What would Kant have made of non-Euclidan geomety?
Given that Kant regarded space as the a priori form of all possible sensible representations, what would he have made of general relativity and non-Euclidean geometry? Is the fact that he considered all conceptions of space to be merely derivative or limitations of pure intuition fatal to his transcendentalism? Has he been proved wrong empirically or can his enquiry into Newtonian science be extended to cover modern science?
Comments (170)
Non-Euclidean geometries are just as intuitive (synthetic a priori) within their contexts as Euclidean geometry is within the context of everyday experience, so I don't see the problem. Kant considered all of mathematics and geometry to be synthetic a priori knowledge, unless I am mistaken.
I see no reason to suppose that it would undermine Kant's notion of the Transcendental Aesthetic, which is that we process raw inputs within a framework of three space dimensions and one time dimension. Kant would not have said that the framework could not be non-Euclidean, since the concept was not known in his day. He may well have mentioned Euclid, but that would have been because at the time Kant was writing, the name of Euclid represented all known geometry. Non-Euclidean geometry only became well-known after Gauss wrote about it in the 1813. So Kantian references to Euclid can be interpreted simply as unqualified references to geometry.
Being a great scholar and physicist, Kant, upon being introduced to non-Euclidean geometry, would have rapidly noticed that an essential feature of the non-Euclidean geometry that is used in physics is that Euclidean geometry is an extremely accurate approximation to the non-Euclidean geometry at non-cosmic scales. Since humans live, work and think in non-cosmic scales, I expect Kant would have been entirely comfortable with the notion that our in-built mechanism for arranging information is an approximation to a paradigm whose differences are only visible at scales that are beyond ordinary human experience.
I don't think non-Euclidean geometries would be considered intuitive or come under the transcendental aesthetic because they are concepts of the understanding and have to be described mathematically or modelled within Euclidean space. So they would be in the transcendental analytic.
A quote from the essay that summarises the problem:
"Impressed by the beauty and success of Euclidean geometry, philosophers -- most notably Immanuel Kant -- tried to elevate its assumptions to the status of metaphysical Truths. Geometry that fails to follow Euclid's assumptions is, according to Kant, literally inconceivable."
Frank Wilczek
The problem is that for Kant space is not merely a three dimensional framework but the very basis of any possible intuition whatever.
Non-Euclidean geometry does follow Euclid's assumptions at the scales that are meaningful to humans. So there is no conflict.
A geometry that is not locally Euclidean would have to be fractal - infinitely wiggly - and most of us (certainly me!) find that inconceivable. I can do formal calculations about infinitely wiggly things (the Ito Process that is used as the basis of pricing most financial instruments is infinitely wiggly), but I cannot visualise them, and I have no intuitions about them.
I highlighted the interesting part, because what does it mean for Kant for something to be beyond ordinary human experience, particularly in context of:
Quoting andrewk
Also, modern physics has entertained higher dimensional space and colliding branes, along with multiple universes. I wonder if those concepts could fit into the transcendental aesthetic since they're so far beyond the normal framework of space/time.
Did he do that? I don't know. I never read the original, being a secondary-source kind of chap. I think we'd need to dig out a quote, both in the original German and a diversity of English translations, and analyse it to see whether we can reach that conclusion from it. My loose observation about Kant scholarship is similar to that commonly made about economics: If we put n Kant scholars in a room there will be n+1 opinions about what Kant meant by any particular passage he wrote.
Yes thank you for clarifying my point. In the transcendental aesthetic Kant speaks of space not as an objective absolute background like Newton, nor as a system of relations like Leibnitz but as an integral part of our perceptive apparatus. If Euclidean geometry is meant to be a necessary and timeless truth constrained by the nature of our perceptions then any deviation from this in nature would seem to prove Kant wrong. However, later on in the critique he does seem to talk about space as a coordinate measurement system so this is rather confusing and I'm not sure if he was consistent.
I'm not sure we can rely on geometry remaining Euclidean at human scales because if one takes general relativity into account then masses are warping spaces and causing gravitational geodesics that support our entire existence.
I believe Kant did as I said, though as with you, I'm relying on a secondary source (and potentially worse, my hazy recollection of that source). Mostly, I find it plausible that my memory is correct here since Kant also made a similar sort of assertion (that turned out to be wrong) about Aristotelian Logic ("we have no need for more logicians", the century before we got Classical Logic, lol) and something similar about biology. Maybe I have a poor recollection and that has colored my view of Kant.
The curvature that gives us our fairly modest Earth gravity is a curvature of spacetime, not a curvature of space. The difference between the two concepts is crucial. IIRC, it is possible to have a spacetime that is curved but for which all spatial slices are flat. The curvature is only in the relation between space and time, not in the space itself.
Kant saw space and time as separate, so curvature in the relations between them, would have meant nothing to him, and could not contradict his 3D + 1D model* that is the Transcendental Aesthetic . Indeed, the maths needed to express that curvature had not even been developed when Kant was around. All that mattered was that spatial slices ('constant-time hypersurfaces') were flat, and that is extremely close to being the case in a site of very small gravity like the surface of the Earth.
* As distinct from Einstein's 4D model.
and they’re in an absolute schemozle for having done so.
There is an old thread about this I opened here:
____________________________________________________________________________________________
START OF QUOTE
"Both Schopenhauer and Kant take space to be an a priori form of representation, applied by the cognitive faculties to the senses. They understand this to mean that the propositions of geometry are synthetic a priori judgements, and are therefore apodeictic - certain.
Schopenhauer follows Kant in conceiving of space (and geometry - the study of space) as transcendentally ideal:
Now Schopenhauer's ontological idealism (and I refer here to the phenomenon/noumenon distinction largely) critically requires that the stage on which experience occurs be transcendentally ideal, for this stage being transcendentally ideal is what enables experience to be called the veil of Maya - appearance - and hence necessitates the noumenon, the thing-in-itself. Without space, time and causality (which constitute the stage in/on which experience occurs) being transcendentally ideal, the distinction between noumenon/phenomenon is in danger, as is idealism - for if at least part of the stage on which experience occurs is real, then Schopenhauer's ontological idealism is false.
Schopenhauer laughed at mathematicians trying to prove Euclid's Fifth Postulate, thinking that it is known from pure perception a priori:
Non-Euclidean geometry came along, and it turns out that we have empirical proof that Euclid's Fifth Postulate is actually false, with regards to space as investigated by physics. Now the curvature of space cannot be perceived - we perceive objects in space - things in space curve - but how can space itself curve - that is anathema to our perception. What does this mean for Schopenhauer? Well for one, Euclid's Fifth Postulate isn't apodeictic, and neither is it a priori - contrary to what Schopenhauer thought. So Schopenhauer was wrong at least about this one truth of mathematics, and if he was wrong about this one, why should the other mathematical propositions that he was certain of be anymore certain than this? Indeed, it is his method that is wrong. Grounding mathematical propositions in a priori perception without appeal to experience is wrong. As Einstein said:
More importantly, there is one feature of space - its non-Euclideanness which is NOT a synthetic a priori, but rather a synthetic a posteriori, and therefore not transcendentally ideal, but empirically real - for it takes experience (physical experiments) for us to know of it. This means that at least part of the stage on which experiences occur isn't imposed on reality as a structure by our cognitive faculties, but rather is empirically real. If part of the stage is empirically real, then Schopenhauer's ontological idealism falls apart."
END OF QUOTE
____________________________________________________________________________________________
To summarize - read the 10 or so pages of the Transcendental Aesthetic. Kant is very clear about this - space and time are the pure forms of the sensibility and they are not derived from the contents of sensation. They are NOT empirical intuitions.
And where do the axioms and postulates get their certainty from?
The answer will be forced to be either synthetic a priori or synthetic a posteriori, and one must be given a reason why. If all axioms are synthetic a prioris, except for the 5th one, then why is it that the fifth one isn't, and yet Kant and Schopenhauer thought it is? Because clearly, non-Euclidean geometry demonstrates beyond a shadow of a doubt that the 5th postulate is not a synthetic a priori, despite what Kant and Schopenhauer thought.
If the axioms are synthetic a posteriori, then geometry is not apodictic - it is not certain. This means that outer space does not have to obey our geometrical constructs (since clearly, those constructs are not some a priori forms of our sensibility that exist in the mind and through which all phenomenal experience is given). Then the whole of Kantian thought collapses since space is not an A PRIORI form of the sensibility, and therefore (probably) must be determined by the senses itself (or at any rate, by the senses + a (reconceptualized) understanding), being a synthetic a posteriori.
That essay is a pile of manure.
>:O And Kant wasted his time saying that the propositions of geometry are synthetic in order to tell us they can be denied without contradiction, because that certainly proves the reliability of mathematics when applied to physics (which is what Kant was trying to do - just open a page of the Prolegomena, instead of reading 20,000 secondary sources who don't know what they're talking about). Of course, this isn't what Kant did. Kant aimed to say that the propositions of geometry don't derive their CERTAINTY (because he took their certainty for granted) due to the law of non-contradiction (hence the synthetic part). Rather they derive their certainty from their a priority, rooted as they are in the pure form of sensation, space.
