…we may infer its nature in accordance with what seems most plausible…. — Janus

Conventionally speaking, true enough. But what of those inferences we seek, regarding the nature of something for which we wish to obtain apodeictic certainty, for which the merely plausible isn’t sufficient?

Can anything we know except those things whose negation would be a logical contradiction be apodeictic?

You should accept the premise of the possible world, since in our relationship with the world, it is shown as something that is not given once and for all (the future is not given). — JuanZu

That's exactly why there is discontinuity. The past is given, the future is not. As you say, "the world is not given once and for all", only the past has been given. Therefore the present constitutes a discontinuity of time.

Your post discusses only the future and the possibilities of the future. Now, what about the actuality of the past, and the discontinuity between the possibilities of the future and the actuality of the past?

That, and the stronger version, that of which the negation is impossible.

Neither of these can refer to things we know, however. There can be no apodeitic certainty in empirical knowledge, at least that given from inductive inference, re: Hume, 1739.

That leaves me wondering what could be the criteria for logical impossibility other than contradiction.

And then what would be the criteria determining whether something would count as a physical or metaphysical impossibilty?

Hmmm. Good question. Insofar as logic regards only what we think, maybe logical contradiction has to do with the relation of conceptions to each other we think in a judgement, whereas logical impossibility has to do with the relation of judgements to each other we think in a cognition.

With respect to physical/metaphysical impossibility, the former has to do with the content, the latter with the form, of propositions in general? The physically impossible is e.g., that proposition in which there can exist no object to which conceptions may belong, and the metaphysically impossible merely exposes that this conception has no relation whatsoever to that conception.

Dunno. What say you?

Perhaps an example might help. It is not logically impossible that Mt Everest might detach form the Earth and float up into the sky. We might say it is physically impossible, given what we understand to be the laws of nature governing this Universe. We could say it is metaphysically possible since a Universe where the laws of nature or the lack of them allowed such a thing to happen is not a logical contradiction.

Is it possible that something could both be Mt Everest and not be Mt Everest in any imaginable world. It would not seem possible, since it is logically contradictory.

In any case, the point of my question was more concerned with understanding whether you think anything which is not logically necessary (or impossible) could be apodeictically certain.

Getting complicated, methinks. Logic talked about is in propositions; logic used in a cognitive method is in judgements, and the first presupposes the second.

The apodeictic certainty in Aristotle rests on either definition or self-evidence in propositions and is empirically demonstrable; the apodeictic certainty in Kant rests on judgement, and is merely thought. In Kant, the certainty which rests on definition or self-evidence, are termed analytic judgements. It follows that judgement antecedes and sets the ground for propositions.

That something is not logically necessary does not say it is not logically possible. The proof of that apodeictic judgement is in the truth of its negation: for that something which is logically necessary, that something must be possible.

Anyway….I think for the not logically necessary, there is apodeictic certainty in its logical possibility.

I agree there is logic used in "cognitive methods" (given that I'm understanding correctly what you mean by that), but that logic is not deductive, so I would say its results cannot be apodeictic. Rather the logic there is inductive (expectation based on observed regularities) and abductive (speculative inferences to what seem to be most plausible explanations, i.e. explanations most consistent and coherent with what has been currently accepted as knowledge).

So, I disagree with Kant that non-analytic judgements can be apodeictic. There can be no synthetic apriori certainty. I think what Kant was doing in working out the forms of intuition and the categories was phenomenology―that is he was reflecting on the nature of perception in order to establish its general characteristics. So, in that sense it's more of an observation-based inquiry. We can be certain of observationally confirmed judgements, but only within the appropriate context―the are not deductively certain and their negations are not logically self-contradictory.

That said I cannot, for example, imagine a non-spatiotemporal visual perception―visual perceptions are strictly defined in terms of spatiotemporality, so anything that doesn't comply would not be defined as such, and it can therefore be said to be, in that sense, an analytic judgement that all visual perceptions must be spatiotemporal.

