions, but also very big questions. There is a description you might sometimes encounter, 'scientia sacra', meaning the sacred science. It is not a popular term, but still has currency amongst the advocates of the perennial philosophy, such as Seyyed Hossein Nasr and others. This is the theme that there are universal, undelying tenets of wisdom which are made manifest in the individual cultural forms throughout history. In the pre-modern world, there was a perceived unity between the human being as 'microcosm' and the universe, Cosmos ('as above, so below', although the traditionalist vision has been undermined by science in some important respects.)

But it's a vast field of study, which can be approached through a number of perspectives. Karen Armstrong is a good source on that. Huston Smith might be another to consider. Joseph Campbell, as mentioned already. James Hillman another. There's also the more up-to-date and contemporary approaches, like Brian Swimme's evolutionary cosmology. Gary Lachmann's books might be of interest also.

In a nutshell, 'mathematical platonism' would suggest people have experienced these higher realities and found mathematics to be existing within them. — Tzeentch

Here is a passage about Augustine which details the Platonist insights that inspired his religious conversion.

During his Manichaean period Augustine’s attention had been focused on the external corporeal world. His thinking had consequently been bound by sensory experience: he could conceive only what he could form a sensory image of. Platonism, however, admonished him to abandon the corporeal world and turn inward, using the eye of his own rational soul. When he did so, he discovered an astonishing new realm. The incorporeality, immutability, and eternity that characterize purely intellectual thought are the clues that led Augustine, by stages, to the divine nature itself.

Augustine begins by establishing a hierarchy that sorts into general categories and ranks the natures that comprise the universe: existence, life, and understanding:

Therefore the nature that merely exists (and neither lives nor understands) ranks below the nature that not only exists but also lives (but does not understand) – the soul of the non-human animals is of this sort. This nature in turn ranks below the nature that at once exists, lives, and understands – for example, the rational mind of the human being. (lib. arb. 2.6.13)

His strategy will be to argue that there is a nature that ranks above the rational mind of the human being, a nature that he will identify as divine (lib. arb. 2.6.14, 2.15.39). In order to discover it, he ascends the hierarchy of natures, turning attention first from bodies (the first and lowest-ranking category in the hierarchy) to the soul (psuche, the nature constitutive of both the second and third categories), and then within his own soul from the sensory (found in both human beings and the non-human animals) to the rational: “a kind of head or eye of our soul ... which does not belong to the nature of non-human animals” (lib. arb. 2.6.13).11

Having ascended as far as reason – that which is highest in us – he focuses on reason’s distinctive perceptual capacities and the distinctive sorts of objects they put us in contact with, the objects of pure thought. By way of example, Evodius, Augustine’s interlocutor in the dialogue, first suggests that they consider “the structure and truth of number,” by which he means arithmetical facts and relationships of the sort expressed by such truths as “seven plus three equals ten” (lib. arb. 2.8.20–21). Augustine himself adds the example of the indivisible mathematical unit that is the foundation of all number. He later introduces into the discussion a collection of a priori evaluative and normative truths such as “wisdom should be diligently sought after,” “inferior things should be subjected to superior things,” and “what is eternal is better than what is temporal” (lib. arb. 2.10.28). He thinks of these truths as constitutive of wisdom itself and therefore normative for anyone who would possess it. Moreover, anyone who is able to contemplate them will recognize their truth. Examination of these various examples leads Augustine to three conclusions: intelligible objects of these sorts are independent of our minds, incorporeal, and higher than reason. Put briefly, the main lines of his reasoning are as follows (lib. arb. 2.8.20–12.34):

1. Intelligible objects must be independent of particular minds because they are common to all who think. In coming to grasp them, an individual mind does not alter them in any way, it cannot convert them into its exclusive possessions or transform them into parts of itself. Moreover, the mind discovers them rather than forming or constructing them, and its grasp of them can be more or less adequate. Augustine concludes from these observations that intelligible objects must exist independently of individual human minds.

2. Intelligible objects must be incorporeal because they are eternal and immutable. By contrast, all corporeal objects, which we perceive by means of the bodily senses, are contingent and mutable. Moreover, certain intelligible objects – for example, the indivisible mathematical unit – clearly cannot be found in the corporeal world (since all bodies are extended, and hence divisible). These intelligible objects cannot therefore be perceived by means of the senses; they must be incorporeal and perceptible by reason alone.

