. Mathematical judgements are always synthetical [a priori]. Hitherto this fact, though incontestably true and very important in its consequences, seems to have escaped the analysts of the human mind, nay, to be in complete opposition to all their conjectures. ... — waarala

This is not true. Mathematical structures rest in nature, for us to be discovered.

The science of natural philosophy (physics) contains in itself synthetical judgements a priori, as principles. — waarala

True, but after the principles are found. The principles lay buried in nature. No a priori in our minds. The only a priori is the scientific attitude.

:up:

There are "pure" or ideal triangles in nature (as sensible matter)??

Notice that these synthetical a priori principles are called "judgements". This is why skepticism as an approach to fundamental principles is very important, as a tool to find mistakes within these judgements. Without skepticism, these synthetical judgements are simply taken for granted as "discovered" principles like Hilary says, when in reality they are synthetical structures created by judgement. Plato is often misunderstood as promoting a position in which we take these principles for granted as eternal truths, when in reality Plato was a skeptic demonstrating the need to question such principles.

Galileo counters the Aristotelian approach not by performing experiments, but by showing that it [e.g. the mathematical fabric of space-time] must be so and not otherwise. In this sense, physics is made to be an a priori discipline of necessary truths. Koyré sums it up as follows: ‘The Galilean revolution can be boiled down … to the discovery of the fact that mathematics is the grammar of science. It is this discovery of the rational structure of Nature which gave the a priori foundations to the modern experimental science and made its constitution possible.

That, I think, is the source of Kant's conviction that physics can be an a priori science - that 'physics, like mathematics, is a body of necessary and universal truth.' Noble sentiment but hardly sustainable in respect of physics since Einstein, I would think. — Wayfarer

When mathematics is "the grammar of science", which formats the way that the structure of nature is revealed to us, then what is understood as "the structure of nature", the phenomenon which bears that name, is literally formed or created by that mathematics.

The issue then becomes the quest to maintain correspondence between "the structure of nature" (as created by the human mind), and intelligibility (as dictated by what the mind can understand). When mathematicians allow the axioms of "pure mathematics" to stray outside the limits of the fundamental laws of logic, identity for example, then there is inconsistency between the grammar of mathematics and the grammar of intelligibility. Since "the structure of nature" is based in mathematics, it is possible that this structure may become completely unintelligible to us, depending on the intelligibility of the axioms employed.

There are "pure" or ideal triangles in nature (as sensible matter)?? — waarala

Yes. And cubes, hexagons, parabola, even hyperbolical space. Spheres, sine waves, fields, groups, you name it. Mandelbrot considered his set a discovery.

How these forms are observed to exist?

Just look at them!

But you can look at them only through the pure intuition!

Hey.....

I remember. I generally agree, taking exception only to your referring to “pure physics” in a Kantian context. As brought to light by , it is clear there is a pure part of physics with respect to the a priori principles which make the science possible, but “pure physics” as a general conception, has not the same distinction as....

“...Before all, be it observed, that proper mathematical propositions are always judgements à priori, and not empirical, because they carry along with them the conception of necessity, which cannot be given by experience. If this be demurred to, it matters not; I will then limit my assertion to pure mathematics, the very conception of which implies that it consists of knowledge altogether non-empirical and à priori....”

....in which we see how he wishes “pure” regarding the “theoretical sciences of reason” to be understood.

Minor point to be sure, but.....you know.....in the interest of the straight and narrow.....
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discovery of the rational structure of Nature which gave the a priori foundations to the modern experimental science

That says more about us than I would ever allow.

Yes but that passage that waarala has pointed out is significant here. Again:

2. The science of natural philosophy (physics) contains in itself synthetical judgements à priori, as principles. I shall adduce two propositions. For instance, the proposition, “In all changes of the material world, the quantity of matter remains unchanged”; or, that, “In all communication of motion, action and reaction must always be equal.” In both of these, not only is the necessity, and therefore their origin à priori clear, but also that they are synthetical propositions. — V. In all Theoretical Sciences of Reason....

What is the import of 'not only is the necessity...'? Is that not that such propositions are actually both a matter of logical necessity and also of physical principle?

I remember. I generally agree, taking exception only to your referring to “pure physics” in a Kantian context. As brought to light by ↪waarala, it is clear there is a pure part of physics with respect to the a priori principles which make the science possible, but “pure physics” as a general conception, has not the same distinction as....

“...Before all, be it observed, that proper mathematical propositions are always judgements à priori, and not empirical, because they carry along with them the conception of necessity, which cannot be given by experience. If this be demurred to, it matters not; I will then limit my assertion to pure mathematics, the very conception of which implies that it consists of knowledge altogether non-empirical and à priori....”