I agree with you that something not being logically necessary does not entail it being logically impossible. If all the events in this world are not logically necessary, they must nonetheless be logically possible, so I also agree that there is apodeictic certainty in establishing what is logically possible―it's basically anything which is a non-contradiction. But the downside of that certainty is that it doesn't really tell us much about anything.

I agree there is logic used in "cognitive methods" (…), but that logic is not deductive, so I would say its results cannot be apodeictic. — Janus

There is a theoretical argument in which parts of the cognitive method, under certain conditions, as means to certain ends, is deductive, but the subject is not conscious of its functioning. I’d nonetheless agree the cognitive method in itself, insofar as it is not susceptible to empirical proof, wouldn't meet the criteria for apodeictic certainty. But the point of speculative metaphysical theory in general only extends to whether the parts of the method reflect certainty with respect to each other. It’s like….if this then that necessarily (the point)…..but…..there’s no proof there even is a this or that to begin with (beside the point).
————-

I disagree with Kant that non-analytic judgements can be apodeictic. — Janus

Non-analytic judgements are synthetic, and it is true no synthetic judgement possesses apodeictic certainty. But synthetic and synthetic a priori while being the same in form are not the same in origin.

There can be no synthetic apriori certainty. — Janus

Of course there can, provided the method by which they occur, which just is that difference in origin, is both logically possible and internally consistent. And is granted its proper philosophical standing.

Case in point: mathematics. How many pairs of straight lines would you have to draw, to prove to yourself you’re never going to enclose a space with them? After you’ve thought about it, maybe even drawn out a few pairs, why do I NOT have to tell you it cannot be done? And if you thought about it more narrowly, you'd discover you wouldn’t need to draw any pairs of lines at all to arrive at that conclusion yourself.

The form of that discovery;
…..(the judgement you made)….
The process by which the discovery manifests;
…..(a priori because you didn’t need the experience of drawing pairs of lines to facilitate the judgement)….
And the content of the discovery;
(The unrelated, thus synthetic concepts, “enclosed space” and “pairs of straight lines”, conjoined in the judgement)

…..gives exactly what you say there cannot be.

So even if this particular method is not accepted, it is still true, still necessarily the case, two straight lines cannot enclose a space. Is there another way, equally valid, to get this apodeictally certain kind of non-empirical knowledge?
—————-

…..doesn't really tell us much about anything. — Janus

True enough. Knowledge proper is in experience. Logic merely guides the system and limits the method by which experience is possible.
—————-

Have you heard about the observation of (the effects of) colliding black holes? Talk about paling in comparison, everything I just said…..

But the point of speculative metaphysical theory in general only extends to whether the parts of the method reflect certainty with respect to each other. It’s like….if this then that necessarily (the point)…..but…..there’s no proof there even is a this or that to begin with (beside the point). — Mww

So, what you seem to be talking about is not certainty, but consistency.

Non-analytic judgements are synthetic, and it is true no synthetic judgement possesses apodeictic certainty. But synthetic and synthetic a priori while being the same in form are not the same in origin. — Mww

I think we agree on this. Do you mean that syntheses are hypotheses, whereas synthetic a priori propositions are phenomenologically derived by reflecting on experience in order to establish its general characteristics?

There can be no synthetic apriori certainty.
— Janus

Of course there can, provided the method by which they occur, which just is that difference in origin, is both logically possible and internally consistent. And is granted its proper philosophical standing. — Mww

Now you seem to be contradicting what you said above. I don't see how either logical possibility or internal consistency can yield certainty. And I have no idea what "proper philosophical standing" could be.

Case in point: mathematics. How many pairs of straight lines would you have to draw, to prove to yourself you’re never going to enclose a space with them? — Mww

It is logically self-evident that a pair of lines cannot enclose a space, so I'd call that analytic, not synthetic.