Nope. This is a mistaken view. Kant was trying to show why mathematics is so effective at describing physical space - if mathematics is just a human construct, its effectiveness cannot be accounted. So Kant resolved the problem by saying that it is not just a human construct. We have this form of pure intuition, space, from which we derive the axioms of geometry. So they are synthetic, since we can imagine the opposite to be the case, but they are a priori, because they stand as being true prior to experience. His point was that I don't need to run a physical experiment to see that Euclid's Fifth Postulate holds (which of course turned out to be false). It is simply impossible, according to Kant, for Euclid's Fifth Postulate not to hold - not due to its synthetic nature, but due to its a priority, and the role the pure form of sensation (space) has in constituting all (spatial) experiences, and hence any possible experimental result.
Quoting Janus
That is correct.
Exactly... If you had, you'd be amazed by how often Kant mentions the "apodictic" nature of geometrical propositions.
Sorry, but I will disagree with you on this.
Quoting Hanover
Reality is a hazy word. Why is the noumenon reality, and the phenomenon not? Don't forget that it is the phenomenon that is the empirically real according to Kant, not the noumenal. 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. It takes Schopenhauer to clarify this aspect of Kant.
Quoting Hanover
Precondition of the sensibility, not of the understanding. Kant talks of space (and time) in the Transcendental Aesthetic, and labels them as forms of the sensibility (as opposed to the matter or content of the sensibility, the sensations themselves), which comes before he goes into the categories of the Understanding.
Also it is a grave error to think that there is any "true reality" to space according to Kant. Space is transcendentally ideal, given by the forms of our sensibility. There very likely is no space at all out there - external to our phenomenal, empirically real experience. The space studied by physics is this phenomenal, empirically real space (which of course is a contradiction - Kant wouldn't claim physics studies space, that would be the job of mathematics).
The essay that posted is actually very interesting in regard to the different types of curvature.
I was wondering about this. Before Kant, a priori truths were considered analytic and so are true by definition. The a priori synthetic truths may be certain from the perspective of our sensations but does this really make them logically necessary? Could there not conceivably be forms of intuition different from ours that allow for different types of space?
Your intended point of disagreement is not clear. I haven't suggested that space is a "human construct", in case that was it.
According to Kant, the a priori synthetic truths must be certain from the perspective of the phenomenon and our experience. One repercussion of this is that you could not do a physics experiment which did not obey the laws of geometry.
Quoting Agustino
I apprehend some inconsistency here. In the first, you describe space as a "form of pure intuition". In the second you describe space as one of the "forms of sensibility".
As a form of "sensibility", I would assume that space is a condition for the possibility of sensation. But "intuition" I would think only arises from a being which has sensation. So if space is an intuition, then sensation would be prior to space as an "intuition".
Therefore one or the other cannot be correct. Either space is an intuition, in which case it occurs after sensation, or, space is a condition for the possibility of sensation, in which case it is prior to sensation and cannot be an intuition, which only occurs to creatures which already have sensation.
Edit: This is probably why there is such variance in interpretation of Kant on this issue.
Kant uses intuition in a technical sense. It's not what we mean by intuition in common language. In Kant, intuition is something closer to what we mean in common language by perception.
And so, in Kant's system, there is no inconsistency though my language was a bit convoluted by saying "form of pure intuition" instead of just "pure intuition". Kant says:
Transcendental Aesthetic - Introduction:
"The pure form of sensibility I shall call pure intuition"
Before Gauss, Lobachewski, Riemann and others developed the notion of non-Euclidean geometry in the 19th century, people tended to regard Euclid's axioms as a job lot, which you accepted or rejected all together, not picking and choosing. Nobody could see a way of accepting some but not others and still coming up with workable, useful notions.
But now, thanks to those 19th and then 20th century mathematicians we have a good understanding of many different possible axiomatisations, all producing workable, useful spaces. The mathematics is called Differential Geometry and the spaces are called Riemannian Manifolds.
What is common between the different spaces/manifolds - Euclidean and Non-Euclidean - is that they all satisfy the axioms of a Riemannian Manifold, the key ones of which are:
- the space is connected, so that from any starting point you can get to any other part of the space via a continuous path
- the space is continuous, so going along a path you won't suddenly find yourself in a completely different, faraway part of space
- the space has a constant dimensionality that is a positive integer n. For our space n=3.
- between any two points there is a measurable 'distance', which is the length of the shortest path from one to the other, and these distances must:
* be non-negative
* are zero iff the two points are the same
* are symmetric, so that the distance from A to B is the same as the distance from B to A
* obey the triangle inequality, so that the distance from A to C does not exceed the distance from A to B plus the distance from B to C
The Riemannian Manifold axioms are enough to give a very strong notion of space as a three-dimensional area in which things are located and can move about. Further, it accords well with our intuitions of space - well, with mine at least!
My hypothesis, which may be dispelled by further direct Kant quotes, is that perhaps it's this more general notion of space notion that Kant was insistent on, not on a notion that added additional axioms to make the space Euclidean. The test would be whether Kant actually directly mentions Euclid's parallel postulate, or something equivalent to it.
Note that mentions of Euclid or Euclidean do not count, because in Kant's time those terms only indicated a reference to geometry generally, not to something that is distinct from Non-Euclidean space - a meaning that only arose in the 19th century.
In the Kant quote that Agustino kindly provided for us (link above), Kant only mentions two specific aspects of space, which are:
(1) the triangle inequality for distances; and
(2) that things can be 'inside, outside or alongside one another'
These are properties that are satisfied by any Riemannian Manifold, not just Euclidean ones. Perhaps he mentions the parallel postulate somewhere else, but he certainly does not do so in the above quote.
One last thing. The parallel postulate says that there exist pairs of straight lines that never meet, and that pairs that do meet only do so at one place. I, and generations of mathematicians before me, do not find that particularly intuitive, whereas Euclid's other axioms do seem intuitive. That's why people wondered for centuries whether that aximo was necessary in order to do geometry at all. Gauss's brilliance was to show that it wasn't.
Another aspect of the parallel postulate is that the three internal angles of a triangle must add to 180 degrees (or to 'two right angles' as Euclid put it). Again, I do not find this at all intuitive. In non-Euclidean geometry the sum of angles can differ - it is more than 180 for elliptical manifolds and less than 180 for hyperbolic ones.
Anyway, TLDR, sorry about that, My question is: did Kant ever specifically insist that Euclid's parallel postulate was part of our a priori processing of intuitions?
Still the same issue, sensation is necessary for perception, so space, as a perception cannot provide the condition for the possibility of sensation. The ;logical order is reversed.
Quoting Agustino
"Perception of space and time" implies that there are these things, "space" and "time" which are being perceived. Either they are perceived through the senses (sense objects), or they are perceived directly by the mind (intuitions). If the latter, then the problem I indicated in my last post stands. These intuitions cannot provide the condition for the possibility of sensation because sensation is prior to intuition.
Quoting Agustino
So this is meaningless nonsense, it's unintelligible, incomprehensible.
Quoting Agustino
This thought experiment is faulty, because he has already had these sensations. So he cannot put himself in a place of never having had these sensations, therefore he cannot make any determinations about the conditions for the possibility of sensations in this way.
So, apparently you don't have a point of disagreement, then.
I think you are misreading Kant, though.
Quoting Agustino
I would say there is no "physical space" for Kant; space is not a physical object. Can you provide a citation that supports your idea that Kant was specifically concerned with showing "why mathematics is so effective at describing physical space"? Of course geometry is effective at describing perceptual space, because it just consists in formulations of our intuitions of the nature of our visual perception.
For Kant space is a pure form of intuition, it is not given empirically, rather it gives the empirical. Euclidean geometry is the direct intuition of the characteristics of perceptual space. Non-Euclidean geometries are not empirically given either but are intuitively derived models of how geometrical principles would diverge form the Euclidean on curved two-dimensional planes. The curvature of space-time is also not empirically given, but is a hypothetical construct, whose predictions have been very precisely confirmed and measured. The point is, though, that spacetime is not the same as space and time understood as pure forms of intuition; it is something else, we know not what, something that we cannot even visualize.
Can you quote a passage from Kant where he clearly claims that all experimental results must be in accord with our synthetic a priori conceptions of the pure forms of intuition? I have read Kant pretty extensively and I don't recall any such claim. The mistake you are making consists in thinking that spacetime is a "(spatial) experience", and that's wrong; spacetime is not perceptual space.
There is a remark about something similar:
In addition, there are the remarks of Kant's student, Schopenhauer, more clearly about the same subject:
So all evidence available seems to point to the fact that Kant (& Schopenhauer) did consider Euclid's parallel postulate to be a synthetic a priori.
