Yes, this is exactly what I'm saying. I'm a bit uneasy about your use of "intuition" there though.It's as if you're saying non-euclidian a posteriori discoveries are true and a priori euclidian intuitions are false. — Hanover
I don't understand the question?How do you know which is true when truth is noumenal? — Hanover
A posteriori experience, and physical experiments are also phenomenal. They investigate the phenomenon, by no means do they investigate the noumenon. A posteriori non-euclidean statements hold with regards to the phenomenon. According to Kant's definitions at least.All we're talking about is phenomenal, and it's perfectly logical to say that euclidian intuitions are necessary for phenomenal experience even if a posteriori knowledge might be discovered that is incomprehensible on an intuitive level. — Hanover
Sure, but we build by building on what exists before, trying to make it more general, and extend it. Like the factorial situation. We don't have negative factorials, so we try to find a way to have negative factorials that is in agreement with what we already have.It need only be internally coherent and can be built up deductively using the rules we put into it. It could be many years later that a practical utility is found for a system that was initially believed to be only a mathematical curiosity. — Perplexed
On what grounds do we judge a geometrical proposition to be a synthetic a priori? — Agustino
(1) Why is it sensible that we could be wrong about the form of the intuition?
(2) Does the form of intuition belong to our subjectivity? If so, is it possible to be wrong about our own subjectivity?
(3) Can we know whether a geometric statement really is a synthetic a priori with certainty? And if so, how?
I don't follow how "we are able to have synthetic a priori knowledge about space due to our knowledge of geometry". Our synthetic a priori knowledge of space is what we codify through geometry.
I also don't follow what you mean by "space isn't intuitively obvious to us". For example, it seems impossible to imagine 4D space. So is the three-dimensionality of space not something intuitively obvious to us? Could we be wrong about that too? And what would that even mean?
If I follow you correctly, your point is the traditional Kantian one that the phenomenal world is organised through the a priori forms of space and time and the categories of the understanding - so in this specific case, space doesn't exist "out there", it is just how we represent the phenomenal world to ourselves. In other words, space continues to be transcendentally ideal per your view?
So if we don't have synthetic a priori knowledge of the form of intuition there are two main questions:
(1) Since the form of intuition is subjective, why don't we have such knowledge? How does acting in the world (empiricism, scientific experiments, etc.) help us gain that knowledge? Aren't we ultimately gaining knowledge about ourselves then?
(2) How do we even know that synthetic a priori knowledge even exists if we do not know when we have it? How can we know if a piece of synthetic knowledge is a priori (Riemmann) or a posteriori (Euclidean)?
I still don't have a lot of time, so since you think this might be the salient point of our disagreement, we probably should focus on defining our terms and thereby hopefully gaining enough confluence to progress the discussion.
"Thoughts without content are empty, intuitions without concepts are blind." This well-worn quotation form Kant I have always taken to be suitably paraphrased as " Conceptions without perceptual content are empty, perceptions without conceptual content are blind". — Janus
But if physical effects external to ourselves can be shown to influence the geometry of space, is this not fatal to the assumption that space is an a priori form of our intuition? — Perplexed
Thanks, I had a quick look and will read in more depth soon.
This doesn't make sense. Either the geometry is a synthetic a priori, or it's not. It cannot merely "count" as a synthetic a priori at one point, and not at another. If it is a priori, then it is always a priori. We can, on the other hand, be mistaken about which geometry is the a priori geometry. And if we can be mistaken, I have to ask that you specify how we can know if we are mistaken about it. And the further question, how can we know that (the geometry we have) it is a priori? Because knowing that, would seem to require infinite time, since a particular geometry (like the Euclidean) can always prove in the future not to have been complete.Originally my thought was that any geometry upon which natural phenomena are predicted would count as the synthetic a priori geometry, whereas previous geometries would be considered approximations of the a priori -- and therefore empirical, since they are no longer necessary (at least for predicting physical phenomena occuring within the forms of space and time). — Moliere
This doesn't make much sense to me. Both geometries are contradictory to each other. Two contradictory statements cannot both be true, hence they cannot both be a priori, since a priori truths are necessary, and hence always true.But now I'm wondering if it's possible that both could be considered synthetic a priori -- since we can demonstrate either geometry within the non-empirical intuition by means of either physics. It would still count as a cognition regardless of the physics we use. — Moliere
