## What would Kant have made of non-Euclidan geomety?

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• 11.3k
No. Just as placing dots on a paper to demonstrate counting or addition, or drawing a triangle to demonstrate a triangle do not make mathematical knowledge a posteriori, so too with light. In terms of the intuition it is no different from using a ruler.
Sure, so tell me how you "place dots on a paper" (or the equivalent) to demonstrate non-Euclidean geometry (specifically intrinsic curvature). What is the "intuition" relevant to non-Euclidean geometry?

Also, how is using the empirical behaviour of light (that it cannot go faster than a certain speed) the same as placing dots on a paper?

So? How does that have anything to do with the modality of the copula, in Kant's logic?
Everything. A necessary judgement is one which must be thought to be true. Euclidean and non-Euclidean geometries cannot both be thought to be true since one allows possibilities that the other denies. Therefore they cannot both be necessary.

Your saying "necessary" means not mistaken -- or, perhaps more strongly, not even possible to be mistaken. I am saying "necessary" means to give assent to by everyone, and hence be objective.
No, I don't agree "necessary" means assent by everyone to Kant. Please quote Kant where he says something like this. As far as I know, this is what later neo-Kantians would claim (ex, Husserl).

According to Kant, the modality of apodeictic judgements involves one necessarily having to assent to the truth of the proposition when considering it. And it is necessary because one appeals to the intuition. So now, you have to show, as I asked you before, how one appeals to the pure intuition to "construct" non-Euclidean geometries.

Yes, we could.
Yes, it can.
I asked you to tell me how that is possible though.

It's like "necessary" and "certain" mean the same thing to you -- if some proposition is necessary then it is not possible for it to be false. But truth and falsity have nothing to do with necessity, in Kant'.
I think this is incorrect. Necessary means that we must think that proposition true, we must assent to it. Do you have a quote to prove that this isn't the case in Kant?

I mean, of course these things can change in Kant's system -- especially considering that necessity, being a category, isn't even time-dependent. What happens in time can change when some proposition is necessary.
I see this the other way around. Precisely because it is not time-dependent, what happens in time cannot change the necessity of the proposition.

He doesn't really go into psychology very much. But mathematics seems to form the heart of his philosophy of science. So it would just be the fact that it's not a science, that we can be wrong, and so forth. It's a mundane answer, but I don't think there is a deep answer. Kant's dealing with the structure of the mind, a structure we all share as compared to the contrast class of an intellectual intuition. It's not really about our subjectivity as much, though Kant uses the word "subjective" in his own way that fits within the philosophy.

The self and subjectivity and all of that just aren't really there to be talked about. And psychology and anthropology are only mentioned in passing.
Sure.
• 11.3k
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.
In principle it could. Since this is at all possible, space cannot be a priori. We're talking about how reality happens to be, not how we encounter it in our limited region of space and time. So yes, the fact that our principles are contradictory means that we can't both be right with respect to reality. Unless you want to claim we inhabit two different realities, that is.

What's your preferred method? I find ouija boards OK but Automatic Writing tends to be quicker.
:lol: - well, I only meant that from my reading of him, I'm sure Kant would have found such an idea ludicrous, and pulling in a completely opposite direction to what is aimed at through philosophy. We aim to reach the truth - not opinions.
• 11.3k
between the thing-in-itself and the phenomena, in Kantian jargon. This, to me, seems to remain quite untouched.
It is touched for the Kantian if the transcendental aesthetic falls apart. For the Platonist, yes, it does remain untouched.

there is no problem if non-Euclidean geometries are purely fictitious. I think they are, for they were "discovered" by fiddling with axioms, not from empirical observation.
But yet, there is empirical observation that confirms such geometries to be the case. How is it possible for them to be purely fictitious given that this is the case? Kant's argument would indeed be unaffected if these non-Euclidean geometries were, as you say, purely fictitious. Kant's position is that geometry is synthetic, so it is possible to form a concept of non-Euclidean geometry, since there is nothing logically contradictory in such a concept. However, Kant would claim that such isn't a science anymore, since it is a purely empty concept, which does not rely on the pure intuition of space.
• 1.9k
We're talking about how reality happens to be, not how we encounter it in our limited region of space and time
I don't agree with the 'In principle it could' claim before this. But I find this quoted bit much more interesting, so I'd like to explore that instead.