Quoting andrewk
Why don't you find it intuitive? When you imagine space, isn't this how you necessarily would imagine it? I lean towards saying that my intuition is thoroughly Euclidean, and non-Euclidean geometry wasn't discovered for so long precisely because we don't have an intuition / direct perception of it. Otherwise, why did it take non-Euclidean geometry so long to be discovered?
What I'm pointing out is that the words 'Euclidean Geometry' have a different meaning now from what they had in the 18th century. In the 18th century they just meant Geometry simpliciter, because Euclid was seen as the father of geometry and was considered synonymous with it, and because no other sort of Geometry was known and people imagined no other sort was possible.
But now the term 'Euclidean Geometry' is used to refer to a subset of Geometry that excludes manifolds with curvature. To argue that Kant intended that meaning, without additional evidence, is to participate in an anachronism, using a meaning of the term that was not the meaning at the time it was used.
The only way to substantiate a claim that Kant was not just referring to Geometry generally (Riemannian Manifolds) is to find a quote where he specifically insists on the importance of the parallel postulate.
I see that @Agustino has just posted a new quote from Kant involving the sum of angles in a triangle, so I'll read that and see where it leads me. Since I find reading Kant really hard work, it'll probably be quite a while before I have anything coherent to say about it.
Or that he didn't at least understand that the parallel postulate was not necessary in order to have all the usual concepts of continuity, connectedness, insides, outsides, points, lines, angles, volumes, shapes? Again, nobody else knew that until 1813, so why should we have expected Kant to realise it?
The following is counterfactual, and hence unfalsifiable and otherwise empty, but no more so than the rest of the discussion:
I suggest that if a mathematician that Kant respected had discovered these things and had explained to Kant in 1781 that you can get all those things without the parallel postulate, Kant would have taken that on board and related his Transcendental Aesthetic to Riemann and his manifolds, rather than to Euclid.
I think the distinction 'pure and not empirical' is significant, as it refers to any principle which is immediately evident to intuition itself without reference to any empirical or sensory object. This reflects the Platonist distinction between the intellectual intuition which is able to grasp ideas directly, with sensory perception which is of a lower order in only grasping its objects mediately. Whereas, empiricism generally wants to start from sensory perception, and validate any proposition with respect to it. This, then, was the historical grounds that Euclidean principles were to provide insight a kind of higher truth. Whereas:
[quote=Wikipedia]The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant's treatment of human knowledge had a special role for geometry. It was his prime example of synthetica priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift.
The existence of non-Euclidean geometries impacted the intellectual life of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements. This curriculum issue was hotly debated at the time and was even the subject of a book, Euclid and his Modern Rivals, written by Charles Lutwidge Dodgson (1832–1898) better known as Lewis Carroll, the author of Alice in Wonderland.[/quote]
(I am sure the Cheshire Cat's grin is relevant here, but can't quite put my finger on how. Perhaps I should try and read his book.)
[quote=wiki version of Thomas Heath's translation of Euclid] Let the following be postulated":
1. "To draw a straight line from any point to any point."
2. "To produce [extend] a finite straight line continuously in a straight line."
3. "To describe a circle with any centre and distance [radius]."
4. "That all right angles are equal to one another."
5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
[/quote]
Unless I'm missing something, the 5th postulate would also be true for an elliptic surface, such as the surface of a sphere. In order to exclude elliptic geometries, the words ',and not on the other side' would have to be added at the end of the sentence.
The same applies to the alternative version of the postulate given in the following section of the wiki article:
It seems to me that 'at most' needs to be changed to 'exactly' in order to exclude elliptic surfaces.
I feel I must be missing something. Can anybody help me find what it is?
Yes, this is an important distinction. There no are truly Euclidean or non-Euclidean objects of the senses, in any case. And perceptual space is definitely not intuited as being curved; in fact we cannot even visualize curvature of a three dimensional space, and non-Euclidean geometries deal only with curvatures of two-dimensional planes.
One way of arguing is that our intuition is still Euclidean. So in spite of non-Euclidean geometry, our form is Euclidean.
Another way of arguing: you could say that our intuition of space is actually non-Euclidean (or whatever happens to be the correct geometry of space, supposing non-Euclidean geometry is superseded), and Euclidean geometry was merely an empirical concept of that form.
I have just been informed by an impeccable source that there are coordinate systems in which space near a large mass like the Earth is locally perfectly flat. Here's the wiki page that describes those systems.
Problem solved! Immanuel Kant has been vindicated. X-)
As an aside, my source pointed out that, even in the Swarzschild coordinate system that is more typically used near a planet, the spatial curvature near Earth would be about one part in a billion, and probably not possible to detect with current equipment.
Quoting Moliere Nice!
You've got this twisted. The "criticism" was simply that if indeed he believed the axioms of Euclidean Geometry were metaphysically necessary, then Non-Euclidean geometries seem to falsify this notion. So I don't understand why you brought up the historical matter of when such maths were developed. I wasn't criticizing him, I just said that (unless I'm missing something) his view on this matter was incorrect. So bringing up that Kant didn't specifically mention the parallel postulate is entirely beside the point since he was referencing the geometry of the day.
Kant didn't believe that anything was metaphysically necessary. His whole project involved refuting rationalist metaphysics such as those of Leibniz, Spinoza, the Scholastics and the Ancients. Although he didn't specifically state it this way, his project aimed to show what was phenomenologically necessary for human experience.
I'm having trouble making sense of the idea of Euclidean geometry as an "empirical concept", other than it being obviously a conceptual scheme derived directly from everyday experience (taken in its broadest sense of both "inner" and "outer" experience). If you mean it in some other sense, perhaps you could explain how it would qualify as such?
Fair enough. I had assumed - wrongly, it now seems - that you were aligning with the group that @Wayfarer identified in this post ( ) that assume the discovery of non-Euclidean geometry undermines Kant’s understanding of a priori truth. If all you are suggesting is that Kant may have had a wrong idea about the necessity of the parallel postulate, then you are not adopting the assumptions of that group. The suggestion seems not to damage Kant's thesis at all, and I do not argue against it.
Since most people in Kant's time believed the parallel postulate was necessary in order to be able to do geometry at all, it is no adverse reflection on Kant, or on the Critique of Pure Reason or the usefulness of the Transcendental Aesthetic, if he believed that as well.
That's what I mean. Surely it's sensible that we could be wrong about the form of inuition. So, supposing non-Euclidean geometry is the true geometry of the space we experience it doesn't seem like a large step to say that we were simply wrong before about the form of intuition. At least not to me. If that were the case, then it would just be an empirical concept, though -- since a priori concepts of space are apodeictic.
OK, the problem I have now is with the notion that we experience space. Space is the pure form of intuition, according to Kant, which means that intuitions (visual perceptions in this case) must take spatial form, and our a priori apprehensions of that space are intuitively obvious to us. Our visual perceptions do not take the form of curved space, and as I said before we cannot even visualize such a thing. So, even if spacetime is curved by mass (whatever that actually means beyond our mathematical models and predictions, I still don't see why we would say that spacetime could be the pure form of our intuitions.
I don't think "intuition" in Kant means the same thing as intuitive. Space isn't intuitively obvious to us. Others have been wrong about space -- like Leibniz and Newton, for instance. So while the examples Kant uses are from Euclidean geometry it seems to me that one could modify the philosophy without losing the core of the aesthetic. It's not that something is obvious, but rather that we are able to have synethic a priori knowledge about space due to our knowledge of geometry. If one geometry is wrong then, just like Newton could be wrong, we could understand such sciences as something which wasn't part of our cognitive faculties but was derived from them, and is therefore empirical in that sense (and not synthetic a priori knowledge, but instead rests upon that)
As far as I understand it "intuition" for Kant means something pretty close to what we would call 'perception'. I think space is intuitively obvious to us, and that's why the axioms of Euclidean geometry are self-evident. But what space is (merely an abstraction from objects for Leibniz versus an empty but absolutely existent container for Newton while Kant disagreed with both) is certainly not intuitively obvious.
You say "we are able to have synethic a priori knowledge about space due to our knowledge of geometry" but if this were true then it would not be "synthetic a priori knowledge" at all but synthetic a posteriori knowledge. I think it is more to the point that we are able to have knowledge of geometry due to our synthetic a priori knowledge of space. I think that is certainly what Kant thought.
I don't think it makes sense to say that Euclidean or non-Euclidean geometries are "wrong"; both are intuitively obvious in their contexts. This is not say that it is, or even can be, intuitively obvious that spacetime is curved, because, to repeat myself, I don't think we have any reason to think that spacetime is the same thing as perceptual space, for the simple reason that we cannot perceive, or even visualize, the curvature of spacetime. Is there any reason you can think of why we must believe they are the same?
Yes, I am already aware of that. It's necessary for me to talk of "physical space" because Kant was wrong.
Quoting Agustino
Quoting Janus
Sure.
Quoting Janus
Well, for Kant, there is only one space and mathematics (geometry) describes it with apodictic certainty.
