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... — Wayfarer

Science isn't committed to the reality of alethic modalities (necessity, possibility, probability) in the devout epistemological sense you seem to imply here, for they are merely tools of logic and language - the modalities do not express propositional content unless they are falsifiable, which generally isn't the case.

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. — Wayfarer

IMO, that is a merely an instance of an inductive argument happening to succeed. A purpose of any theory is to predict the future by appealing to induction -- but there is no evidence of inductive arguments being more right than wrong on average. Indeed, even mathematics expresses that it cannot be unreasonably effective, aka Wolpert's No Free Lunch Theorems of Statistical Learning Theory.

Humans have a very selective memory when it comes to remembering successes as opposed to failures. Untill the conjecture is tested under scrutiny, it can be dismissed.

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. — Wayfarer

Let's slow down on this one. Kant doesn't speak about "content" in the [Prolegomena] (where the 7+5 example is discussed). He says that the concept of "12" is not the same as the concept of "7+5". According to him, we need an "intuition" ("perception" would probably be our way of saying it today) of the physical in order to discover "12". (He suggests that our five fingers, and then seven fingers, would do the trick.) "Hence our concept is really amplified by the proposition 7+5 = 12, and we add to the first concept a second concept not thought in it." What Kant regards as analytic here is the judgment that 7 and 5 must add up to some number -- but this does not tell us what particular number.

The place where this can be challenged, I think, is the reliance on intuition. If this is truly the case, don't we have to question whether the judgment is indeed a priori? Kant addresses this in Sec. 281: "How is it possible to intuit anything a priori?" I don't want to take this any farther, except to say that the case for math as a series of synthetic a priori judgments, even on Kant's own terms, is far from closed.

Need the year of publication, for whatever text you’re saying has Groundwork in its title. The Groundwork I’m familiar with is a treatise on moral philosophy, having nothing to do with mathematical judgements, and 7 + 5 is not discussed as far as I could determine, but that a categorical "ought" implies a synthetic a priori proposition, is.

And sec 281 doesn’t Google.

Thanks.

Oh good lord, sorry, I meant the Prolegomena. :grimace:

Quite, the Prolegomena. That’s where I read it.

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.
— Wayfarer

IMO, that is a merely an instance of an inductive argument happening to succeed. A purpose of any theory is to predict the future by appealing to induction -- but there is no evidence of inductive arguments being more right than wrong on average. Indeed, even mathematics expresses that it cannot be unreasonably effective, aka Wolpert's No Free Lunch Theorems of Statistical Learning Theory. — sime

But inductive arguments are a posteriori by definition. Dirac’s prediction of antimatter was not an inductive guess but a deductive consequence of the mathematics of the electron. It’s a perfect case of the synthetic a priori: by synthesising the elements of the theory, he saw that negative counterparts must exist — long before observation confirmed it.

It’s a perfect case of the synthetic a priori . . . — Wayfarer

The debate about this often centers on how "prior" the a priori is supposed to be. What is the ideal situation in which an a priori judgment is imagined to take place? Prior to what, exactly, can we know that 7+5=12? Prior to what can we know that antimatter exists? Prior to (or independent of) observations, perhaps, but prior to any experience of the world whatsoever? Priori to knowing how to count? "A priori" and "a posteriori" have conventional interpretations, and a better Kantian scholar than I could perhaps tell us precisely what Kant envisaged, but the division doesn't feel like a "natural kind" to me.

What is the ideal situation in which an a priori judgment is imagined to take place? Prior to what, exactly, can we know that 7+5=12? — J

A priori means “prior to experience.” If you tell me you have seven beers in the fridge and I bring to another five to give you, I can know you have twelve beers without opening the fridge door. That’s a trivial example, but it illustrates the point: the truth of 7+5=12 doesn’t depend on checking the fridge.

This is where Kant’s answer to Hume comes in. Hume divided truths into those that are true by definition (analytic) and those known only from experience (synthetic a posteriori). Kant showed there’s a third, crucial category: synthetic a priori judgments, which extend knowledge while still being necessary. That’s how mathematics and mathematical physics are possible — Dirac’s deduction of antimatter being a dramatic case in point.

Kant didn’t believe in Plato’s innate ideas, but he did argue that the forms of intuition (space and time) and the categories of the understanding are innate conditions of human reason. Later, Quine challenged these distinctions in Two Dogmas of Empiricism, but that’s a separate debate.

Cool. I figured, but confirmation is always best.