My understanding of the TA is that it is not about how reality happens to be but about how humans shape the raw sensory input received into a usable form. I don't see how Kant would be likely to make any claims about how reality happens to be since, for him, reality is noumena, about which we can know nothing.

Can you elaborate about what you mean by this reference to how reality happens to be?
• 11.3k
My understanding of the TA is that it is not about how reality happens to be
Only if you take reality to be the noumenon. But if reality is the phenomenon, or the empirically real, then what you're saying here is false.

Many people do not get this very well. They imagine we have representations, and then there is this noumenon which causes the representations, which is actually very similar to the representations themselves. That's wrong as Schopenhauer illustrates. Since space and time are pure intuitions, they cannot apply to the noumenon, so the noumenon is neither spatial nor temporal. So "reality" (if by that we understand the noumenon) is neither spatial nor temporary. Physics doesn't deal with "reality". It deals only with the empirically real, with the phenomenon, which is exactly what the form of intuition of space applies to.
• 11.3k
Can you elaborate about what you mean by this reference to how reality happens to be?
So to be more clear, by reality I mean the empirically real, that which physics addresses and that which we encounter in experience, whether directly through our sense organs, or mediately, through scientific instruments. I don't mean Kant's noumenon.
• 11.3k
You should listen to the last few minutes of this for a more detailed explanation on the same matter that I was talking about above:

• 11.3k
Our expositions, consequently, teach the reality (i.e., the objective validity) of space in regard of all which can be presented to us externally as object, and at the same time also the ideality of space in regard to objects when they are considered by means of reason as things in themselves, that is, without reference to the constitution of our sensibilities.
We maintain, therefore, the empirical reality of space in regard to all possible external expereince, though we must admit its transcendental ideality; in other words, that it is nothing, so soon as we withdraw the condition upon which the possibility of all experience depends and look upon space as something that belongs to things in themselves
— Kant
• 1.6k
I should be able to eventually. My local library has access to a lot of academic journals. It'll just take some time since, like, I have to actually go there and stuff. :D
• 11.3k
Thanks.
• 3.2k
It is touched for the Kantian if the transcendental aesthetic falls apart. For the Platonist, yes, it does remain untouched.

You're missing the forest for the trees, or perhaps for what you regard as a few dead trees in the Kantian forest. Here is Schopenhauer, from which I derive the thought you quoted of me:

What Kant says, in its essentials, is the following: "Time, space, and causality are not determinations of the thing in itself, but pertain only to its phenomenon, insofar as they are nothing but our cognitive forms. Since, however, all plurality and all arising and passing away are only possible through time, space, and causality, it follows that they too attach only to the phenomenon and in no way to the thing in itself. Because, however, our cognizance is conditioned by those forms, the whole of experience is only cognizance of the phenomenon, not of the thing in itself; therefore, neither can its law be made to apply to the thing in itself. These assertions extend even to our own I, and we are cognizant of it only as phenomenon, not with respect to what it may be in itself."

But now Plato says: "The things of this world, which our senses perceive, have no true being whatsoever: they are always becoming, but never are; they have only a relative being, all of them existing only in and through their relations to one another; one can thus just as well call their entire existence a kind of non-being. They are, consequently, not even objects of any real cognition. For the latter can be only of that which has being in and for itself and always in the same manner; they are, by contrast, only the object of opinion occasioned by sensation. So long, then, as we are limited to perception of them, we are like men who sit so tightly bound in a dark cave that they could not even turn their heads and, by the light of a fire burning behind them, would see nothing but, on the wall in front of them, shadowy images of actual things made to pass between them and the fire; and even of each other, indeed of themselves, but would see only just the shadows on that wall. Wisdom for them would consist in predicting the succession of those shadows as learned from experience. What, by contrast, can alone be called truly existent, because they always are and never become nor pass away, are the real archetypes for those shadowy images: they are the eternal Ideas, the original forms for all things. No plurality pertains to them, for each is in its essence only One, being the very archetype whose copies or shadows are all named after it: individual, transitory things of a given kind. Nor does arising or passing away pertain to them, for they are truly existent, never becoming nor perishing like their constantly vanishing copies. (In these two negative determinations, however, it is necessarily contained as a presupposition that time, space, and causality have no meaning or validity with respect to them, and that they do not exist within the latter.) Of them alone is there thus any real cognizance, since an object of the latter can only be that which has being always and in every respect (and so in itself), not that which is while it again is not, depending on how one views it."