Quoting Janus
Nope. This is wrong on two counts. (1), our perception may not be Euclidean. Parallel lines do meet, in our perception, at the horizon. So if you want to argue for this point (that our natural intuition of space is Euclidean), with which I actually agree, you cannot appeal to the "nature of visual perception". (2), there is no "perceptual" space as differentiated from "physical" space (the space we encounter when we do our physical experiments) in Kant - there is only one space.
Quoting Janus
Yeah, or rather, the empirical is given by means of space. Space is the form, and the empirical is the content or matter of that form.
Quoting Janus
I've already tackled this above.
Quoting Janus
This is incoherent. Can you perceive non-euclidean geometries? If you can't, then they are not intuitive per Kant's understanding. andrewk has still not told us how he "intuitively" perceives that Euclid's parallel postulate is not a priori.
Quoting Janus
Space-time is empirically given, that's why it can be empirically validated.
Quoting Janus
No, and you can't give any to the contrary.
But in Kant's system I can tell you for certain that it can be no other way.
"Therefore in one way only can my intuition anticipate the actuality of the object, and be a cognition a prioir, viz., if my intuition contains nothing but the form of sensibility, which in me as subject precedes all the actual impressions which I am affected by the objects"
Kant conceives the intuition as "anticipating" the actuality of objects, since it already has the cognition, a priori, of space, which precedes & organises the sense impressions of the objects. So objects are given in space, and we know space a priori. Thus, objects necessarily must conform to this space. It follows from this that physics - or anything else - cannot invalidate or cause us to revise our conception of space, if it is a priori given. The only way we could revise it is if it's not a priori. At minimum, space must be more than transcendentally ideal.
I don't see why "and not on the other side" must be added when "on the same side" and "on that side on which are the angles less than the two right angles" already exists. This is implicit.
Quoting andrewk
And globally?
Quoting andrewk
:-}
Quoting andrewk
But it still exists, hence invalidating Kant.
And you still haven't explained:
Quoting Agustino
You keep repeating that Euclid's parallel postulate is not intuitive, but you don't explain why.
Ummm, no. Plato's intellectual intuition goes more with Kant's Understanding and the categories than with the forms of sensibility. The forms of sensibility ARE sensuous or sensory in nature. So space and time are not like, say, causality, which is a category of the Understanding. And the forms of sensibility are in no way "lower order" or "higher order" - there is a difference in kind between the content of sensibility and the form of sensibility. The latter is a form - it is the organising principle of the matter, the matter is given through it. And the former is the matter or content itself. On the other side, the Understanding provides the organising principles of our judgements.
You keep doing a mishmash of Plato and Kant and this is totally wrong - these two thinkers are not on friendly grounds in most regards. The Eastern / Schopenhaurian Platonic reinterpretation of Kant fails along with the failure of transcendental idealism.
Yeah, this isn't controversial.
I don't agree that it does, but I was wondering who might be an example that strong view that Wayfarer mentioned some people holding. So now I know.
Quoting Agustino If I could explain it, it wouldn't be an intuition.
The other postulates seem obvious and undeniable to me. That one doesn't. I suppose it must be just the way my brain's wired.
At least I can tell you why it took so long to discover the other geometries though. It's because it wasn't just a question of removing the parallel postulate. It needed to be replaced by something, otherwise we're taking away too much. In fact, what was needed was a complete re-axiomatisation, starting with a completely new set of axioms that does not resemble the existing ones at all. In fact a completely new language was needed, involving things called manifolds, vector spaces, tensors and metrics.
That was a very difficult task, and needed to wait for some extremely clever people to first realise that's what was needed, then secondly work out how to do it.
For Kant intuition means something closer to perception. So I assumed you were using that term, otherwise, it has no bearing on what Kant was writing about anyways. So is non-Euclidean geometry perceptible in your mind's eye / imagination?
Quoting andrewk
Why doesn't the parallel postulate also seem undeniable? There must be a reason for it, otherwise, I think we will have to attribute it to habit. Are you a mathematician? If so, perhaps you have trained for long enough in non-Euclidean geometry that this training has become second-nature to you.
For me, Euclid's parallel postulate seems undeniable. I am not a mathematician, and from early on I was taught geometry according to Euclid. Euclid's postulates are second nature to my practice of geometry and thinking about geometric problems. Indeed, this may be exactly why I find it easy to perceive the truth of Euclid's parallel postulate, and not to perceive the opposite. It may just be habit, as Hume said, that has entrenched these unprovable things that we take for granted, that Kant now pulls out of the hat as a magician, and calls them "synthetic a prioris". There have also been other critiques of Kant along Marxist lines which claim that Kant's Categories themselves are conceived in the praxis of economic exchange, and then philosophy (& science) is misled to consider them properly basic.
Quoting andrewk
So this new reconceptualisation was not a pure intuition as per Kant's definition of the term? It arose by means other than intuition, such as conceptualisation, right? It took several minds to adjust the conceptualisation so that it all made sense.
Yes, I mean what ordinary people mean by intuition, not what Kant means . He uses words too weirdly for me.
Quoting Agustino
Yes. It may be, as you say, cos I'm a mathematician. Or maybe I'm a mathematician cos I look at things that way.
Quoting Agustino
I think the concepts are a lot easier than the axiomatisation. The concepts are intuitive (again, maybe only to me), but the axioms are not.
(1) What ordinary people mean by intuition cannot be used to defend Kant, who uses that word differently, and thus means different things by it than ordinary people.
(2) The way ordinary people use "intuition" and others words that are derived from it is extremely vague. In ordinary language, an "intuition" is just when I throw up my hands and tell you "I know it is this way, but I can't say why". Very often, habit can entrench thoughts, principles, and the like in people's mind, and they easily recall them, and feel very certain in them, but are unable to give justification for them.
Quoting andrewk
Which way?
I disagree with this, but I'll touch on it in replying to your third paragraph. Probably gets to the crux of our disagreement though.
Quoting Janus
I don't disagree with that interpretation of Kant here. This is why I think non-Euclidean geometry is problematic, just not destructive to the aesthetic. It can be "saved", that is -- and still feel reasonable rather than ludicrous.
So, following my second strategy, Euclidean geometry could be interpreted as synthetic a posteriori knowledge while non-Euclidean geometry could be interpreted as syntehtic a priori -- and the same would apply to any other geometry which predicts the events of the phenomenal world.
Quoting Janus
It's not our perception of space that's at issue, I'd say. The propositions of geometry are closely tied to physics, by my reading. Because our intuition follows mathematical laws we are also able to apply those mathematical laws to objects, which are themselves within our intuition.
Strictly speaking it's not perception which intuition is trying to explain, but rather intuition is one half of the elements of cognition which explains how knowledge of objects is possible. Clearly there are relations between perception and cognition, and granted the intuition's description relies heavily upon visual imagery (like a lot of Western philosophy), but the reason why mathematical laws are able to be posited and discovered in the phenomenal world is because our cognition relies upon this form. It sort of explains why we are able to make predictions which are actually caused -- meaning the "necessary connection" between two events -- in the first place, rather than merely the constant conjunction of non-related events believed by force of habit.
So if it turns out that Euclidean geometry is not the form of intuition it would seem to upend the notion that we have synthetic a priori knowledge of the form of intuition. Same goes for the physics based upon that synthetic a priori knowledge. However, if Euclidean geometry were merely empirical, an approximation of our cognitive faculties as Newton was an approximation, then I'd say that the aesthetic is saved.
But in either case, it's not how we perceive that's at issue. It's how we are able to know math and why it applies to the objects of our perception in the first place. Kind of a hair-thin distinction, but I'd say it's important because in one case we are dealing with phenomenology and psychology, and in the other we are dealing with the possibility of knowledge which seems to fit more in line with the whole Critique.
Ptolemization, I see X-) - when it doesn't work, we'll add new fudge factors to make it work... Kind of ironic, given that this was supposed to be a Copernican revolution >:O
Yes and that is the space of human perception. Spacetime, whatever it is, is not that space; that has been my point all along.
Quoting Agustino
Parallel lines in perceptual space do not meet, otherwise trains could not operate. They only appear to meet, and it very well understood why that happens. We can build rail lines extending thousands of kilometers and the rails are (not perfectly, but on average, parallel). Can you think of any reason, other than practical limitations, why rail lines could not extend indefinitely?
(2) There is no "physical space" in Kant, as you have already acknowledged; so it seems you have fallen back into confusion again. Spacetime is a hypothetical construct; there is no actual spacetime that we can intuitively understand, as we can our perceptual spaces.
Quoting Agustino
Of course we can; we can intuitively understand them when they are visually represented on two dimensional curved surfaces. The analogy from curving or warping of two dimensional surfaces into the third dimension (which we can visualize) to curving of three dimensional space into a hypothetical fourth dimension is the only way we can get any notional sense of it; it is not something we can directly represent visually to ourselves at all.