....in which we see how he wishes “pure” regarding the “theoretical sciences of reason” to be understood. — Mww

I believe there is no such thing as "pure" a priori. The a priori is always conditioned by the basic intuitions, space and time, which are inherently dependent on experience. Even in mathematical principles, if we attempt "the pure", we remove ourselves from an applicability with the consequence of useless fiction.

What is the import of 'not only is the necessity...'? — Wayfarer

The importance of necessity resides in the condition Kant requires for pure a priori manifestations in his transcendental system. Necessity, along with universality, are the conditions determining whether or not some conception/judgement/cognition/knowledge is a priori or empirical.

“....Necessity and strict universality, therefore, are infallible tests for distinguishing pure from empirical knowledge, and are inseparably connected with each other....”

The synthetical part comes from the relation of conceptions in a proposition to each other, but that is beside the point of the grounds for determining what kind of proposition is under consideration.

This is all just groundwork, setting the stage, for the rest of his speculative metaphysics. Ever notice how little time he spends on stuff like this, compared to the non-empirical stuff on which he did elaborate, seemingly to no end, that the empiricists of the day utterly neglected? He writes for 98 pages on the empirical, but writes for 610 pages on the non-empirical. All set up by a mere 14 pages in the introduction, from which all the above is a part.
———-

Is that not that such propositions are actually both a matter of logical necessity and also of physical principle? — Wayfarer

This has to do with origins, not examples, or proofs. The matter of physical principle in play here is induction, which lessens the importance of “strict universality”, relegating it to “as far as we know”, but leaves necessity to condition the proposition as a priori as opposed to entirely empirical.

Referencing the quote, have to keep in mind “the science of natural philosophy” is merely that which the human does in accordance with certain criteria he himself constructs, in order to make sense of observations. The physical doesn’t contain principles, it abides by them, at least as far as our kind of intelligence decides it does.

I have a deep confusion about why philosophy sees this disconnection between logical necessity and physical causation — Wayfarer

They are different categories. Logic consists of correct rules of reasoning. Causation is a physical phenomenon, reflecting a physical relation.

Law realists (e.g. Armstrong, Tooley, and Sosa) solve the problem of induction by proposing that there are laws of nature, not merely relations between objects (as Hume suggested). A law is a physical relation between types of things.

I believe there is no such thing as "pure" a priori. — Metaphysician Undercover

That’s fine, no problem. In effect, you’re dismissing, or at least disputing, the fundamental ground of Kantian metaphysics, which has been done since he published it. Buried in the subtleties, though, is the “first time for everything” qualifier, which, because people know so much about the world these days and science has taken us merrily down the empirical path, we tend to overlook as cognitive prerequisites. It’s like....boiling water. Big deal. Boiled water since I was a kid. That I gotta put water in the pot first just comes with the territory. And turning on the stove. And making sure I got electricity. And making the electricity that runs the stove. And building the dam. And mining the gypsum.

How boring. The real fun starts in going the other way. Everybody thinks; no one knows how thinking happens. So...there ya go, ripe for theoretical musings.

Yes, but that intuition has been shaped by countless encounters with them. Already in the womb your brain is stimulated by concentrically converging circle shapes.

How wonderful the nature is! Full of complicated geometrical figures!

Every mathematical shape has a counterpart in nature or can be created by us. Of course an infinite dimensional vector space has no counter part, and it's the question of path integrals of the Lagrangian truly exist, but it should be possible to physically make one. Like Möbius bands.

The physical doesn’t contain principles, it abides by them, at least as far as our kind of intelligence decides it does. — Mww

'Our kind of intelligence', compared to what?

Note the quotation on my profile from Chris Fuchs, author of ‘quantum Qbism’, ‘Quantum mechanics is a law of thought.’

Law realists (e.g. Armstrong, Tooley, and Sosa) solve the problem of induction by proposing that there are laws of nature, not merely relations between objects (as Hume suggested). A law is a physical relation between types of things. — Relativist

I believe in the concept of 'laws of nature', but I don't believe they can be described as physical. They precede the physical, they are what first must exist in order for there to be anything physical.

'Our kind of intelligence', compared to what? — Wayfarer

Hmmmm....dunno, really. I don’t think comparisons are possible. I mean, all we have to compare with, is our own, so what we we learn from it, except what ours tells us?

‘Quantum mechanics is a law of thought.’ — Wayfarer

I like it!!

I believe in the concept of 'laws of nature', but I don't believe they can be described as physical. They precede the physical, they are what first must exist in order for there to be anything physical. — Wayfarer

Where do laws of nature exist? In the mind of God? Platonic "third realm"? How do these nonphysical laws influence physical things?

In nature. Where else? The laws describe how natural stuff behaves so it's in the behavior of nature where they reside.