A priori means “prior to experience.” If you tell me you have seven beers in the fridge and I bring to another five to give you, I can know you have twelve beers without opening the fridge door. That’s a trivial example, but it illustrates the point: the truth of 7+5=12 doesn’t depend on checking the fridge. — Wayfarer

No, it's true by definition that if he has seven plus five beers he has twelve beers, and to know that seven plus five equals twelve it's either through having remembered the addition tables, counting mentally or on your fingers or whatever. In no way is your knowing prior to experience, other than in its analytic aspect, and even analytic definitions are learned.

A priori means “prior to experience.” If you tell me you have seven beers in the fridge and I bring to another five to give you, I can know you have twelve beers without opening the fridge door. That’s a trivial example, but it illustrates the point: the truth of 7+5=12 doesn’t depend on checking the fridge. — Wayfarer

Right, that's the standard interpretation, but think about it: Prior to how much experience? Can I know about the 12 beers if I don't know what beer is? Can I know it without knowing about counting? Can I know what 7 or 5 or 12 anythings are without lived experience? So where do we imagine the "a priori judger" standing, so to speak, when they make their judgments? (BTW, you can see immediately that this is yet another place where Rodl's important questions about propositions surface.)

I think your knowledge of Kant is deeper than mine, so please say if you don't agree with my interpretation of these passages in the Prolegomena.

You could say the role of the synthetic a priori in science is precisely to bridge the gap between logical necessity and empirical causation. Logic alone gives tautologies, while experience alone gives contingent observations. Kant’s point is that principles like “every change in velocity has a cause” are synthetic a priori: they enable prediction, but also hold necessarily for all possible experience. That’s what allows physics to be both law-governed and universally valid.

Of course, it’s said that much of this comes to grief in quantum physics (but that’s a separate topic and even there the debate turns on how to interpret the a priori structures of knowledge, not on whether they exist at all.)

Kant’s point is that principles like “every change in velocity has a cause” are synthetic a priori: they enable prediction, but also hold necessarily for all possible experience. That’s what allows physics to be both law-governed and universally valid. — Wayfarer

We don't know if physics is law-governed and universally valid. It's universal validity is merely an assumption and the laws may have evolved as habit (pace Peirce). Same with causation―we can only explain changes in terms of causation, and it doesn't necessarily follow that all changes are caused. Imputing causation is an inveterate habit of thought―even some animals do it.

Careful what you ask for. I don’t have a problem with the Prolegomena because I don’t consider it the relevant text for the current discussion.

300 years after the fact, all there is, is opinion. My opinion is, most everybody, in concentrating on this or that, overlooks transcendental philosophy as a whole.

I can explain til I’m blue inna face, but there remains a serious problem: there’s no need for mathematical judgements or their synthetic a priori classification, when I’ve known all about them since I was knee-high to a grasshopper, thanks to my 1st grade teacher. It’s extremely difficult to comprehend the reason for them when rote instruction has removed the consciousness of their applicability. That being said….

1)….the human being has not evolved out of the condition he was in 300 years ago: he still perceives and he still thinks, from which it follows the tenets of transcendental philosophy still hold;
2)….from 1), regardless of current opinion concerning the system prescribed by transcendental philosophy with respect to human cognition, each part of the system remains fully dependent on all the others;
3)…..from 2, mathematics being synthetic judgements a priori is merely an example of what they are, where they reside in the system, and what they do for the system, but rely on something else for sufficient proof of their possibility.

It makes no difference if synthetic judgements a priori are accepted or not; within the theory they are required, which just means to reject that part is to reject the whole. Which is fine, things do move on, after all.
————-

In the case of the conception of a priori itself, Kant did not mean it with respect to time as such, but with respect to placement in the system as a whole. The systemic procedure in a nutshell, for knowledge of things, is perception through to experience. Kant allows a priori to be pure or impure, but stipulates….probably for the sake of his editors…when he writes the word, he means the pure version, always, without exception. The pure/impure signifies whether or not the subject under consideration is empirical, subject being the propositional form thereof, indicating what he’s talking about at the time: impure means, e.g., the subject conception is represented by a real thing, while pure, on the other hand, means, “….not such as is independent of this or that kind of experience, but such as is absolutely so of all experience….”.

Now, given the only two possible ways for the human cognitive system to work, either from perception of things, which is all the empirical side, or, from mere thinking of things, which is all the rational side, it follows that “independent from experience” makes explicit the term is restricted in its use to the rational side alone.

So, a priori means within, or restricted to, any internal systemic function in which there is nothing having to do with empirical predication. To then say a priori, as it relates to time is before experience, is not quite right, insofar as pure thought absent empirical conditions, is already that for which there never will be any experience anyway, so before experience or before the time of experience, in such case, is superfluous.