Quoting Agustino
If you think anything can be no other way in Kant's system then I would conclude that you have not read Kant, or if you have, have not understood him. Kant scholar's have been arguing over just what he meant for centuries.
I'm sorry to do this, but I have little time at the moment, so I will direct you to my response to Agustino as I think it deals with some aspects of what constitutes our ongoing disagreement. I'll try to return to address your post more fully latter. :)
My intention is to defend not Kant, with whom I disagree on many important things (although I do have enormous admiration for him), but what I see as the amazing insight and usefulness of his notion of the Transcendental Aesthetic (TA). In the discussion over whether you and I find non-Euclidean geometries unintuitive, I see that as just a reflection on your and my particular cognitive processes, rather than about the TA, which is suggested to be universal to autonomous humans.
My interpretation of the TA, which has evolved in the course of this discussion (thank you everybody - this forum can be such a learning experience), is that humans process sensory input in a framework consisting of two Riemannian manifolds: a 3D one that we call 'space' and a 1D one that we call 'time'. That Kant did not describe it this way I ascribe to the fact that the language necessary to express that did not exist in his time.
Space as a 3D Riemannian manifold gives us points, lines, shapes, volumes, angles, directions, relative positions, insides and outsides, and distance.
As I see it, that, together with time, is enough for us to navigate, imagine and discuss the world. At most I would add a requirement that any curvature not be too extreme, because if that were the case we might find ourselves back where we started if we walked one metre (if the space were elliptic), That requirement is completely consistent with the region of the universe in which we evolved, and which we now inhabit.
It may well be the case that for some people the space manifold is also perfectly flat (ie no curvature, not even if unmeasurably small), as you report to be the case for you. But I suspect that is an individual variation, rather than a universal feature. For my own case, It is not necessary in order to obtain all the concepts listed in bold text above.
Quoting Agustino In a way that does not require the space manifold I use to be perfectly flat.
This got me thinking. How would we build a rail line to circumnavigate the equator, if there were a 5m wide land bridge all the way that followed the great circle of the equator? Say the land bridge is perfectly level (constant altitude above mean sea level) and extends at least 2.5m to either side of the equator at every point.
I'm pretty sure that the answer is that the rails would always be parallel and equidistant, but what we'd have to give up is the requirement that they be 'straight' - what's called a 'geodesic' in tech terminology. Say the gauge is standard and the centre of each rail is always 717.5mm away from the equator - one in the Northern and one in the Southern hemisphere. Then neither rail can follow a great circle but instead is constantly curving away from the equator at an incredibly small, constant rate.
So the lines would be parallel and a constant distance apart, but they would not be perfectly 'straight'. However a train could run along them with no difficulty at all.
Why can't the rails both be straight? Because a straight line on the surface of a sphere is a great circle, and any two great circles will intersect at two antipodal points. It would however be possible to make one of the rails a great circle and the other one not - eg if one rail followed the equator and the other were in the Southern hemisphere..
What is "human perception"? Is this not the same space as the space in which our bodies act and live? Before you said visual perception - that's not correct. We can have a notion of space through touch alone, for example.
Quoting Janus
So then this is not visual space - what you see in front of your eyes, but rather something else. You admit that in visual perception, the lines appear to meet at the horizon.
Quoting Janus
What is spacetime? And how does it relate to the space we intuit?
Quoting Janus
So what about light rays travelling in straight lines but bending around planets? We cannot perceive that or?
Quoting Janus
But we cannot intuitively understand them in three-dimensions, except by analogy, no?
Quoting Janus
So then we really don't have an intuitive understanding of it? We have an understanding by proxy of 2D objects curved in the 3rd dimension. Furthermore, I think in mathematics, @andrewk should correct me, the notion of intrinsic curvature does not require the existence of another higher dimension for the space to curve into. So the 2D objects curving in another dimension - that's extrinsic curvature, and we can have an intuition of it. But we can't have an intuition of intrinsic curvature - in the Kantian sense of intuition.
Quoting Janus
I am aware there are Kant scholars who disagree - they are free to do so. But those who disagree, do such violence to Kant's system, that it is essentially unrecognisable, or otherwise a Ptolematization. I've seen and read scholars who don't take Kant's transcendental idealism seriously enough, and who buy into Kant's confused idea of the noumenon, and there being a real space out there (that physics figures out), and adapt Kant's ideas to take into account their naturalism, etc. - that's not philosophy if you ask me, that's nonsense. Schopenhauer understood Kant rightly, and at least set the noumenon bit straight, and avoided the pitfalls of naturalism.
If you have any Kant scholar who follows in the footsteps of Schopenhauer and deals with the issue of non-Euclidean geometry, feel free to let me know, and I will look into them.
How do you imagine a 3D, non-flat space? How do you imagine intrinsic curvature? Hopefully, you won't say that you do via analogy to extrinsic curvature.
That's true; they would not be straight in the vertical plane, because they would curve to remain parallel to the curvature of the Earth. What if we could build a rail line into space; it could be straight and parallel in both planes I think.
I already said it is a hypothetical construct.
Quoting Agustino
We cannot directly perceive light rays at all. On account of our explanatory theories about what we do observe we can infer that they are bending. We can further infer that the bending is caused by curvature of spacetime in accordance with other theories.
I keep getting sucked back into these discussions and sometimes they just take up too much time, I don't have much time right now, so...really gotta go...
What does it mean that it is a hypothetical construct?
Okay, answer when you have time then :P
Quoting Janus
We perceive them via instruments, that is still perception. It's like looking at a cell with a microscope - still counts as percieving, even though not directly (I don't see how that is relevant though). Even so called direct perception is mediated through our eyes - if we're color blind, we perceive things differently. So... Whether mediated through eyes, or telescopes or whatever - makes no difference as far as I see it. We basically see that they are bending.
Anyway, apart from that point, do you believe atoms exist? We also infer the existence of atoms from related evidence. So spacetime and its curvature isn't just a theory, it really exists.
I'll describe below how I imagine it, but that's beside my point, which is that I think we don't need to imagine it. I think all we need to cognise the world is the bolded list of items in my previous post, and we get that from any Riemannian Manifold, whether flat or curved. When we use those things in navigating the world we can remain uncommitted as to whether the space is slightly curved or perfectly flat.
Now to reply to your specific question. You are right, it is weird to imagine. Here's a couple of ways:
1. Two spaceships set off on a journey, travelling initially parallel and starting 1km apart, going at the same speed and steering straight ahead. If the space is flat they will remain 1km apart. If not, they will subsequently measure that they are getting further apart if the space is hyperbolic, or closer if it is elliptic.
3. Set up three space stations 1, 2, 3 in deep space, each firing a laser beam at the next: 1 to 2, 2 to 3, 3 to 1. Each measures the angles between the incoming and outgoing beam. The stations are floating freely, not firing rockets to accelerate. The three angles will add to 180 degrees if the space is flat, less than that if hyperbolic and more than that if elliptic.
I will address this later when I have more time.
Quoting andrewk
I don't see how you're imagining anything. To imagine is to create a visual, tactile, or in any case sensory picture or image of intrinsic curvature in your mind. To imagine isn't to come up with some experiments that would prove or disprove the hypothesis.
To further illustrate what imagination is, when I imagine a curved line, I imagine that line curving right in front of my eyes. Basically I see what anyone would see if they were to draw a curved line on a piece of paper. So if someone asks me about extrinsic curvature, that's what it is - that's how you imagine it. Now show me that you can imagine intrinsic curvature in the same way.
If the rail line were stationary relative to Earth, the lines could not be both straight and parallel, because in that reference frame the spatial slices are curved. Since parallelness is necessary in order for the train to be able to run but straightness is not (trains can go around curves), we would have to give up straightness, rather than parallelness.
It may be useful to be clear what we mean by parallel. What I mean is that if we draw a straight line perpendicular to one track then it meets the other track at right angles.
Interestingly, if the track were in free fall towards Earth then it may be possible for the lines to be both straight and parallel. That's because, subject to a few other initial conditions being met, its reference frame could be the one I referred to earlier as one in which curved spacetime can have flat spatial slices. It would make the devil of a mess when it hit the Earth though.
http://faculty.poly.edu/~jbain/spacetime/lectures/13.Kant_and_Geometry.pdf
Say there are three poles, coloured red, yellow, blue, at distances of 3 1/3 steps away from one another, in a straight line in the direction I'm looking. There can't be more than three because the fourth pole would be where the first one is.
As I look along the line I see an infinite series of poles: red, yellow, blue, red, yellow, blue, etc. Next to every red pole is an image of me, seen from the back. The images of poles and of me diminish in size as they move along the line of vision, just as a series of poles beside a long, straight road does.
It would be somewhat similar to what one gets when one stands between two opposing mirrors, except that the view of myself would always be from behind. You may be interested in this essay I wrote about something like this - what happens when we point a TV camera at its monitor, inspired by a comment Alan Watts made in one of his talks. There are some pictures and videos in it that I find quite cool.