Have you heard about the observation of (the effects of) colliding black holes? Talk about paling in comparison, everything I just said….. — Mww

I hadn't, so I searched on it...very interesting, said to confirm predictions made by Einstein and Hawking. The interesting thing about scientific theories is that it seems they cannot be confirmed to be true, even on account of their predictions being confirmed by observation.

Our intentional acts, as they are thrown into the possible and the non-given of the world, imply operationally a continuity between the measuring apparatus and that which is measured. For, after all, to act in a non-given world is to act in relation to something other than the presence of the present and the present of consciousness. There is, then, a relationship between our operational actions and the non-given of the world. That is, because non-consciousness is involved in operativity, there is a continuity between the measuring apparatus and that which is measured. This continuity is therefore beyond consciousness, just as the future is beyond the present (which is the form of consciousness). And our intentional acts such as "measuring" are also thrown beyond consciousness and the presence of the present.

There is no place here to talk about the past, since conscious and intentional acts occur in relation to a possible future. There may be a discontinuity in time (I leave the question open), but there is no discontinuity between the measuring device, that which is measured, and our intentional-operational act. All three are beyond what is given to consciousness. That is why subjectivist interpretations of quantum physics are wrong. Measuring here is seen as an act that is thrown into the possible and the not-given for consciousness. It is no coincidence that operativity is closely related to the body, and that the body has long been conceived as the other of the mind and consciousness.

The non-given, the body, intentional acts (measuring), and non-consciousness are all intertwined.

Our intentional acts, as they are thrown into the possible and the non-given of the world, imply operationally a continuity between the measuring apparatus and that which is measured. — JuanZu

I don't dispute the continuity between the measuring device and the physical world being measured. Both are part of the given world. The discontinuity is between the non-given possibilities of intention, and the givenness of the sensed world.

Nor do I dispute that there is a "relationship" between the non-given and the given. However, I assert that the relationship is one of discontinuity. In fact, the description as two distinct things, given and non-given, with a relationship between the two, itself implies a discontinuity. If there was continuity, it would be one continuous thing.

There is no place here to talk about the past, since conscious and intentional acts occur in relation to a possible future. — JuanZu

How can you say this? The reality of what you refer to as "the measuring apparatus and that which is measured" is supported by their existence in the past, and sense observation of them, in the past. Without their past existence, they are only future possibilities, needing to be created in a physical presence. "Physical presence" is a product of past observation, having no reality without past observation.

Any "measurement" itself, as the "quantity" or "value" derived, exists in the realm of intentionality, the non-given. And, there is a discontinuity between this, the non-given, and the givenness of the apparatus and object to be measured

I don't dispute the continuity between the measuring device and the physical world being measured. Both are part of the given world. — Metaphysician Undercover

The measuring device and that which is measured enter into a teleological operational dynamic. Here, it is the act of measuring. That is why neither can be excluded from the non-given of the world. This is even more so when things in quantum physics are decided from one moment to the next with the intervention of the measuring device.

How can you say this? The reality of what you refer to as "the measuring apparatus and that which is measured" is supported by their existence in the past, and sense observation of them, in the past. — Metaphysician Undercover

I can say this because it is essential for their participation in scientific practice that the world is not given once and for all. The creation of the entire experiment depends on it. Experimenting implies a relationship with the future, and so we create the conditions for an experiment just as we create a measuring device.

Experimenting implies a relationship with the future, and so we create the conditions for an experiment just as we create a measuring device. — JuanZu

Correct, but that relationship between the past and the future is discontinuous. That's why "the prediction" is never a statement of necessity, and this is fundamental to experimentation.

It is logically self-evident that a pair of lines cannot enclose a space…. — Janus

Since the ancients that has been the case. It just makes sense that two straight lines cannot enclose a space but no one ever thought about the rational mechanism by which two unrelated, non-empirical conceptions can be conjoined to construct its own evidence, since Nature is never going to provide the universality and absolute necessity required for its proof.