That is Plato's doctrine. It is obvious and in need of no further demonstration that the inner sense of both doctrines is entirely the same, that they both describe the visible world as a phenomenon that is in itself nothing and has a meaning and borrowed reality only through that which is expressing itself in it (for one of them the thing in itself, for the other Ideas), while to the latter, to what which is truly existent according to both doctrines, all and even the most general and most essential forms pertaining to that phenomenon are altogether foreign.
— Schopenhauer

there is empirical observation that confirms such geometries to be the case. How is it possible for them to be purely fictitious given that this is the case

I don't think so. Einstein's model of the universe, for example, makes use of certain non-Euclidean geometries, but that doesn't mean the model is accurate. Astronomers don't know for certain whether the universe is Euclidean or non-Euclidean.
• 11.3k
I don't think so. Einstein's model of the universe, for example, makes use of certain non-Euclidean geometries, but that doesn't mean the model is accurate. Astronomers don't know for certain whether the universe is Euclidean or non-Euclidean.
What do you mean it doesn't mean the model is accurate? If that model makes certain predictions (such as light bending around massive objects) and we go out there and test that, and the test confirms the predictions of the model, in what sense is the model "not accurate"?

You're missing the forest for the trees, or perhaps for what you regard as a few dead trees in the Kantian forest. Here is Schopenhauer, from which I derive the thought you quoted of me:
I don't see how this part of Schopenhauer is relevant. I claim that Kant's way of deriving the phenomenon/noumenon distinction is not valid, though Plato's is. As Schopenhauer likes to say, the right conclusion, from the wrong premises ;)

Time, space, and causality are not determinations of the thing in itself, but pertain only to its phenomenon, insofar as they are nothing but our cognitive forms.
If space was real, then transcendental idealism cannot hold. Space must be a faculty of the mind and must be imposed by the mind, in order to be able to claim that the plurality found through (amongst others) space is not, in the end, real (just like space itself).

So all the machinations of physics in both relativity and quantum mechanics - and certain aspects of quantum mechanics, such as the proposed loop quantum gravity theory which doesn't even have continuous space and has no time at all - must be accounted for by a transcendentally ideal philosophy.

Now, my opinion, if I were to try to defend Kant, is that we must insist that our pure intuition is Euclidean, and even non-Euclidean geometry we represent based on our Euclidean geometry. I asked Moliere to show me the equivalent of the "sketching figures" which is the intuition we use with regards to coming up with Euclidean geometry when it comes to non-Euclidean. I don't think he, or anyone, will be able to. Indeed, if you think how we construct non-Euclidean geometry, we do so by analogy, from within Euclidean geometry. We look at the properties of geometric shapes on the 2D surface of a sphere which curves in the 3rd dimension and infer from that, by analogy, what it would mean for a 3D surface to curve in the 4th dimension (and further dimensions from there). Now, this non-Euclidean geometry is not revealed to us by our pure intuition - we deduce it by analogy and extrapolation based on our pure intuition. So Euclidean geometry remains synthetic a priori.

What about non-Euclidean geometry? What is its mathematical status, if Euclidean geometry is synthetic a priori? Well, granted that we know that non-Euclidean geometry must always be at least locally Euclidean, and at any rate, non-Euclidean geometry always presupposes the Euclidean one in its derivation, then we know for sure that it cannot be synthetic a priori. So it must be either synthetic a posteriori, or analytic a priori. But non-Euclidean geometry isn't something that requires experience in order to be derived. Therefore it is a priori, and it must be analytic, built by analogy to Euclidean. This means that there is no grounding either in experience or in the forms of intuition for non-Euclidean geometry (except, as it were, by analogy).