I can only repeat what I said above, that we don't need to imagine it. Cognising space as a Riemannian Manifold is not non-Euclidean, but aEuclidean (think of the difference between immoral and amoral). It is uncommitted as to whether the space may be curved, as long as it is not heavily curved.
I would call the experiments I described a way of 'imagining' a non-Euclidean 3D space. But I feel no need to argue if you don't consider that imagining.
See, I am tired of "reinterpretations" of Kant such as:
From here.
If this interpretation is correct, Kantianism is dead anyway. Schopenhauer would rightly find the notion of a space and time beyond perceptual space and time (the Euclidean ones) abhorrent to Kant's doctrine, and rightly so. If you ask me, such reinterpretations are pathetic, and they exist because people can't abandon a dead doctrine, and try to change it to fit the facts, when it really should be let go of.
Actually... I misread your solution initially. At least you seem to understand what the problem is. So here are my comments again:
Quoting Moliere
On what grounds do we judge a geometrical proposition to be a synthetic a priori?
Quoting Moliere
(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?
Quoting Moliere
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?
Quoting Moliere
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?
and
(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)?
To go back to your initial question, your solution doesn't appear like a cop-out, but there are a lot of things to flesh out.
We don't observe light rays or curvature of space in the way we see cells through a microscope, though. We observe other phenomena about which light rays and curvature of space are explanatory theories.
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".
So curvature of space, if it exists at all (recall the observation above that there exist coordinate systems (reference frames) within which entire regions of space are flat), cannot be directly observed even with extremely high tech equipment. One needs to be proficient in GR, and very patient and determined and have a lot of time on one's hands, even to do the calculations that might indicate a curvature.
I suggest that, if curvature of space cannot be directly observed, but only inferred from long, complex calculations that most people would not understand, it interferes with or invalidates our intuition of space in the TA not one whit.
Consider a line segment AB of length 1cm, with a line L1 going through point A at right angles to AB and another line L2 going through point B at right angles. The parallel postulate says that L1 and L2 never meet.
Now replace L2 by a line L3 through point B, that is at an angle that differs from 90 degrees by such a tiny amount that it intersects L1 at point C, a distance of a billion megaparsecs from AB.
Can you imagine triangle ABC? I can't. If we look at the AB end, what we see looks like one end of a rectangle. If we look at the C end, what we see looks like a single straight line. I cannot hold the whole triangle in my mind's eye.
The parallel postulate says that L2 meets L1 but L3 does not. But I cannot distinguish in my mind's eye between AB with L1 and L2 attached and AB with L1 and L3 attached.
So, for me, the parallel postulate is not something that can be visualised.
Another way of saying that is that 'infinity is a very long way'. It is such a long way that the difference between 'these two lines will never meet' and 'these two lines will meet at a point a billion megaparsecs away' is meaningless - to me at least. Of course I can do calculations with it that have different consequences for the two cases. But that is not visualising it, and I suggest it is not intuiting it either.
So I submit, your honour, that the parallel postulate is not intuitive.
But this is not what Kant held. He held that our intuition of space and time represented the world as it is experienced by us. The Euclidean conception of space is the conception of space of our everyday perception of the world. We do not perceive the space of the world as curved. Admittedly we don't perceive it as "straight" either; it is simply neutral. But neutral is straight, and in fact spacetime overall is Euclidean, it is curved only in the proximity of massive objects.
In any case your example of perspective effects is not apt, because our perceptual space in the comprehensive sense is not merely a single view from the ground. All the perspective effects cancel themselves out; if we look at the rail from one end it diminishes to the other; if we look at it from the other, it diminishes to the first, so we know the lines are parallel. The salient point is straightness in any case.
"What would Kant have made of non-Euclidan geomety?"
He'd have said it was obvious if you decide that triangles can exist across 3D space, or on the surface of spheres.
He was smart enough to realise that Euclid assumed 2D.
Move along now.... nothing to see here!
The parallel postulate is simply true by the law of non-contradiction. Lines which do not meet are parallel lines. Lines which are not parallel meet. Anything else would be contradictory.
The parallel postulate does not say what, based on your post, you appear to think it says.
So, say two lines met a billion parsecs away. In that case do they start out truly parallel? If they do, then at what point do they cease to be parallel? If they don't, then does that rule out the possibility of two lines being parallel tout court? If so, on account of what exactly would that be the case? So, take the case of a Globe like the Earth; we have a great circle at the Equator; is a circle precisely one meter north or south of that not concentric, and if it is would not the two lines qualify as parallel against an imaginary plane 90 degrees to the horizontal?
Of course there are no perfect circles, perfectly straight or parallel lines in nature, but that would seem to be an entirely separate issue.
Yes, it says just what I thought it says, I looked it up before I posted to make sure. It's all a matter of definition, and non-contradiction.
So if we want an intuitive parallel postulate, I imagine it would have to be something like:
and this postulate would be met by any space that is no more than very slightly curved, which would include our real world space.
A line exists by definition, it is a defined thing. You can't see what it looks like, nor can you even imagine it. I think this throws everyone off, you can only know what it is, by its definition. The principles of geometry are definitions which must be adhered to in creating geometric forms. In the case of the parallel postulate we have the definition of "line", and also some defined relations between lines. There is the defined relation of parallel, and the defined relation between parallel lines which intersect another line. The parallel postulate holds, (is valid) only if one respects the various definitions which support it.
The idea of a "plane", and the relationship between planes is something created by definition, it cannot be sensed nor can it be imagined. In common use, "intuited" means directly apprehended by the mind. This is how these principles are understood. The key thing is to "understand" them, because whether or not they are "true", and how they relate to the physical world is irrelevant to understanding them. They just need to be understood to be used in construction in the physical world. But the fact that they are useful in the physical world justifies the claim that there is some sort of "truth" to them.
Are you saying that Kant would have denied that any physics experiment could reveal that spacial geometry was not Euclidean?
Since we do not have, and cannot have access to infinity then the premise can only work at a mundane level. Since we can never know if the universe is infinite, or tell if it might or might not coalesce into a pinprick then it might be the case that all lines, including apparently parallel ones might at some point meet.
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?
As far as I'm aware Kant did not mention the geometry of the surface of spheres but in any case that is merely a subset within Euclidean space and as such would have no bearing on the form of our intuition.
No, the parallel lines never meet, it is impossible, because the definition indicates this. If they meet they are not parallel. The point being that we must accept the definitions and adhere to them whether or not there is any such thing as infinite lines, or parallel lines in the physical world.
What we see, sense, and even what we imagine in our minds, is completely different from what we know by the acceptance of definitions. However, we apply the things we know through acceptance of definitions, to the sense things of the physical world, by relating them, and if the definitions are useful this helps us in understanding and using the physical things in the sense world.
I am not saying he did.
I am saying that he could. Any child can see the difference between a flat triangle and one which is plastered on the side of a ball.
The parallel postulate does not define parallel lines. They are defined in Book I Definition 23, as being two lines that never meet.
What the parallel postulate does is assert that two lines that cross either end of a line segment at non-right angles are not parallel. It doesn't actually say 'not parallel' but rather gives a property that is equivalent to being not parallel.
So the postulate neither defines parallel lines, nor asserts that there are any. I presume the existence of at least one pair of parallel lines must be a theorem that is deduced from that and other postulates. Although, as I said much earlier, I believe that statement of the postulate is incomplete, and needs the words 'and only on that side' to be added. Otherwise the postulate does not exclude elliptic geometries, where all non-coincident pairs of lines meet in two places.
I disagree. We can set up experiments where we send a beam of light in a straight line passing by the sun and the set up detectors on the other side to see where the beam lands. If it lands not in a straight line, but in a curve, then we have seen the light rays bendings. We don't see the curvature of space, but the curvature of space is that which explains the bending of the light rays, just like atoms (which we don't see) are what explain phenomena such as brownian motion, which we do see. You are a pragmatist, so how did the Peirce go - the whole of the effects is the whole of the conception, or something of that sort, anyways.
Quoting Perplexed
Yes, I'm quite sure he would have. If he found out about non-Euclidean geometry, he would have tried to re-adjust his theories.
Well it's not strictly that the angles are "non-right angles", because a parallelogram need not have right angles. What is necessary is that the angles where parallel lines cross the line segment must be equal. If the angles are not equal then the two lines are not parallel, or as you say "equivalent to being not parallel". This must be accepted "as defined", and the definition of "parallel" must be accepted "as defined", in order for the parallel postulate to hold.
Quoting andrewk
The parallel postulate follows logically from accepting the definitions, and accepting the law of non-contradiction. It is produced from these definitions: the definition of parallel, that two parallel lines never meet, and the definition that if two parallel lines cross a line segment, they have equal angles. If you accept the definitions, the postulate holds. If you do not accept them, it does not hold.
That is a claim, not a definition. Observe how it does not say 'we define X to mean' or any of the equivalent forms of words that flag a definition.