….so I'd call that analytic, not synthetic. — Janus

Usually a judgement is termed tautological insofar as it is true by definition irrespective of its conceptual content, whereas analytical merely indicates that the subject/predicate conceptions as the content in self-evident judgements belong to each other, or that one contains the other within it.

The conception of a straight line, on the other hand, does not contain the conception of number, nor can the conception of a number be thought as belonging to the conception of a line, hence the judgement with “two straight lines” as its subject, is termed synthetical. In the judgement “every body is extended”, the conception in the subject is related to the conception in the predicate, in that you cannot think a body without the extension connected to it, so is analytical.
—————-

I don't see how either logical possibility or internal consistency can yield certainty. — Janus

Neither do I; in themselves they don’t. They are the conditions necessary in the form of a judgement, for the certainty in the relations of the conceptions which are its content. They don’t yield, or produce, certainty, so much as make it possible.

Sorry for the delay. I got doin’ Her Satanic Majesty’s Request, if ya know what I mean. Flower beds, of all things. The kinda thing the average joe’s hardly likely to get right.

It just makes sense that two straight lines cannot enclose a space but no one ever thought about the rational mechanism by which two unrelated, non-empirical conceptions can be conjoined to construct its own evidence, since Nature is never going to provide the universality and absolute necessity required for its proof. — Mww

Think about the ancient shepherds—if they put in just two parallel fence lines, it would have been obvious that would not keep the sheep in or the wolves out. Or a building with just two walls and a roof and the ends open, or even three walls—it is immediately evident that the spaces have not been enclosed.

Usually a judgement is termed tautological insofar as it is true by definition irrespective of its conceptual content, whereas analytical merely indicates that the subject/predicate conceptions as the content in self-evident judgements belong to each other, or that one contains the other within it. — Mww

I don't see how a tautology could be independent of its conceptual content—can you give an example?

Neither do I; in themselves they don’t. They are the conditions necessary in the form of a judgement, for the certainty in the relations of the conceptions which are its content. They don’t yield, or produce, certainty, so much as make it possible. — Mww

Right, certainty certainly could not be possible with logical possibility and consistency.

Sorry for the delay. I got doin’ Her Satanic Majesty’s Request, if ya know what I mean. Flower beds, of all things. The kinda thing the average joe’s hardly likely to get right. — Mww

No need for an apology, we all have other commitments and there are many things more important than philosophy. Getting flower beds right is something I take for granted, given that my profession as landscape design and construction. I also know that the downsides of not yielding to the requests of "Her Satanic Majesty" can be great. 'Their Satanic Majesties Request' is also an underrated Stone's album, one of my early favorites.

I don't see how a tautology could be independent of its conceptual content—can you give an example? — Janus

I said irrespective, not independent. It is impossible to even think, judge, cognize, reason…any of that stuff, independently of conceptual content. Even so, I probably could have worded it better, in that, while being independent of conceptual content is impossible, the fact the judgement warrants the title “tautology” indicates only a certain relation between them.

I meant to say it isn’t the conceptions themselves that earn the title, but the relation of them to each other. For those conceptions that don’t relate the title is lost, that’s all.
————-

if they put in just two parallel fence lines, it would have been obvious that would not keep the sheep in or the wolves out. — Janus

Actually, a good example. It shows since Day One, humans had this cognitive capacity, as simply a part of its general intellect. Those academically/philosophically/scientifically illiterate shepards knew something without having to do the work for experiencing it. All that was needed, a few centuries later when somebody stopped to think about it, was a system in which that knowledge was possible, followed by a theory to demonstrate how the system works.

It never was a question of it being done, but, how it is done. Kant’s Claim to Fame is that he assembled the first definitive exposè for this particular inherent human capacity, and no one has done it any better since. “Better” here meant to indicate more complete, beginning to end, front to back, top to bottom.