Where do laws of nature exist? In the mind of God? Platonic "third realm"? How do these nonphysical laws influence physical things? — Relativist

To me, that is THE most important question in philosophy. I could write a lot, but I will confine myself to this observation: in relation to these kinds of order, what does it mean to say that they exist?

Consider a number - pick any number, 7 will do. In what sense does '7' exist? Well, you might say, you're looking at it. But what we're looking at is a symbol. It could just as easily be denoted 'VII' or 'seven'. What is denoted by that symbol is a mental operation, a count. It is discernable only to a rational mind, a mind capable of counting. Yet for any such mind, it is invariable; 7 = 7 in all possible worlds.

My view is, all of these primitive or basic intellectual operations such as number and logical principles underpin the process of rational thought and language. We're not conscious of them, as we see through them, and with them, they're the architecture of reason. But as our culture is overwhelmingly empiricist in outlook, then we don't consider them real, as they don't exist 'out there somewhere'. And for empiricism, what is 'out there somewhere' is the touchstone of what is real.

This is a revisionist form of platonism. See the discussion in this article. I'm with James Robert Brown, representing Platonism in that article (so much so, I bought his book, which has not been that useful, regrettably.)

I mean, all we have to compare with, is our own, so what we we learn from it, except what ours tells us? — Mww

That was my point! You said

...at least as far as our kind of intelligence... — Mww

What other kind is there?

I'm beginning to see why there is this dogma that logical necessity and physical causation belong to different domains. It's the underlying mind-body dualism that is still at the basis of our modern outlook - post-Cartesian dualism, which operates in our thinking whether we know it or not. I'm reading Husserl's Crisis of the European Sciences, and it is all laid out clearly in that book, in the chapters on Galileo and 'the mathematicization of nature'.

Hmm. Stuck in Kant again, aren't we.

Recall Quine’s Alternative?

perhaps has something like that in mind.

But for my part, the answer to

In what sense does '7' exist? — Wayfarer

is that it is part of how we talk about the world, not so much part of the world. They are real more in the way of money and property than of rocks and thunder.

Things happened in philosophy after Kant.

How boring. The real fun starts in going the other way. Everybody thinks; no one knows how thinking happens. So...there ya go, ripe for theoretical musings. — Mww

Well, I'll qualify my statement then. Instead of saying that there is no such thing as pure a priori, I'll say that if we seek such, we find a big division between the intuition of space, and the intuition of time. And this division manifests itself in the principles of mathematics. Time is fundamentally order, and space is represented by lines. But order is of discrete units, while a line is a continuity.

Space, being an intuition dealing with how we relate to the external, cannot be a priori. This is because it can only be created from the individual seeing oneself as an individual, distinct or independent from one's environment, and this condition is posterior to the most primitive experience of the human being, being born. If we look back to the prior condition, the earlier condition, which is being in the womb, at that time the developing individual is united to, as a part of something larger. So there is no self with an external, as the thing which will be a self, is simply an internal part of something else. This is the condition which some mystics guide us toward, the position in which we are simply united within a grander whole, and there is no proper individual self, and no proper "external".

From this position, there can be no intuition of space, as this the original condition, in which there is no external, therefore no space, and this experience is prior to the intuition of space. And this is why, within the manifestation of mathematics. spatial conceptions lack in necessity. Geometrical principles can be altered as we see fit, so that, for example, parallel lines may meet in a curved spacetime. Our spatial intuitions, which lead us into geometry are contingent on how we conceive of time, which is derived from our temporal intuitions.

This places the intuition of time as deeper than, and prior to, the intuition of space. It manifests as the most basic of mathematical principles, order. The issue which Wayfarer points us to with this thread is the question of necessity in order, which is understood through "causation". But the real question is whether necessity can be removed from order, in a way similar to how I described that necessity can be removed from spatial conceptions.

Modern mathematics uses axioms which deny the necessity of order (a set of objects without an order for example). To validate such an axiom, "order" would have to be contingent on an even deeper intuition. We'd have to intuitively apprehend something deeper which order is dependent on. But I cannot find any deeper intuitions, to say that order is dependent on something else, and validate removing necessity from order. So I find that such axioms which attempt to remove necessity from order, are extremely counter-intuitive. How this intuition of time, manifesting as order, is itself grounded, whether it is grounded in experience, or something more fundamental than experience, as prior to experience, and a condition for the possibility of experience, is probably an issue of how we define the terms.