The claim is only true if we adopt the parallel postulate or something equivalent to it.
Any definition can be stated as a claim. That two parallel lines do not meet can be said to be a claim. The so-called claim here, is derived from the relationship between different planes and this relationship is defined by angles, it is not claimed.
Edit: Planes only exist by definition.
I puzzled at the way you seem to view this idea.
All maths is a conceit. It's a means by which humans are able to describe space. There are no points, straight lines, nor perfect shapes in nature.
It's not that he 'did not know about it' because it had yet to be invented by Gauss, by the time Kant was DEAD.
Is it invented or discovered?
What do you mean by applied to the world?
Why would an empirical discovery (an a posteriori truth) impact an a priori one? If a priori truths are determined by a posteriori truths, then they're not known prior, but post experience, and therefore are defeated definitionaly, not empirically. The point being that an a priori truth need not comport to an a posteriori discovery. Neither type of truth is primary or of higher order. And by "higher order," I mean more consistent with reality, which is noumenal anyway.
No, this gives you the wrong idea of mathematics. If you go back 2000 years ago, the math we had back then was completely different from the math we have today. Math has developed over the centuries - parts of new developments were kept, others were thrown out. For example, some cultures have thought that zero was not a number. It was a huge conceptual breakthrough to introduce negative numbers! Imagine what is a negative quantity? Makes no sense. Imaginary numbers? Give me a break! And complex numbers too?! Trigonometry applied outside of triangles? Limits? Calculus? Etc.
These are all mathematical inventions - and these are only the small set of inventions that were accepted over time. But there are a whole host of other ones that were rejected. Of course, you don't take all those into account. Human beings have created math, and continue to create math - and they keep what is useful, and throw everything else away.
Now, you and I, are affected by a bias - Kant too. Namely, we look at all the math that has proven useful and we have kept, and we say "Ahh what an achievement! How was this at all possible?" But we forget all that we've thrown away along the way - all the math that didn't work. We also forget about the ridiculousness of some mathematical concepts like imaginary numbers, which are still used, to this day, and practically applied in physics and engineering. Imagine for a moment that you did not know about the state of mathematics, and someone started telling you about imaginary numbers! You'd kick him out. What about stuff like 0! = 1? n! = 1*2*3*...*n . So how does it make sense that 0! = 1! = 1? But it does! Again - these are just modes of behaving and thinking about things that have proven useful. We tend to celebrate successful "method" and ignore the productive disclosing that made it possible to begin with. We are blinded by success.
https://mathoverflow.net/questions/10124/the-factorial-of-1-2-3
Simple - the latter isn't a priori, it was only mistaken to be a priori.
Kant was good at thinking 'Copernican turns'. You might want to think about your question backwards?
Maths and the world is a dialogue. Whilst we invent maths, we draw our instances from the world.
But if you are yet to be convinced, please show me PI or any other irrational number in the world.
The world seems to be a round hole and maths is a square peg, as it relies on integers, which also do not exist. 1=1 might be true. But an orange is never equal to an orange.
Yes, this is exactly what I'm saying. I'm a bit uneasy about your use of "intuition" there though.
Alas - if a so-called a priori claims something that contradicts what an a posteriori reveals, then obviously the latter is true, and the former is false. You cannot have two contradictory statements hold as true with regards to the same subject matter at the same time.
Quoting Hanover
I don't understand the question?
Quoting 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.
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.
Thus far I've read these two articles to remind me of the finer points of the aesthetic -- though I am fine with cracking open the source material too if we need to. Regardless, these are both pretty darn good articles to read with respect to the original question:
https://plato.stanford.edu/entries/kant-mathematics/
https://plato.stanford.edu/entries/kant-spacetime/
@Janus and @Perplexed too for the above articles, though the rest of this is directed at Agustino.
Quoting Agustino
After reading the above articles I might make a modification, actually. I'm sort of toying between two ideas. 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).
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.
"Sensible" as in why does it make sense and is reasonable?
If so, then it would just be a matter of the fact that propositions are truth-apt. We can say "All intuitions are empirical", and that would be false (via Kant, at least). There is a subject, "is", and a predicate, and the category of "Allness" appended so it fits within the logic of Kant. And surely it is truth-apt, since Kant argues against the truth of the proposition.
Yes and yes.
"Subjectivity" understood within the context of Kantian philosophy, of course. Definitely a tricky word, but yes and yes.
Well, toying with the two ideas I talked of above...
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.
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.
I'd very much disagree with this assertion. 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.
Eh, bit of a side issue, but I would say that 3D space is not intuitively obvious to us.
Regardless, though, "intuition" is not the same as "intuitive" -- "intuition" is one half of the mind which operates differently from understanding, where "intuitive" is more akin to meaning obvious or easily comprehended without instruction or something along those lines.
My main point here is that "intuition" does not mean the same thing as "intuitive", and that is an understandable mistake, but a mistake all the same.
More or less, yes. Not sure about "out there" in your reply, so I say "more or less", but everything else is what I'd say (that Kant says, at least).
??? I don't mean to imply we have no synthetic a priori knowledge. Is it of the form of intuition? I'm not sure because things get funny when we are looking at the frame (is it knowledge, at that point? Or simply what we must accept in order that knowledge be possible? Is it really a cognition anymore?). But mathematical knowledge relies upon the form of intuition.
We do have that knowledge. 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. In a way you could argue we are learning about ourselves, but in a way that also doesn't make sense to say -- because it's not really about our identity or psychology but rather the possibility of knowledge.
I think in the above I answered these. Let me know if you disagree.
No worries about time. As you see I can take a bit of time to respond too. (sometimes too much time! Sometimes I run out of ideas, too...) Take as much time as you need.
I don't think your paraphrase is off. I also don't think the well-worn quotation is Kant at his most rigorous. I think perception is a good stepping-stone for getting a handle on the critical philosophy, but when speaking strictly 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.
I don't think so. Specifically because since the physical effects are demonstrated then they already fall within the form of intuition. "outside us" is not something that relies upon our intuitive perception of space, but rather "outside" -- meaning the noumenal world -- is outside of all possible knowledge. If we demonstrate that space behaves in accordance with one or another geometry then I'm inclined to say either 1) we have determined the "correct" geometry which conforms with the intuition, or 2) that both geometries are "checked against" the form of intuition, and hence both count as knowledge.
Thanks, I had a quick look and will read in more depth soon.
Quoting 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. 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.
Quoting 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.
Quoting Moliere
Yes.
Quoting 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".
Quoting Moliere
Why? Why does the fact that physics "rests" (what does that even mean?) on it guarantee it certainty?
Quoting Moliere
Can you explain how you can be wrong about your own subjectivity, and what you mean by that idea?
Quoting 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.
Quoting 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?
Quoting Moliere
So, if space is transcendentally ideal, then there is no noumenal space, correct?
Quoting Moliere
Ok.
Quoting 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?
Yes, true. I just mean how we categorize something, not what it is.
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.
So it would seem to me that any geometry should at least count as a priori simply because it's not something we come to by way of inferring from experience but is seated in the understanding, first.
What then makes it synthetic is that the propositions of geometry rely upon more than the principle of non-contradiction. There is something added to the subject. And how we do that, as humans, is through the understanding being used to judge our form of intuition (in the case of geometric propositions, that is)
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.
And it's that fact that the very form of our intuition is what we are coming to know which then guarantees certainty of how objects relate to one another and also how they react. Since it is the form of our intuition that we are learning about it applies to everything within that form. Hence we can also be certain of our knowledge.
Whether some set of propositions is incomplete doesn't really influence whether or not it is a priori. And it is fair to say that Euclidean geometry is an approximation of space, if we are to take non-Euclidean geometry as the sort of true representation of space. Newtonian mechanics, after all, still work within certain parameters -- we just weren't always aware of what those parameters were.
Quoting Agustino
What is contradictory in them? Perhaps we are wrong in thinking that.
Quoting Agustino
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).
Quoting Agustino
Because mathematical knowledge and scientific knowledge are closely tied together. And isn't a guarantee just another way of expressing that we are certain?
Apodeictic certainty just means that a proposition must be true to the consciousness who holds it to be true.
So if we have one kind of geometry which predicts physical phenomena then we have a reason for holding that it must be true. It is universal and necessary.
In fact, if we have two kinds of geometry, and one geometry does a reasonably good job of predicting some phenomena, but the other geometry predicts all phenomena then we know which one is universal and necessary, and which one is merely necessary (and therefore empirical, i.e., not a priori).
It's not that math and physics are identical to one another, but it is our knowledge of math which guarantees the physics. If it turns out that our physics needs different maths then we were merely approximating in our first estimations the form of the intuition.
Quoting Agustino
What exactly are you after here?
I mean, it doesn't strike me as controversial at all. Especially at the level of abstraction that Kant is working at. To use your question later on -- why was Aristotle wrong about the categories? Why did it take so long for someone to formulate the critical philosophy and identify transcendental errors?