This is all fine and good, but notice what happened. Just like the dogmatists before Kant were led into metaphysical (or transcendental) illusions because they applied their concepts outside of the area of the phenomenon, so too this non-Euclidean geometry is a mathematical illusion that comes about when the concepts of Euclidean geometry are applied outside of their rightful realm of application, where they can be grounded in the form of the pure intuition. Because non-Euclidean geometry, according to Kant, is as you say, a fiction.

The weird thing that begs for explanation though is, how come that this fiction is useful in predicting events in the real world? How come we can use this fiction to determine how a ray of light bends around the sun? (and I mean determine its exact path!) Is this use merely instrumental (as andrewk speculated here: https://www.physicsforums.com/threads/spacetime-doesnt-really-exist-does-it.487794/ )? If so, how come that it works - we would, by all means, not expect a fiction to tell us about reality. And if it's not merely instrumental, and it really describes the structure of empirical reality, how is this at all possible?

If Kant warned about transcendental illusions, then it seems fair that a warning about mathematical illusions is also necessary. And what are all the physicists doing who have built entire explanatory frameworks based on these mathematical models? They claim to be describing the structure of empirical reality - how is it possible that such mathematics apply to the structure of empirical reality? There are also Euclidean models which can explain all that general relativity explains, the issue is just that they are more complex. So what is happening? Are we using what is convenient for faster calculation, and not what is most likely to describe the actual structure of reality? Or are we describing the actual structure of (empirical) reality?

How does bending spacetime, or quantum entanglement, etc. cash out?

Or, was Kant wrong, to begin with, in restricting the sphere of application of our concepts? And so, non-Euclidean geometry is derived from Euclidean (and there is nothing wrong with this) just as the metaphysical conclusions of dogmatists are derived from concepts that are extrapolated beyond possible experience, and there is nothing wrong with them?

Other interesting material:
• 1.9k
I'm very happy to go along with your definition of reality as the phenomena rather than the noumena as, not being a materialist, that is the idea of reality that is most natural to me as well.

With that definition, why do you think there's a problem with the idea that different people have slightly different ways of processing phenomena, even if we describe that as having slightly different empirical realities? To me such a suggestion is not only plausible but seems the most natural thing to assume, even if we had never come across the works of Kant, Gauss, Lobachewsky or Einstein. All it says is that things appear slightly differently to different people. As long as that doesn't lead to conflicting decisions or predictions, there is no difficulty. Where it does lead to conflicting predictions, we say that the person who made the wrong prediction was 'suffering an illusion' (although sometimes it is just an error of calculation instead).
• 11.3k
I'm very happy to go along with your definition of reality as the phenomena rather than the noumena as, not being a materialist, that is the idea of reality that is most natural to me as well.
Well, it is not only my definition it is also Kant's definition of the empirically real. Certainly, Kant would never have thought that physics, or whatever other things that we can do empirically, can ever lead us to knowledge of the noumenon. That is out of the question. So for the most part, barring Schopenhauer's advances and insights that can be achieved through meditation, direct revelation, prayer, etc. we'll leave it that all that reality is, is empirical reality, the phenomenon.

With that definition, why do you think there's a problem with the idea that different people have slightly different ways of processing phenomena, even if we describe that as having slightly different empirical realities?
I don't think that is a problem (people having different experiences of some empirical phenomena). But we were discussing (pure) geometry, which according to Kant is a science a priori and not empirical. This geometry must be objective (true for all), since it is a priori. It is not like other matters of experience (eg. color, which can be different for different people).