Yes."Sensible" as in why does it make sense and is reasonable? — Moliere
Propositions may be truth-apt, but if something is true in an a priori fashion, then it follows that it cannot fail to be true, regardless of what happens in the world. Like "it is raining or it is not raining".If so, then it would just be a matter of the fact that propositions are truth-apt. — Moliere
Why? Why does the fact that physics "rests" (what does that even mean?) on it guarantee it certainty?My first inclination was to say that any geometry upon which the physical sciences rests would count as the synthetic a priori geometry, and would thereby be certain. — Moliere
Can you explain how you can be wrong about your own subjectivity, and what you mean by that idea?Yes and yes. — Moliere
I doubt this. Kant does talk about apodeictic certainty innumerable times with regards to mathematics. Part of the TA project, as far as I see it, is to secure where the certainty of mathematics comes from - and for Kant, it comes from the (synthetic) a priority of its propositions.But now I'd also note that certainty isn't quite as important in Kant as other epistemologies. Certainty is obtained subjectively or objectively -- and the difference between the two is subjective certainty is where one person holds something to be true, and objective certainty is when everyone does. I had to look up certainty in A Kant Dictionary to come up with that, though. It's by Howard Caygill, and just like the above articles certainly relies upon a certain interpretation to help readers through Kant -- but unfortunately I didn't mark in my CPR where Kant talks about the conditions of certainty :D. So I found it hard to find. — Moliere
Right, I definitely agree with you here. This is undoubtedly correct from a Kantian point of view. So then, our geometrical judgements (Euclidean geometry) can be wrong. What exactly is the relationship between the intuition and the understanding that causes us to be capable of forming wrong concepts based on the former?Though space is the form of outer intuition, and is so for everyone with an intellect like ours, knowledge cannot be obtained except by the use of both our understanding and intuition. Space is an intuition, and knowledge of space only comes about by use of the understanding. — Moliere
So, if space is transcendentally ideal, then there is no noumenal space, correct?More or less, yes. — Moliere
Ok.But mathematical knowledge relies upon the form of intuition. — Moliere
So why is it that it took so long for us to discover non-Euclidean geometry? According to this development of Kant, we gain knowledge by comparing our concepts with our intuition. Do you claim that, in our intuition, we knew that non-Euclidean geometry is possible? If we did, then why did it take so long for us to compare our concepts (Euclidean geometry) with our intuition, and find out that they were different?The way that acting in the world helps us gain the knowledge is that we compare concepts to our intuitions, and the form of intuition is the basis for geometry. — Moliere
This doesn't make sense. Either the geometry is a synthetic a priori, or it's not. It cannot merely "count" as a synthetic a priori at one point, and not at another. If it is a priori, then it is always a priori. — Agustino
We can, on the other hand, be mistaken about which geometry is the a priori geometry. And if we can be mistaken, I have to ask that you specify how we can know if we are mistaken about it. And the further question, how can we know that (the geometry we have) it is a priori? Because knowing that, would seem to require infinite time, since a particular geometry (like the Euclidean) can always prove in the future not to have been complete.
This doesn't make much sense to me. Both geometries are contradictory to each other. Two contradictory statements cannot both be true, hence they cannot both be a priori, since a priori truths are necessary, and hence always true. — Agustino
Propositions may be truth-apt, but if something is true in an a priori fashion, then it follows that it cannot fail to be true, regardless of what happens in the world. Like "it is raining or it is not raining". — Agustino
Why? Why does the fact that physics "rests" (what does that even mean?) on it guarantee it certainty? — Agustino
Can you explain how you can be wrong about your own subjectivity, and what you mean by that idea? — Agustino
I doubt this. Kant does talk about apodeictic certainty innumerable times with regards to mathematics. Part of the TA project, as far as I see it, is to secure where the certainty of mathematics comes from - and for Kant, it comes from the (synthetic) a priority of its propositions. — Agustino
What exactly is the relationship between the intuition and the understanding that causes us to be capable of forming wrong concepts based on the former? — Agustino
So, if space is transcendentally ideal, then there is no noumenal space, correct? — Agustino
So why is it that it took so long for us to discover non-Euclidean geometry? — Agustino
According to this development of Kant, we gain knowledge by comparing our concepts with our intuition. Do you claim that, in our intuition, we knew that non-Euclidean geometry is possible? If we did, then why did it take so long for us to compare our concepts (Euclidean geometry) with our intuition, and find out that they were different?