It's because we're stupid, on the whole. Humanity can perform feats of intellectual might which are very impressive. But, generally speaking, we aren't all that smart and we make mistakes and we believe false things all the time.
Quoting Agustino
Yes he does but it doesn't mean the same thing, exactly, either. Mathematical propositions must be true -- therefore they are certain. There is no more to it than that.
Quoting Agustino
Why on earth would it not be possible to be wrong?
The relationship, as I see it, is one of flow. I see the understanding as flowing down and the intuition as flowing up. In the middle is the schemata, which connects the two different parts of the mind -- kind of like a baptism of the categories into the forms of intuition. The categories are distinct still, of course, it is only the schemata which is the union between the two.
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.
Why exactly do you think we can't be wrong? Simply because the knowledge is universal and necessary?
In Kant's time, Euclidean geometry was both of those things. It wasn't until we developed different physics, different geometries, that Euclidean geometry -- as the mathematical model of physics (plus other maths, while we're at it) -- were seen to be approximations.
Quoting Agustino
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.
Quoting Agustino
Sort of a repeat question again, but why shouldn't it take us so long to discover non-Euclidean geometry?
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.)
Also: certainty is definitely not as central in Kant's philosophy as it is in Descartes'.
I don't think the intuition knows anything. Knowledge is not generated without both parts of the mind.
Quoting Moliere
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?
We determine if a proposition is a priori if it is universal and necessary. Universality applies across all space and time. Necessity is a category which modifies the copula in the logic. It is a modal category which deals with how we assent to some statement. Those three categories are problematic, assertoric, or necessary. Necessity does deal with certainty. But certainty is holding-to-be-true for everyone, i.e., objective. That's the meaning of certainty in Kant -- it's more about intersubjective agreement, and our attitude towards a proposition than anything else.
So, in Kant's time the propositions of Euclid were universal and necessary, hence a priori. Furthermore, the subject and the predicate of those axioms were synthetic in that the predicate was not contained within the subject.
Now, we may be able to still say they are universal and necessary, which is the idea I'm sort of toying with. But my original thought was that since we have non-Euclidean geometry which is universal, and now know that Euclidean geometry is not universal because of that, then Euclidean geometry would not have the sufficient conditions for being a priori -- it would still be necessary, but not universal. And we would know that because we have a geometry to compare it to which is universal.
The first approximated the second geometry.
Since they were so close we simply missed it. Plus, we developed other means of probing the shape of space which were not available in Kant's time -- without which we wouldn't have noticed the difference.
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.
"Now, in the first place, if we have a proposition which contains the idea of necessity in its very conception, it is a if, moreover, it is not derived from any other proposition, unless from one equally involving the idea of necessity, it is absolutely priori [...] Necessity and strict universality, therefore, are infallible tests for distinguishing pure from empirical knowledge [...] But as in the use of these criteria the empirical limitation is sometimes more easily detected than the contingency of the judgement, or the unlimited universality which we attach to a judgement is often a more convincing proof than its necessity, it may be advisable to use the criteria separately, each being by itself infallible."
Does it mean that it is a posteriori if we have to "probe the shape of space" for it? :s
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 )?
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.
So then this is just about categorising statements, not about how things really are?
Quoting Moliere
The correct answer would be due to universality and/or necessity according to Kant I think.
Quoting 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?
If we are wrong, it seems to follow as the night follows day that they were not universal and necessary. And yet, lo and behold, we can be mistaken about statements being universal and necessary. But we determine that they are universal and necessary by appealing to our intuition. So then our intuition must be wrong. Or we must have appealed to something other than our intuition, when we thought we were appealing to our intuition.
Quoting 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.
Quoting 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.
Quoting Moliere
Ehmmm X-) - I don't think he was.
Quoting 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.
Quoting Moliere
Yes.
Quoting 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).
Quoting 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.
This is the standard and, I might add, Schopenhauerian criticism of Kant: that he claimed the noumenon causes the phenomenon. But this is an error so obvious and egregious that it offends the principle of charity greatly to assume a philosopher of Kant's caliber wasn't aware of it. I think he was and that the contradiction you impute of him here is only apparent. 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.
Haven't looked at the rest of the thread, as it's beating a very dead horse. I've said my piece about space.
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!).
The way I see it, that either means that:
1. at least one of us is wrong about its necessity; or
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.
In my way of thinking there is perception, (animals also perceive) and then there is understanding; which includes various kinds of understandings: commonsense, practical, scientific, geometrical, mathematical, statistical, dialectical, and so on. This seems to be consonant with Kant's scheme if the following diagram from the SEP article you linked, which deals with Kant's views about space and time, is any indication.
Of course. all these terms can be equivocal.
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.
Intuitions are inherently subjective. One intuition says Euclid's fifth is correct, another says it is not.
Quoting Agustino
Take a long hard look at the way Kant uses "necessity". You know the principal use of "necessary" is the use which implies "needed for", and I think Kant's "necessity" is a derivative of this. So if in some geometries it is necessary for Euclid's 5th to be true, and in other geometries it is necessary that it is not true, there is no real contradiction, just like one logical possibility doesn't contradict the opposite possibility. The different geometries are simply based in different intuitions.
Yup, I agree with your reading of Kant. I hadn't thought of your argument though. I suppose I missed your point, originally, but I think I understand you now.
I think the tendency is to think of them in conflict because of the universal nature of space, and how we come to know about space through Euclidean geometry. That's what I was thinking, at least. It's universal and so either one or the other must apply.
But now I'm sort of wondering if Euclid, though it was one of Kant's primary examples of synthetic a priori knowledge, could be taken as just that -- an example. And perhaps both Euclidean space and non-Euclidean space could be thought to apply universally. That is, any geometry could count as knowledge, even universally, insofar that we are at least applying it to the form of intuition (so that it's not just analytic and true by definition, but true by virtue of a concept matching its object).
I'd be hesitant to say spacetime just doesn't apply. But I hear you when you are saying that because it is an empirical model it simply wouldn't be the space that Kant is talking about. That makes perfect sense to me.
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?
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.
Quoting Agustino
So? How does that have anything to do with the modality of the copula, in Kant's logic?
I think your use of "necessary" differs from Kant's use. Even in the introduction you quoted Kant downplays necessity stating that universality was the more impressive proof -- that necessity follows from universality. I agree that he links the two there, but if we're talking about "saving" the Kantian system then I don't think we'd have to hold that link. Though maybe so. It's been awhile since I've read the text, so I'm not sure.
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.
Quoting Agustino
Yes, we could.
Yes, it can.
This is why I think you're harboring some Cartesian sympathies here. 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'. Truth is when some concept matches its object. Necessity is about the quality some judgment has -- so we judge a proposition as necessary when it is objective, i.e., it holds for everyone.
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.
Quoting Agustino
That's what I was doing in that exchange, yes. I was categorizing statements.
Quoting Agustino
They turn out to be false. It's something of a cheeky answer, but really I don't think it goes much more in depth than that. Some people claimed to square the triangle, in ages past. That was false too.
Quoting Agustino
Maybe we are dealing with a limited situation in this case. So both can be true, if not necessary and universal.
Quoting Agustino
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.
Where?
In a multitude of PMs to Agustino and in the big thread he made a while back. My current thought is this, I suppose: 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.
I'm not a very sophisticated math person, but my understanding is that they evolved as investigations of what would obtain if you curved the usual two-dimensional plane of Euclidean geometry. It seems the results are intuitively obvious (graphically derived, I guess) in the context of hyperbolic, elliptical and spherical surfaces.
I guess non-Euclidean geometries are universally possible, just as the shapes that yield the various curves are possible shapes in three-dimensional space. I don't know, though; I'm not confident that that is not a pretty dumb thing to say, given my lack of expertise in the subject.
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.
Ok.
Quoting Thorongil
I don't think so. At minimum, I think you should read my exchange with Moliere, starting from here:
https://thephilosophyforum.com/discussion/comment/152158
Non-Euclidean geometry strikes at the very possibility of the Transcendental Aesthetic.
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.
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?
Quoting Moliere
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.
Quoting Moliere
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.
Quoting Moliere
Quoting Moliere
I asked you to tell me how that is possible though.
Quoting Moliere
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?
Quoting Moliere
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.
Quoting Moliere
Sure.
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.
Quoting andrewk
: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.
It is touched for the Kantian if the transcendental aesthetic falls apart. For the Platonist, yes, it does remain untouched.
Quoting Thorongil
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.
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?
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.
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.
https://www.degruyter.com/view/j/kant.1970.61.issue-1-4/kant.1970.61.1-4.5/kant.1970.61.1-4.5.xml
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:
Quoting Agustino
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"?
Quoting Thorongil
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 ;)
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:
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).
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.
Quoting andrewk
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).
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.
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.
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.
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.
Quoting Agustino
True, but that's a pretty big "if" in my opinion.
http://www.socsci.uci.edu/~jheis/bio/Heis,%20K%20Parallels%20HOPOS.pdf