It is the entire point of transcendental philosophy, is to combat Hume’s reluctance to pursue pure rational thought as the ground of knowledge. In order to be successful, Kant had to demonstrate those conditions under which ALL knowledge stems, and that from the very condition Hume’s resolution was to “….consign it to the flames…”.
————-

To answer your question when do we know 7 + 5 = 12, we know it when we represent it to ourselves by empirical example. Yet beforehand, we know a priori there is nothing contained in the conception “7”, or in the conception “5”, from which we are given the conception “12”.

Because that is known with apodeictic certainty….
(when all you have is boards over there and nails over there)
….yet the are mathematical statements we know with equal certainty from experience….
(yet there’s houses everywhere you look)
….it remains to be undetermined how to get from one to the other….UNLESS….the cognitive part of the system as a whole, and in particular the part which reasons, does something with the two given conceptions…
(hammer the nails into the boards is the way to build a house; synthesis the “7” and the “5” in understanding is the way to judge the relation of two given conceptions having nothing to do with each other)

Full stop. You hammer all day long, you still don’t have a house; you synthesis the conceptions, you still don’t have the conception “12”. Now we see synthetic judgements a priori are only representations of a very specific cognitive function, a synthesis done without anything whatsoever to do with experience, and of which we are not the least conscious. It is all an act of reason, which is that systemic faculty not so much involved in knowledge itself, but provides the principles by which it becomes possible. At this point we don’t care about the 12, just as we don’t care the house isn’t done yet. All we want is proof for a way to get the house built, and proof of a way to get to whatever the relation of 7 and 5 gives us.

We think nothing of combining 7 and 5. We don’t think anything of the combining of them. But we stop dead in our cognitive tracks, when the very same synthesis is just as necessary but for which immediate mental manipulation is impossible. The rote mechanism of mere instruction doesn’t work for a vast majority of us, when the synthesis is of, like, numbers containing many digits, or of a different form of synthesis altogether, i.e, calculus. The principle is the same, though, for all of them.

And all that, is only half the story….

In the case of the conception of a priori itself, Kant did not mean it with respect to time as such, but with respect to placement in the system as a whole. — Mww

Good.

To then say a priori, as it relates to time, is before experience, is not quite right, — Mww

Yes, that's what I was suggesting.

Now we see synthetic judgements a priori are only representations of a very specific cognitive function, a synthesis done without anything whatsoever to do with experience, and of which we are not the least conscious. — Mww

But then why does Kant say:

We must go beyond these concepts by calling to our aid some intuition which corresponds to one of the concepts -- that is, either our five fingers or five points . . . -- and we must add successively the units of the five given in the intuition to the concept of seven. — Prolegomena 268

But we stop dead in our cognitive tracks, when the very same synthesis is just as necessary but for which immediate mental manipulation is impossible. — Mww

Kant notes this in the same section: "larger numbers . . . however closely we analyze our concepts without calling intuition to our aid, we can never find the sum by such mere dissection."

the cognitive part of the system as a whole, and in particular the part which reasons, does something with the two given conceptions… — Mww

I still read this "something" as requiring intuition. Do you not see it that way?

“…find the sum…” is the something reason directs understanding to do, in the synthesis of given conceptions; what the sum is requires intuition, because only from sensibility can an object representing what reason requires. Herein is counting, for the easy math, the development of formulas and equations for the not-easy, thereby obtaining empirical knowledge of that which originated in thought alone.

This is Kant's “…mathematical cognition….”, in which is what he calls the “….construction of conceptions….”, as opposed to philosophical cognitions, in which is the “… spontaneity in the production of conceptions…”, herein whatever intuition represents the synthesis of the two given constructed conceptions, which will eventually be constructed that elusive “12”.
(Constructed conceptions arise immediately as schemata of the categories, not mediately as representations belonging to mere thought)

There are no numbers naturally in Nature; all of them are put there by us, as objects of sensibility, hence numbers, when employed by understanding in mathematical cognitions, originate as intuitions a priori. How did that happen, you ask. Well…cuz reason switched gears on us, of course, by insinuating presupposing categorical schemata as a real object for what is usually mere phenomena given from a naturally occurring object.

The transcendental aesthetic prescribed the method required for the beginning of empirical knowledge. In keeping with that, if one were to use his fingers for counting, how did he get 1, 2, 3…and not finger, finger, finger….