Sure. Analytic statements are a priori because we don't need to appeal to experience to know that they are true. If geometric statements are also a priori, then we don't need to appeal to experience to know that they are true, correct? And if we don't need to appeal to experience to know that they are true, then experience cannot disconfirm them. But experience is able to disconfirm Euclid's 5th postulate. Thus it cannot be a priori, and yet we have mistaken it for a priori. How is it possible to know if the other geometric postulates we have aren't also mistaken to be a priori, when in truth, they really aren't? And if we can't know that they are a priori, then on what basis can we claim that space is a form of our intuition?I think you're conflating a priori with analyticity here. The principle of non-contradiction is the hallmark criteria, for Kant, of analyticity. a priori just means without experience. All analytic statements are a priori, but not all a priori statements are analytic (according to Kant). — Moliere
Thus it would be a priori according to Kant. It is sufficient for a proposition to be either necessary or universal to be a priori.it would still be necessary, but not universal. — Moliere
Although I disagree it would be necessary. Again - Euclid's 5th postulate contradicts non-Euclidean geometries by not allowing cases that non-Euclidean geometry does allow. Therefore, it cannot be necessary. So why was it that we thought it necessary in the first place? How is such a mistake at all possible (to use Kant's transcendental language :P )?it would still be necessary, — Moliere
Another one. Apparently, geometrical principles are united with the consciousness of their necessity - I don't see how that is the case with Euclid's 5th postulate. If we had the consciousness of its necessity, then we couldn't be wrong, could we? That consciousness cannot just vanish can it? So my prior question remains significant - how is it even possible to be mistaken about our a priori cognition as it relates to our pure intuition? This cognition is necessary, if it is necessary, then we cannot be mistaken about it - that seems to follow, necessarily, if I may say so.Geometry is a science which determines the properties of space synthetically, and yet a priori.
What, then, must be our representation of space, in order that such a cognition of it may be possible?
It must be originally intuition, for from a mere conception, no propositions can be deduced which go out beyond the conception, and yet this happens in geometry. But this intuition must be found in the mind a priori, that is, before any perception of objects, and consequently must be pure, not empirical,
intuition. For geometrical principles are always apodeictic, that is, united with the consciousness of their necessity, as: "Space has only three dimensions" — Kant
So then this is just about categorising statements, not about how things really are?Yes, true. I just mean how we categorize something, not what it is. — Moliere
The correct answer would be due to universality and/or necessity according to Kant I think.Well, it's a priori because it does not rely upon particular experience -- it is non-empirical. Space itself is classified as non-empirical. We don't come to know it through inference. Space, like time, is unique in this way: that it is both part of our intuition, and that it is non-empirical. — Moliere
I've asked this before, but for completeness sake, I'll ask it again to the above: how can we be wrong about judgements which are universal and necessary?What makes it knowledge is that we then compare our propositions generated in the understanding to the form of intuition. And since it is knowledge of the form of intuition it is also universal and necessary. — Moliere
Euclid's 5th postulate precludes forms of geometry that are actually possible in non-Euclidean geometry. Thus the two must be contradictory. If one is true, then the other cannot be true, except, maybe, in a limited situation.What is contradictory in them? Perhaps we are wrong in thinking that. — Moliere
If it must be true in the consciousness that holds it as true, and if it is true in virtue of appeal to the form of intuition, then it cannot ever cease to be true.Apodeictic certainty just means that a proposition must be true to the consciousness who holds it to be true. — Moliere
Ehmmm X-) - I don't think he was.To use your question later on -- why was Aristotle wrong about the categories? — Moliere
Here you forget that we cannot appeal to experience at all to justify a priori knowledge, and hence neither can we appeal to experience to disconfirm it.It seems to me that we can be wrong about all manner of things, though. And in this case, with geometry, if the two geometries appear very similar within the world as we are presently living in it then there simply wouldn't be a reason to think there is another one. But then we lived in a different way and someone had some ideas and it turned out to be that we were wrong in some of our predictions. — Moliere
Yes.Why exactly do you think we can't be wrong? Simply because the knowledge is universal and necessary? — Moliere
Agreed for the sake of this discussion. (I take the most coherent version of Kantianism to be the one outlined by Schopenhauer, so I actually disagree here).Of the noumenal world nothing is known, period. So the proposition "The noumenal world is lacking space", while truth-apt, cannot be judged. There is no basis upon which such a judgment can rest. The noumenal world may have space, it may not. We simply do not know nor can we judge in either direction. To believe something along those lines would be to be doing metaphysics, which our understanding is incapable of turning into a science. — Moliere
No, not really. I'm fully aware that we don't have knowledge of our own subjectivity in many regards (Freud's unconscious, etc.). However, I wanted you to explain how this works according to Kant.It seems to me, paired with your balking about being wrong about subjectivity, that you're harboring some Cartesian sympathies for knowledge of the self.
That's fine and all, but if we're talking about Kantian philosophy then the self is not so central in his philosophy. Subjectivity is. But knowledge of the self is not given priority. It is not more certain. In fact, the most certain knowledge in Kantian philosophy is of mathematics and physics, and not psychology :D. (Kant didn't even think chemistry was a science proper.) — Moliere
Kant's notion of a noumenon, at any rate, is confused. He talks of the noumenon causing the phenomenon, which is nonsense, since causality is a category of the understanding, and hence can only apply to the phenomenon. — Agustino
This sounds an interesting avenue to explore. As I recall our discussion of a few days ago, you are conscious of its necessity, and I am conscious of its non-necessity (importantly, that is not the same thing as being conscious of the necessity of its negation!).If we had the consciousness of its necessity, then we couldn't be wrong, could we? — Agustino
I'd caution against thinking of intuition as perception. Intuition is just one half of the mind which acts entirely differently from the other half of the mind -- the understanding.