But, with the exception of space, there is no representation, subjective and referring to something external to us, which could be called objective a priori. For there are no other subjective representations from which we can deduce synthetical propositions a priori, as we can from the intuition of space. (See § 3.) Therefore, to speak accurately, no ideality whatever belongs to these, although they agree in this respect with the representation of space, that they belong merely to the subjective nature of the mode of sensuous perception; such a mode, for example, as that of sight, of hearing, and of feeling, by means of the sensations of colour, sound, and heat, but which, because they are only sensations and not intuitions, do not of themselves give us the cognition of any object, least of all, an a priori cognition. My purpose, in the above remark, is merely this: to guard any one against illustrating the asserted ideality of space by examples quite insufficient, for example, by colour, taste, etc.; for these must be contemplated not as properties of things, but only as changes in the subject, changes which may be different in different men. For, in such a case, that which is originally a mere phenomenon, a rose, for example, is taken by the empirical understanding for a thing in itself, though to every different eye, in respect of its colour, it may appear different. On the contrary, the transcendental conception of phenomena in space is a critical admonition, that, in general, nothing which is intuited in space is a thing in itself, and that space is not a form which belongs as a property to things; but that objects are quite unknown to us in themselves, and what we call outward objects, are nothing else but mere representations of our sensibility, whose form is space, but whose real correlate, the thing in itself, is not known by means of these representations, nor ever can be, but respecting which, in experience, no inquiry is ever made. — Kant
• 1.9k
I don't think that is a problem (people having different experiences of some empirical phenomena). But we were discussing (pure) geometry, which according to Kant is a science a priori and not empirical. This geometry must be objective (true for all), since it is a priori.
While I don't share Kant's willingness to say what is going on in other people's heads, it at least seems plausible to me that people share a common geometry that they use to process sensory inputs. My suggestion is that geometry would be that of a Riemannian manifold that has no discernible curvature. As mentioned above, one way to axiomatise this is to replace the parallel postulate by one that says that two lines that cross either end of a line segment at angles that are discernibly non-right will intersect within a visualisable distance of the segment.

The space we inhabit has that property.

Kant could not have expressed it this way because the concepts and vocabulary to express it did not exist in his time. Nor were they known to Schopenhauer when he was writing about the parallel postulate. Both Kant and Schopenhauer would have believed that a geometry that matched what we experience was impossible without Euclid's parallel postulate.

It was only with the discoveries of Gauss, Lobachewsky et al, later in the 19th century, that it became apparent that our experience can be matched by a geometry with a less prescriptive version of the postulate, like the one I give above.
• 5.2k
The space we inhabit has that property.

What you describe are not properties of space, they are properties of "lines". And lines exist by definition.

We can give "space" any properties we want to, because it is nothing but infinite possibility. It is only when we seek to understand "the space we inhabit", that we have to allow for the existence of real physical objects within this space, and it follows that our geometrical constructs are thus constrained. But considering this constraint, we are now not talking a priori, but a posteriori.

In other words, we can make geometrical constructs however we want, they are a priori and true by definition, but the reality of "the space we inhabit" restricts us such that the ones we end up choosing for belief, are a posteriori, proven by experience.

"Space", in its pure a prior sense is universal and necessary, necessary as the condition for the possibility of all geometrical constructs. First, we assume "space" as the fundamental intuition, and this provides the possibility for whatever constructs we might dream up, the only condition that they are logically consistent. We may end up rejecting them though, if experience does not prove them to be necessary and universal.
• 3.2k
What do you mean it doesn't mean the model is accurate? If that model makes certain predictions (such as light bending around massive objects) and we go out there and test that, and the test confirms the predictions of the model, in what sense is the model "not accurate"?

What I mean is that models can be empirically adequate, in that they can fit what we presently observe, but then by definition they can say nothing about what is unobservable, such as space itself. You can postulate a mind-independent physical space as empirically adequate, and I can accept that it is, but that doesn't oblige me to believe in such a thing in the slightest. For that, you need separate philosophical arguments. Appealing to scientific theories is insufficient.
• 3.2k
I don't see how this part of Schopenhauer is relevant. I claim that Kant's way of deriving the phenomenon/noumenon distinction is not valid, though Plato's is. As Schopenhauer likes to say, the right conclusion, from the wrong premises ;)

Fair enough. But I thought you were now a realist. Do you reject Kant's conclusion or only his premises? You seem to recognize here that a rejection of Kant's premises doesn't therefore entail that realism is true.

If space was real, then transcendental idealism cannot hold.

True, but that's a pretty big "if" in my opinion.
• 11.3k
For those still interested in this topic, I have found this very interesting presentation built on Kant's writings:

http://www.socsci.uci.edu/~jheis/bio/Heis,%20K%20Parallels%20HOPOS.pdf
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