Same with lining up rocks in aggregate with respect to the quantity: when you count rocks you don’t think, rock, rock, rock….

Stick an object up in front of your face, you experience all that from which is intuited in that object. Stick that object of experience now called a hand in front of your face, but this time, while still perceiving fingers as incorporated in the object called hand, you think them as numbers. Not only numbers, but numbers in succession, in exact relation to alternate fingers. Coolest part is…..you’re not the least confused by contradicting your own antecedent experience (finger) by determining something which should be impossible from it (number).

This is the construction of conceptions, and from them are the empirical intuitions a priori, and why this whole shebang must come from reason herself, a transcendental faculty, for if this arrangement originated in any cognitive, or discursive, faculty, we would be oh-so-confused by conflicting experiences, and in fact, most likely couldn’t even function in such manner at all.
(Keeping in mind, reason has nothing to do with knowledge as such but only provides the rules and principles, through “transcendental ideas”, for knowing successfully, that exclusively the purview of the logical faculties of cognition, re: understanding)

The differences in the text is so subtle.
….In the Aesthetic, we have intuitions which are given as “the matter of objects”;
….In judgement of mathematical cognitions, we have “….exhibition à priori of the intuition which corresponds to the conception…” for which the matter would be irrelevant;
….In judgement of philosophical cognition we have conceptions which conform to the intuition insofar as “…the intuition must be given before your cognition, and not by means of it.…”.

Now we see what ALL mathematical judgements are synthetic and ALL are a priori. Pretty simple really: we observe relations in Nature, the a posteriori, but represent them to ourselves with that which isn’t observed in Nature at all, the a priori.

Added bonus: because the intuition of number is exhibited a priori in correspondence to the conception from which it is given, that intuition can contain nothing more than that which is contained in the conception. Hence arises the apodeictic certainty of mathematical judgements.

Dunno if any of this helps or not, and it is all opinion, so…..

Dunno if any of this helps or not, — Mww

I appreciate it a lot, thanks.

The differences in the text is so subtle.
….In the Aesthetic, we have intuitions which are given as “the matter of objects”;
….In judgement of mathematical cognitions, we have “….exhibition à priori of the intuition which corresponds to the conception…” for which the matter would be irrelevant;
….In judgement of philosophical cognition we have conceptions which conform to the intuition insofar as “…the intuition must be given before your cognition, and not by means of it.…”. — Mww

Clearly these are differences, as you say. I'm focused still on the discussion in the Prolegomena, where Kant says:

In one way only can my intuition anticipate the actuality of the object, and be a cognition a priori, namely, if my intuition contains nothing but the form of sensibility, antedating in my mind all the actual impressions through which I am affected by objects. [Kant's italics] — Prolegomena 282

How do you interpret this? How might it apply to 7+5 and the use of fingers?

What is an intuition? Empirically, it is the synthesis of the matter of a given appearance, with a form, the representation of which, is phenomenon.

“…. That which in the phenomenon corresponds to the sensation, I term its matter; but that which effects that the content of the phenomenon can be arranged under certain relations, I call its form. (…) It is, then, the matter of all phenomena that is given to us à posteriori; the form must lie ready à priori for them in the mind, and consequently can be regarded separately from all sensation….” (A20/B34)

This is what I was talking about above, where the matter of an object is irrelevant, because there isn’t an object, in mathematical judgements a priori. But it is a judgement, which requires a relation of conceptions.

Now, the matter of an empirical intuition is conditioned by space, but the form is conditioned by time, hence the two pure intuitions one hears so much about. Absent the need for the condition of space for lack of an appearance, but retaining the condition of time, we “…exhibit an empirical intuition a priori…” to ourselves, in order to cognize a relation of synthetical conceptions, which are represented in the judgement.

What is cognition? It is presentation to the subject the consciousness of a judgement, from which follows that mathematical judgements a priori are not in themselves yet cognitions. The missing piece is the intuition, which in the case of mathematical judgements in order to be cognitions, must get their intuition a priori as form alone.

Incidentally enough, there is a definitive conjunction here: the categories are all relations of time, and number is a schemata of the category of “quantity”, so it naturally follows that form is a representation of time. Not represented in time, but of time.

Remember, we were discussing a certain kind of judgement. By involving intuition we’ve moved on from mere synthesis of unrelated conceptions. While we are certainly authorized to think all connected to something like 7 + 5 = 12, thinking does not present any objective validity, and for which is required the invention and use of real objects.