Or, another way of putting this would be to say there's the understanding and sensibility, and intuition is a part of sensibility. I'm not sure which way to put it, myself.
In either rendition though perception is a psychological phenomena -- it deals with how a particular mind, and how many similarly wired particular minds, come(s) to recognize some phenomena as that phenomena.
But in the case of Kant we're dealing more with how all particular minds arranged Kant-wise (just to be cheeky -- I forget the exact term Kant uses, but our minds are contrasted with an intellectual intuition to give an idea of what sorts of mind he means {EDIT: and, just in case, an intellectual intuition is something like the mind of god, where thinking something creates reality -- not trying to talk down, just trying to make sure I cover my bases}) come to have knowledge about the world -- and in particular synthetic a priori knowledge. It's not about how we see a particular phenomena as that phenomena, but rather how it is possible for us to know some subject is attached to a predicate synthetically and without having to rely upon particular experience.
Linguistically -- perception's "link" is the word "as", and knowledge's "link" is the copula. — Moliere
So we appeal to the intuition and find out that Euclid's 5th postulate is true. And yet, you claim that we were merely wrong about our intuition... but it doesn't make sense to be wrong about our intuition when it is our intuition to which we appeal to determine whether Euclid's 5th is true or not, isn't it? — Agustino
Again - Euclid's 5th postulate contradicts non-Euclidean geometries by not allowing cases that non-Euclidean geometry does allow. Therefore, it cannot be necessary. — Agustino
My original point was that space and time are the pure forms of intuition and are not themselves empirical, for Kant (and I think I remember reading you agreeing with this somewhere in this thread); whereas spacetime is an empirical model that predicts what will be observed. On that basis I can't see how, in the context of Kant's philosophy, the two can be thought to be the same. — Janus
Does it mean that it is a posteriori if we have to "probe the shape of space" for it? — Agustino
Although I disagree it would be necessary. Again - Euclid's 5th postulate contradicts non-Euclidean geometries by not allowing cases that non-Euclidean geometry does allow. — Agustino
Therefore, it cannot be necessary. So why was it that we thought it necessary in the first place? How is such a mistake at all possible (to use Kant's transcendental language :P )?
Another one. Apparently, geometrical principles are united with the consciousness of their necessity - I don't see how that is the case with Euclid's 5th postulate. If we had the consciousness of its necessity, then we couldn't be wrong, could we? — Agustino
That consciousness cannot just vanish can it?
So my prior question remains significant - how is it even possible to be mistaken about our a priori cognition as it relates to our pure intuition? This cognition is necessary, if it is necessary, then we cannot be mistaken about it - that seems to follow, necessarily, if I may say so.
So then this is just about categorising statements, not about how things really are? — Agustino
I've asked this before, but for completeness sake, I'll ask it again to the above: how can we be wrong about judgements which are universal and necessary? — Agustino
Euclid's 5th postulate precludes forms of geometry that are actually possible in non-Euclidean geometry. Thus the two must be contradictory. If one is true, then the other cannot be true, except, maybe, in a limited situation. — Agustino
However, I wanted you to explain how this works according to Kant. — Agustino
Where? — Moliere
What do you make of non-Euclidean geometry? Let's leave spacetime out of it entirely, and just focus on the mathematics. Do you think only one kind of geometry could hold universally for space? — Moliere
Not only Kant, but I also, find suggestion (2) to be ridiculous. Reality has to be a certain way, and it is the job of philosophy to investigate it. It would be contradictory if geometry was different for each different person, since then we would be unable to share a world and communicate at all.2. its necessity is individual-dependent, so that we could both be correct and it is necessary for you but not for me.
I find the second one palatable but my secondary sources tell me that Kant was adamant that his a priori intuitions like the TA were not subjective. If that's correct then I think he'd roll in his grave at suggestion 2. — andrewk
Ok.If causality is an a priori form of the understanding, then we cannot but conceive of the noumenon causing the phenomenon, even though such a relation may not obtain in reality. — Thorongil
I don't think so. At minimum, I think you should read my exchange with Moliere, starting from here:Haven't looked at the rest of the thread, as it's beating a very dead horse. I've said my piece about space. — Thorongil
That doesn't follow. The difference between a perfectly flat space and one that is curved very, very slightly would make no difference at all to the ability to communicate.It would be contradictory if geometry was different for each different person, since then we would be unable to share a world and communicate at all. — Agustino
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