Why two straight lines cannot enclose a space, is no longer a mystery. Even if it isn’t the case, it is still a perfectly logical explanation.

I meant to say it isn’t the conceptions themselves that earn the title, but the relation of them to each other. For those conceptions that don’t relate the title is lost, that’s all. — Mww

Right so one idea cannot on its own be tautologous, but rather a statement is tautologous that redundantly states something about two ideas that both share the same salient content.

Why two straight lines cannot enclose a space, is no longer a mystery. Even if it isn’t the case, it is still a perfectly logical explanation. — Mww

I'm not sure what you mean by "even if it isn't the case".

I'm not sure what you mean by "even if it isn't the case". — Janus

As you say, there are no synthetic a priori judgements, but as Kant says, that logical construct (proposition, judgement), in which the conceptions have no relation to each other but are connected in thought, are called synthetic a priori judgements, and are used by the cognitive faculties as principles. You may be correct, in that it isn’t the case the cognitive faculties use such judgements as principles, because there isn’t any such thing.

Thing is, even though I cannot prove the tenets or conditions supporting the theory, you cannot disprove them either. Of the two, my position is nonetheless stronger, in that the logic used in the construction of the theory cannot be shown to be self-contradictory, which is the only way to falsify the theory itself. Best you can do, is start over, with different sets of initial premises and thereby come to a different conclusion.

What will you do, then, to escape the fundamental starting point, the least likely to be wrong initial condition, that the shepard simply “sees” a two-poled fence won’t work? Granting that point, which seems the most reasonable, all that remains is the explanation for what it is to “see”.

Same as it ever was…..as my ol’ buddy Brain Eno used to say.

As you say, there are no synthetic a priori judgements, but as Kant says, that logical construct (proposition, judgement), in which the conceptions have no relation to each other but are connected in thought, are called synthetic a priori judgements, and are used by the cognitive faculties as principles. — Mww

Might be a good place for a recap of what the synthetic a priori is, and the role it has in Kant's COPR. The phrase “synthetic a priori” is one of the pivots of the entire text. Kant thought the whole problem of pure reason could be summed up in the question: how are synthetic a priori judgments possible? (B19).

The types of judgements are:

• Analytic a priori = true by definition, like “all bachelors are unmarried.”
• Synthetic a posteriori = adds new content, but known through experience, like “this apple is red.”
• Synthetic a priori = adds new content, but is knowable independently of experience.

That last category was Kant’s unique insight. Mathematics is built around it — “7+5=12” is not analytic, because “12” isn’t contained in “7+5,” but it’s still a priori. Geometry also: “the shortest distance between two points is a straight line.” And physics relies on principles like “every change has a cause,” which aren’t derived from experience but condition how we can have experience of a lawful world in the first place (indeed are central to the whole idea of there being physical laws.)

This is why Kant recognized the synthetic a priori judgment as fundamental to science. They’re what make mathematical physics possible — and because physics underpins so many other domains, they’re indirectly what make large parts of the other sciences possible. Without them, knowledge would collapse into either tautology (analytic truths) or mere observation (synthetic a posteriori). The ability to make predictions based on axioms is central to scientific method, and that ability depends on having judgments that are both synthetic (they extend knowledge) and a priori (they hold universally and necessarily).

Kant’s point was that the mind isn’t just passively recording facts, but actively structuring experience according to a priori forms and concepts. That’s how we get laws of nature that are universal and necessary, rather than just habits of expectation.

So when people debate whether the “synthetic a priori” really exists, it’s worth remembering: Kant wasn’t spinning an abstraction — he was trying to explain the actual success of mathematics and Newtonian science. His claim is that you can’t make sense of those successes without granting that there are truths that are both synthetic and a priori.

Nowadays, there is debate over whether there really are laws of nature (see Nancy Cartwright, No God, No Laws.) There is also the tendency to regard such Kantian posits as aspects of psychology or to relativise them in other ways (Quine's Two Dogmas of Empiricism for instance.) But in Kant's terms, the idea of the 'synthetic a priori' is basic to the entire project of the Critique, and without it the possibility of mathematics and natural science as objective knowledge would be left unexplained.