3. Intelligible objects must be higher than reason because they judge reason. Augustine means by this that these intelligible objects constitute a normative standard against which our minds are measured (lib. arb. 2.5.12 and 2.12.34). We refer to mathematical objects and truths to judge whether or not and to what extent our minds understand mathematics. We consult the rules of wisdom to judge whether or not and to what extent a person is wise. In virtue of their normative relation to reason, Augustine argues that these intelligible objects must be higher than it, as a judge is higher than what it judges. Moreover, the intrinsic nature of these objects shows them to be higher than reason. They are eternal and immutable; by contrast, the human mind is clearly mutable. Augustine holds that since it is evident to all who consider it that the immutable is superior to the mutable (it is among the rules of wisdom he identifies), it follows that these objects are higher than reason.

...By focusing on objects perceptible by the mind alone and by observing their nature, in particular their eternity and immutability, Augustine came to see that certain things that clearly exist, namely, the objects of the intelligible realm, cannot be corporeal. When he cries out in the midst of his vision of the divine nature, “Is truth nothing just because it is not diffused through space, either finite or infinite?” (FVP 13–14), he is acknowledging that it is the discovery of intelligible truth that first frees him to comprehend incorporeal reality. — Cambridge Companion to Augustine

Popper's "Third world" differs from Plato's world of forms in that it is entirely an artefact of language and culture and is thus constantly changing. This is in contrast to the changeless world of Plato's forms. — Janus

True. Although there is considerable debate about what 'Plato's world of forms' actually is or means. In any case, the reason I mentioned it, is because Popper grants a kind of irreducibility to those things that constitute the third world.

:up:

There really is privileged metaphysical structure; we're just not sure about the terms to use. — J

That's why I suggested that essay about Frege. I'm no expert in Frege - in fact that essay is about the sum total of my knowledge - but it explores the idea of a 'third realm', somewhat similar to Popper's idea with the same name. Those kinds of ideas are all generally Platonistic.

You seem to be suggesting that one of these logics is correct. — Michael

If you mean, I believe that there is a truth to logical laws that is not dependent on one or another philosophical doctrine, then yes, I do believe that. I think the law of the excluded middle, for instance, describes something inherent in the structure of reality—not something contingent on whether anyone happens to conceive of it. It is a 'metaphysical primitive,' i.e., something that can't be reduced further.

There's a subtle point at issue here—the ontological status of such principles that are not created by the human mind but can only be grasped by a rational intellect. These principles, while independent of any particular mind, require the rational intellect to apprehend them—highlighting the unique role of reason in discerning universal truths. Whereas in today's culture there is an inherent tendency to try and account for those principles naturalistically, as a result of evolutionary neurology, etc (i.e. 'naturalised epistemology'). But this again relativizes them or makes them contingent facts. Would you agree with that?

Your view seems to be a form of transcendental idealism, which is about how we understand reality fundamentally through mental ideas (and cognitive pre-structures) and thusly is a form of epistemic idealism---not ontological idealism. — Bob Ross

Good analysis Bob.

As for the decomposition problem, Kastrup does address that through his theory of 'dissociated alters'. He proposes that reality comprises a universal consciousness ('mind at large'.) This universal mind is analogous to a field of subjectivity, from which all individual experiences arise by dissociation.

Dissociation: Individual conscious beings, like humans, are seen as dissociated "alters" of the universal mind. Just as alters in dissociative identity theory are partitioned segments of a single psyche, the individual consciousness is a localized expression of the universal mind, dissociated from its broader unity.

This is very similar to the philosophy of Advaita Vedanta, which Kastrup has acknowledged in dialogues with Swami Sarvapriyananda, the head teacher of the Vedanta Society of New York. And a similar idea is expressed by Albert Einstein, of all people.

A human being is a part of the whole, called by us "Universe", a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the rest — a kind of optical delusion of his consciousness. The striving to free oneself from this delusion is the one issue of true religion. Not to nourish the delusion but to try to overcome it is the way to reach the attainable measure of peace of mind. — Albert Einstein, Letter of Condolence

But I would add the caveat that the whole concept of 'mind at large' is problematical if it is conceived as something objectively existent in a way analogous to matter or energy. (I wrote an (unpublished) Medium essay on that topic which can be reviewed here.)

There is no scientific evidence for dualism - verifiable separability of mental stuff and physical stuff. It is also not metaphysically parsimonious and borderline incoherent. So which is it? Mental stuff or physical stuff? — Apustimelogist

If you read the OP carefully, you will note that I discuss that problem in paragraph four. I emphatically do not posit any conception of 'mind stuff' or 'spiritual substance' which i regard as an oxymoronic conception, to wit:

To say the world is made of experience in the same way as houses are made of bricks also doesn't avoid the hard combination problem... — Apustimelogist

The second objection (to idealism) is against the notion that the mind, or ‘mind-stuff’, is literally a type of constituent out of which things are made, in the same way that statues are constituted by marble, or yachts of wood. The form of idealism I am advocating doesn’t posit that there is any ‘mind-stuff’ existing as a constituent in that sense. — Wayfarer

All due respect, you're viewing the issue in the wrong register. As I say at the outset, the approach is perspectival, it is not an essay about what 'things are made of.' That is a job for physics and chemistry. But the nature of our own first-person experience is real on a different level and the question of its nature has to be approached in a different way. That's what I mean by 'perspectival'. I know from reading your post here and elsewhere, you view the issue through a certain perspective, and that challenging one's assumed perspectives is difficult. But the philosophical perspective the OP advocating is of a different kind or order.