The drawing of numbers or figures, associating them in accordance with operative demands, is the method of proof. When we draw a figure or number, that becomes the appearance, and that, conditioned by space, combined with time already established as present in the mind, and we have an actual phenomenon. Now the synthesis in intuition is space and time, the synthesis in understanding is phenomenon and conception, and experience of the determined mathematical cognition is given.

Oh what a tangled web we weave….right? While metaphysics cannot be a science, this is how it can be treated as if it were.

“…. That which in the phenomenon corresponds to the sensation, I term its matter; but that which effects that the content of the phenomenon can be arranged under certain relations, I call its form. (…) It is, then, the matter of all phenomena that is given to us à posteriori; the form must lie ready à priori for them in the mind, and consequently can be regarded separately from all sensation….” (A20/B34) — Mww

Here again the echo of Aristotle's form-matter dualism. He transposes Aristotle’s schema from the level of substances to the level of cognition. Instead of matter/form being ontological constituents of objects, they are now epistemic constituents of experience:

Sensation provides the raw material (matter).

Space and time provide the form that makes it intelligible.

The two together yield phenomena — objects for us.

When we draw a figure or number, that becomes the appearance, and that, conditioned by space, combined with time already established as present in the mind, and we have an actual phenomenon. — Mww

That is very helpful - it helps me understand much better Kant's connection of time with number and space with geometry. :100:

That is very helpful - it helps me understand much better Kant's connection of time with number and space with geometry. :100: — Wayfarer

Simply put, Kant associates space, as the outer intuition, with external objects, and time, as the inner intuition, with internal objects. So, I would understand Mww's example like this. Time is already required, as the internal intuition, prior to writing a number, then when it is written, it is apprehended through the external intuition as having a spatial presence. This specific example is consistent with Plato's cave allegory, where the external object (sensible in Plato's terminology) is posterior to, and a reflection of the internal, which is the higher degree of reality.

I don't understand how

Time is already required — Metaphysician Undercover

in terms of Kant's language. He made a claim of how little we can know about it since it is how we experience what we do.

Perhaps Kant is not accepting the speculation of your model.

I would understand Mww's example like this. Time is already required, as the internal intuition, prior to writing a number, then when it is written, it is apprehended through the external intuition as having a spatial presence — Metaphysician Undercover

The way I took it is that addition of numbers is sequential - first, 7, then 'add 5' giving the result '12'. It is the fact of the sequential order of mental operations that assumes time. The spatial representation (writing the numbers down) is only a useful aid; the grounding of number itself is in time, not space.

in terms of Kant's language. He made a claim of how little we can know about it since it is how we experience what we do.

Perhaps Kant is not accepting the speculation of your model. — Paine

I think that if you read Kant's Critique of Pure Reason, you'll find that he characterizes space as the outer intuition (required for the appearance of outward sensations I assume), and time as the inner intuition (required for the appearance of inward sensations I assume).

Claims about "how little we can know about it" do not equate with 'we can know nothing'. And, Kant did make this distinction between inner and outer. I'd look it up for you, but you could Google it if you are interested. I believe the important point which Kant makes with this distinction is that even though both space and time are a priori, the intuition of time is in a sense prior to space. Time is fundamental to the being itself, as essential to internal processes, required for all types of experience, whereas space is necessary for a specific type of experience, the one we understand as the separation between myself and what is other than me.

The way I took it is that addition of numbers is sequential - first, 7, then 'add 5' giving the result '12'. It is the fact of the sequential order of mental operations that assumes time. The spatial representation (writing the numbers down) is only a useful aid; the grounding of number itself is in time, not space. — Wayfarer

Yes, I believe that's pretty much what Kant intended with the distinction between the inner and outer a priori intuitions. Internal ideas, as pure intellectual objects, are grounded in temporal order, therefore not requiring spatial features for understanding. The appearance of phenomena, on the other hand, requires that external, spatial aspect as well as the temporal aspect.

the echo of Aristotle's form-matter dualism. — Wayfarer.

250 years ago, Aristotelian logic ruled academia, from 1770 Kant held the chair of metaphysics and logic at U. of K., so could hardly dispense with it altogether. Thanks to Leibniz in the one hand and Newton on the other, though, Kant did, as you say, move the standardized matter/form duality from an ontological to an epistemological condition. He took it away from the object and gave it to the subject.

And his treatment of time…fascinating. At the expense of real things, no less, that which could actually kill us, relinquishes its importance to something having not the least effect on us at all.

Ballsy move, ya gotta admit, considering the relatively recent advent of the hard sciences, and it took 35 years or so (WWR, 1818) for a decent comprehension of what just happened.