That last category was Kant’s unique insight. Mathematics is built around it — “7+5=12” is not analytic, because “12” isn’t contained in “7+5,” but it’s still a priori. — Wayfarer

It may be said to be analytic because 7+5 is one of the 6 ways that 12 can be divided up into two groups. Just as 'bachelor' is a name for an unmarried man, so '7+5' is one of the seven 'paired' ways of defining '12' (if you don't count 12+0). Or it may be said that it is a posteriori, because if you have twelve pebbles you can divide them up into all the possible pairs. It comes down to different possible ways of looking at things, so there is no absolute fact of the matter as to whether mathematics is analytic or
synthetic.

What Kant writes here can be interpreted to support the idea that mathematics is analytic:

In all judgments in which the relation of a subject to the predicate is thought (if I only consider affirmative judgments, since the application to negative ones is easy) this relation is possible in two different ways. Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case, I call the judgment analytic, in the second synthetic. (1787 [1998], B10)

the predicate concept '7+5' certainly seems to be (covertly) contained in the concept '12'.

Nowadays, there is debate over whether there really are laws of nature — Wayfarer

There are obviously invariances in nature. The "laws of nature" simply codify the observable invariances. Also, as per Peirce, what look like invariances to us, whose horizons of observation, both temporally and spatially speaking, are so tiny, may not be timeless and fixed, but evolved habits.

You’re right that one can treat “7+5=12” as analytic by stipulation, or check it a posteriori with pebbles. But Kant’s point is that neither account explains why mathematics is both necessary and informative. If it were analytic, it would be tautological; if empirical, it would be contingent. The synthetic a priori is his way of capturing that “in-between” character. It also has bearing on how mathematics is 'unreasonably efficacious in the natural sciences.'

I think you are affording Kant less ambiguity than his actual writings display. In the SEP article Kant's Philosophy of Mathematics is this:

"The central thesis of Kant’s account of the uniqueness of mathematical reasoning is his claim that mathematical cognition derives from the “construction” of its concepts: “to construct a concept means to exhibit a priori the intuition corresponding to it” (A713/B741) "

What Kant seems to gloss over is that this kind of a priori reasoning is distilled from perceptual experience, so it counts as a kind of a posteriori insight regarding not particulars but generalities. "7+5=12" is tautological just as 'all bachelors are married" by virtue of the meanings of the words "7+5" "equals" and "12".

You should read Quine's 'Two Dogmas of Empiricism" where he, among other interesting critiques, questions the very distinction between the synthetic and the analytic.

What Kant seems to gloss over is that this kind of a priori reasoning is distilled from perceptual experience, — Janus

Other creatures also have perceptual experience, and some can even discriminate small quantities. But they don’t go on to develop arithmetic. That shows arithmetic is not just “distilled” from perception, but depends on something prior in our cognitive framework — the capacity to represent number as such, and to apply operations universally and necessarily.

I'm familiar with Quine's argument, which is why I mentioned it, but my aim here, as the 'synthetic a priori' was mentioned, was simply to recap what they are.

That shows arithmetic is not just “distilled” from perception, but depends on something prior in our cognitive framework — the capacity to represent number as such, and to apply operations universally and necessarily. — Wayfarer

I don't think it shows that arithmetic is not distilled from perception at all. Of course to have arithmetic, as a systems procedures and rules and you also need symbolic language.