You can believe that numbers and other abstracta really and truly exist without being a mathematical platonist. You merely assert that they exist because we have created them, and they will cease to exist if we also cease. — J

What about the laws of logic, like the law of the excluded middle? Does that cease to obtain in the absence of rational sentient beings? I’m more inclined to the understanding that it is discovered by rational sentient beings, and with it the realisation that it must be true in all possible worlds. The alternative is to subjectivize such principles, which reduces them to social conventions. Meaning whatever reality they possess is contingent - so they can’t ‘really and truly exist’.

I tend towards objective idealism - that logical and arithmetical fundamentals are real independently of any particular mind, but can only be grasped by an act of rational thought. I believe that’s more in line with classical metaphysics.

See Frege on Knowing the Third Realm, Tyler Burge.

Do mathematical objects exist in some exotic realm, awaiting discovery? — jgill

As I said, I think ‘exist’ is problematical in the context. Not that they don’t exist, but the way in which they’re real is different to empirical objects. They are ‘objects of mind’ rather than ‘objects of sense’, but I don’t think the philosophical lexicon has an appropriate term. I tried this out on ChatGPT recently and it suggested ‘transcendentally objective’, although that is hardly an elegant expression.

And it has changed character from a descriptive and predictive tool to an enormous game, unbounded in some aspects, with recently formulated foundational rules. — jgill

Consider synthetic chemistry and genetic engineering. These too are grounded in traditional chemistry and biology but now have dimensions that would never be found in nature herself. It’s analogous in some ways.

I've always thought of these little critters as part of the metaphysics of mathematics — jgill

Maybe they are to natural numbers as viruses are to organisms ;-)

rather I am expressing skepticism towards those who would claim mathematics is 'objectively real', and also pointing out the contradiction in the term 'mathematical platonism'.

Does that make sense? — Tzeentch

It makes sense, but I would also suggest that it’s based on a common misconception. The idea of a ‘realm of Forms’ is often misconstrued as an ‘ethereal realm’, like a ghostly palace. But consider ‘the domain of natural numbers’. That is quite real, but the word ‘domain’ has a very different sense to that of a ‘place’ or ‘world’ - even if there are some numbers ‘inside’ it and others not. ‘Domains’ and ‘objects’ are metaphors or figures of speech which are easily but mistakenly reified as actual domains or objects. But that for which they are metaphors are real nonetheless.

The point about ‘truths of reason’ is that they can only be grasped by reason. But due to the cultural impact of empiricism we are conditioned to believe that only what is materially existent - what is ‘out there, somewhere’ - is real. But numbers, and other ‘objects of reason’, are real in a different way to sense objects. And that is a stumbling block for a culture in which things are said to either exist or not. There is no conceptual space for different modes of reality (leaving aside dry, academic modal metaphysics). Which is why we can only think of them as kinds of objects, which they’re actually not. They’re really closer to kinds of acts.

See this post

:ok:

I’m interested in what you mean, regardless.

If Platonism seems to ‘undercut’ empiricism, it does so only by occupying the opposing pole of the binary implicating both physicalism and platonism within the same tired dualistic subject-object metaphysics. — Joshs

I don't know if I agree with your diagnosis that the opposition to Platonism arises from 'subject-object metaphysics'. I think it goes back to the decline of Aristotelian realism and the ascendancy of nominalism in late medieval Europe. From which comes the oxymoronic notion of mind-independence of the empirical domain, when whatever we know of the empirical domain is dependent on sensory perception and judgement (per Kant). Hence those objections in that passage I quoted, 'The idea of something existing “outside of space and time” makes empiricists nervous'. Anything real has to be 'out there somewhere' - otherwise it's 'in the mind'. That is the origin of subject-object metaphysics.

see both numbers and physical things as pragmatic constructions, neither strictly ideal nor empirical, subjective nor objective, inner nor outer, but real nonetheless? — Joshs

But there are imaginary numbers, and also imaginary objects, even imaginary worlds. There are degrees of reality, and there is a such a thing as delusion, and delusions can be very deep indeed, in today's panoptical culture. Agree with the constructivist attitude overall, but still want to honour the epistemology of the Divided Line.