You said...at least as far as our kind of intelligence...
— Mww

What other kind is there? — Wayfarer

As many other kinds as there are other kinds of brains? Other kinds of CNS’s? I’m not about to say ours is the only kind of intelligence there is, but it is certainly the only kind that’s of any use to us.
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I'm beginning to see why there is this dogma that logical necessity and physical causation belong to different domains. It's the underlying mind-body dualism that is still at the basis of our modern outlook — Wayfarer

Different domains/mind-body dualism....ok. Nature of the human beast, methinks.

Dogma....ehhh, sorta ok. Dogma with proper criticism, fine; dogmatism, use of dogmatic systems without the built-in mechanisms for proper self-criticism, dangerous.

Depends on how you mean the term to be used, I guess. Most use it with pejorative connotations, and I don’t want to imply that’s what you’re doing.

Are you saying our modern outlook shouldn’t have a basic underlying mind/body dualism?

Hmm. Stuck in Kant again, aren't we.

Recall Quine’s Alternative? — Banno

Where I've encountered that argument is in The Indispensability Argument in the Philosophy of Mathematics. What Quine wants to do is 'naturalise' mathematics - part of the general process of naturalised epistemology. And by 'naturalised', what is meant is 'conformant with standard, neo-darwinian materialism' (although Quine may not say so explicitly) But it's the motivation for that which I am calling into question.

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 debar any knowledge of mathematical objects.

And why do 'our best' epistemic theories seem to debar any such knowledge?

Some philosophers, called 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.

Whereas I reject the assertion that humans are 'physical creatures whose capacities for learning are exhausted by our physical bodies'. Instead, I am more inclined to the traditional, rationalist claim that the faculty of reason discerns an order which is not explicable in physical terms. That ties into the the 'argument from reason', which we've discussed previously: the ability to grasp intellectual objects, such as number, is precisely that which sets humans apart from other sentient beings. We have the kind of nous that our simian forbears lack. (Heresy, I know. Somewhere, Hilary Putnam has an essay on the impossibility of naturalising reason, which I must get around to reading. However a large part of Nagel's work is about exactly that point also - see his Evolutionary Naturalism and the Fear of Religion.)

//aha @Banno - found this essay on Putnam - haven't had time to read it, but it mentions Davidson and Quine.

What Quine wants to do is 'naturalise' mathematics — Wayfarer

Oh, probably. But it is not as if he could reduce mathematics to physics!

But it's the motivation for that which I am calling into question. — Wayfarer

Whereas I would rather play with the actual arguments. A difference in style, I guess. I suspect that the what might be missing from Nagel and Putnam is that rationality is a group enterprise; since it is dependent on language, it is an aspect of our institutional world.

So again, we can understand the kettle not heating up while over the flame, and that such an occurrence would be problematic for physics, but not for logic. But the kettle not being a kettle is a problem for logic. Physical cause is a different thing to logical necessity. The desire is there to apply the supposed certainty of modus ponens to physics, but it's a false use.

. I suspect that the what might be missing from Nagel and Putnam is that rationality is a group enterprise; since it is dependent on language, it is an aspect of our institutional world. — Banno

But to 'explain reason' is to invariably sell it short! As soon as you account for it in anything other than it's own terms, then you're denying the sovereignty of reason. I'm beginning to suspect that the very existence of reason is actually an inconvenient truth for a lot of analytical philosophy.

Physical cause is a different thing to logical necessity. — Banno

But they meet all the time. They can be separated by abstracting them, but in practice, everything we do is predicated on the fact that existence has a certain logic. We don't put the kettle on thinking it will turn into an elephant and trample us.

My view is, all of these primitive or basic intellectual operations such as number and logical principles underpin the process of rational thought and language. We're not conscious of them, as we see through them, and with them, they're the architecture of reason. But as our culture is overwhelmingly empiricist in outlook, then we don't consider them real, as they don't exist 'out there somewhere'. And for empiricism, what is 'out there somewhere' is the touchstone of what is real. — Wayfarer

IMO, the touchstone of what is real is the physical world and the physical stuff in it. I'm not inclined to assume non-physical things exist if the relevant phenomena can be adequately accounted for in physicalist terms. That makes it superfluous. Humans are adept at abstract reasoning, rooted in the way of abstraction, whereby we consider properties of things independently of the things. Our ability to discern redness does not imply redness existing independently of red objects. Same with numbers: there exist groups of 3 objects, but this doesn't imply "3" exists independently of the things that exhibit the "threeness" property. There are logical relations between the numeric abstractions (like 2+2=4), but again, this doesn't entail the independent existence of these numbers.

Not only do these abstractions seem superfluous, their independent existence requires accounting for how they relate to the physical world. I have 4 marbles in my hand. Does this fact depend on some obscure relation between an amorphous set of marbles and the number "4"?

I agree they exist in nature, within the objects that exhibit them. I have a problem with assuming they have independent existence, because that raises more unanswered questions.