So I don't agree with this:

But in Kant's terms, the idea of the 'synthetic a priori' is basic to the entire project of the Critique, and without it the possibility of mathematics and natural science as objective knowledge would be left unexplained. — Wayfarer

At least if "But in Kant's terms" is left out. I don't know what that phrase is doing there, unless it is meant to indicate something like "this is what Kant thought". But again, we don't have to agree with Kant unless his arguments are convincing; which I think in light of developments in philosophy since Kant, they are not. In my view, he makes too little of what can be derived from experience in combination with symbolic language. We don't need an "agent intellect" whatever that could even be―the ability to perceive patterns gives us the capacity to recognize regularities and generalities in Nature, and symbolic language gives us the capacity to reflect further on those and abstract out generalities and generalize laws.

Instead of thinking of the subject as a kind of dimensionless point of consciousness, I prefer to think of the subject as processually embodied in a world of other bodies and processes which present us with number, pattern and invariance, such that we could hardly fail to recognize them.

Oddly enough I didn't notice you had mentioned the Quine paper. Have you read it?

Briefly, although it wasn’t on the curriculum of the courses I did.

In my view, he makes too little of what can be derived from experience in combination with symbolic language — Janus

Kant in no way denied the fundamental role of language, I don’t think that would have ever occurred to him.

The ‘empirical doctrine of mathematics’ is associated with John Stuart Mill, although as I understand it, very much a minority view.

But Kant’s point is that neither account explains why mathematics is both necessary and informative. If it were analytic, it would be tautological; if empirical, it would be contingent. The synthetic a priori is his way of capturing that “in-between” character. It also has bearing on how mathematics is 'unreasonably efficacious in the natural sciences.' — Wayfarer

Or rather, it explains why mathematics is simply efficacious - mathematical conventions are arbitrary and independent of facts and hence a priori, and yet the mathematical proofs built upon them require labour and resources to compute, which implies that the truth of mathematical theorems is physically contigent and hence synthentic a posteriori. Hence the conjecture of unreasonable effectiveness is not-even-wrong nonsense, due to the impossibility of giving an a priori definition of mathematical truth.

Kant in no way denied the fundamental role of language, I don’t think that would have ever occurred to him.

The ‘empirical doctrine of mathematics’ is associated with John Stuart Mill, although as I understand it, very much a minority view. — Wayfarer

Of course he wouldn't deny the role of language―he presents his ideas in language after all. But he didn't acknowledge or emphasize that language enables the elaborations of rules constructed on the foundation of what is perceptually recognized. (By "perception" I includes what is cognized just on the sheer basis of embodiment, I.e., proprioception and interoceptions of bodily sensations.

The idea that number is perceptually encountered is not exclusive to Mill, and I doubt that it is a minority view. Do you have statistics to back that claim up? In any case, even if if it were a minority view, so what? What matters is whether it is plausibly in accordance with experience.

If number is inherent to existence, which it seems it must be if more than one thing exists, then it seems plausible to think that its effectiveness is on account of its ontologically inherent and immanent nature

If mathematics were merely convention, then its success in physics would indeed be a miracle — why should arbitrary symbols line up so exactly with the predictability of nature? And if it were merely empirical, then we could never be sure it applies universally and necessarily, which is precisely what science assumes (hence the endless invective directed at things which are said to fall outside physical laws). Kant’s claim is that mathematics is neither arbitrary convention nor empirical generalisation, but synthetic a priori: it extends knowledge while still being based on necessary truths.

A nice case of the “unreasonable effectiveness” is Dirac’s prediction of anti-matter — it literally “fell out of the equations” long before there was any empirical validation of it. That shows mathematics is not just convention or generalisation, but a way of extending knowledge synthetically a priori. (Of course, as Sabine Hossenfelder reminds us, mathematics can also mislead if we take the beauty of equations as a substitute for empirical test.)

I’m objecting to the theory that mathematical knowledge can be attributed to the generalisation of or abstraction from experience. We have to have the ability to count and perform various other mental operations on concepts in order to grasp maths. And even then there is an enormous range of skill that can be observed amongst people, with a Terrence Tao at one end of the spectrum, and those with a rudimentary ability in mathematics at the other. No amount of experience can close that gap, if the ability is not there.