Why Kant was completely right about space
Why Kant was completely right about space
Helge Malmgren
Emeritus professor in theoretical philosophy
University of Gothenburg
Helge Malmgren
Emeritus professor in theoretical philosophy
University of Gothenburg NNPS 2017
Copenhagen
Plan for this lecture Plan for this lecture
What did Kant say about space?
Which are the standard objections to his view?
Why all the standard objections are wrong
Did Kant exclude other geometries than that of Euclid?
The subjectivity of space
Does Kant’s apriori conflict with empirical data?
What did Kant say about space?
Which are the standard objections to his view?
Why all the standard objections are wrong
Did Kant exclude other geometries than that of Euclid?
The subjectivity of space
Does Kant’s apriori conflict with empirical data?
Why Euclidean space is a necessary condition for all empirical knowledge
What is spatial intuition?
What are the consequences of this necessity for the necessity of the synthetic apriori?
Why Euclidean space is a necessary condition for all empirical knowledge
What is spatial intuition?
What are the consequences of this necessity for the necessity of the synthetic apriori?
What did Kant say about space?What did Kant say about space?
Space does not belong to the world of things-in-Space does not belong to the world of things-in-
themselves, but is contributed by our understanding themselves, but is contributed by our understanding ((VerstandVerstand))
Space is the form of intuition (Space is the form of intuition (AnschauungsformAnschauungsform) of outer ) of outer sense (
sense (Sinnlichkeit,Sinnlichkeit, sinnlighet) sinnlighet)
The axioms of Euclid are synthetic, but still necessaryThe axioms of Euclid are synthetic, but still necessary
They are necessary, because Euclidean space is a They are necessary, because Euclidean space is a necessary condition for all empirical knowledge necessary condition for all empirical knowledge
Spatial intuition is not itself a kind of knowledge. Neither Spatial intuition is not itself a kind of knowledge. Neither is it conceptual in nature, since it is prior to all concepts is it conceptual in nature, since it is prior to all concepts
as well as to knowledge. Neither is it an experience.
as well as to knowledge. Neither is it an experience.
(So what, then, (So what, then, is is spatial intuition?) spatial intuition?)
Which are the standard objections to Kant’s view?Which are the standard objections to Kant’s view?
Kant’s theory makes him a subjectivist and idealist about
space because according to it, propositions concerning spatial relations are made true by facts about our own consciousness
It also makes him anti-empirical, in the sense that putative a priori knowledge is allowed to override empirical evidence
Kant did not recognize the existence of non-Euclidean geometries, or even their possibility
The existence of consistent non-Euclidean geometries shows that Euclid’s axioms are not necessary
Kant’s theory of space is not compatible with Einstein’s
theories of relativity, which offer the true story about space
Kant’s theory makes him a subjectivist and idealist about
space because according to it, propositions concerning spatial relations are made true by facts about our own consciousness
It also makes him anti-empirical, in the sense that putative a priori knowledge is allowed to override empirical evidence
Kant did not recognize the existence of non-Euclidean geometries, or even their possibility
The existence of consistent non-Euclidean geometries shows that Euclid’s axioms are not necessary
Kant’s theory of space is not compatible with Einstein’s
theories of relativity, which offer the true story about space
Why are the standard objections wrong, 1:
Did Kant exclude other geometries than that of Euclid?
Why are the standard objections wrong, 1:
Did Kant exclude other geometries than that of Euclid?
By classifying our knowledge of Euclidean geometry as
synthetic apriori knowledge, Kant excluded the possibility that Euclidean geometry is true on logical or conceptual grounds
In other words, Kant did not hold that the axioms of Euclid are logically or semantically necessary
Hence, Kant’s theory does allow for the logical and conceptual possibility of non-Euclidean geometries
By the same token, the existence of consistent non-Euclidean geometries does not show that Kant was wrong in saying that the axioms of Euclid are necessary (in his sense of that term)
By classifying our knowledge of Euclidean geometry as
synthetic apriori knowledge, Kant excluded the possibility that Euclidean geometry is true on logical or conceptual grounds
In other words, Kant did not hold that the axioms of Euclid are logically or semantically necessary
Hence, Kant’s theory does allow for the logical and conceptual possibility of non-Euclidean geometries
By the same token, the existence of consistent non-Euclidean geometries does not show that Kant was wrong in saying that the axioms of Euclid are necessary (in his sense of that term)
Why are the standard objections wrong, 2:
The subjectivity of space
Why are the standard objections wrong, 2:
The subjectivity of space
Kant’s empiricist adversaries (Reichenbach 1927, Nagel 1960) seem to agree with him on the subjectivity of space.
Reichenbach argues that the Euclidean axioms are not the only ones that can be used to describe physical space.
He concludes that Kant is therefore wrong in holding that these axioms are necessary.
But his argument involves admitting that Euclidian geometry is among the geometries that can be used to describe actual
physical space – even considering Einstein’s discoveries
Reichenbach and Nagel also hold that the choice between geometries is a matter of convenience.
The physical laws get much more complicated if you choose the Euclidean alternative, but physics does not contradict Euclid.
Hence, choosing an Euclidean geometry would not mean letting apriori propositions override empirical observations
Kant’s empiricist adversaries (Reichenbach 1927, Nagel 1960) seem to agree with him on the subjectivity of space.
Reichenbach argues that the Euclidean axioms are not the only ones that can be used to describe physical space.
He concludes that Kant is therefore wrong in holding that these axioms are necessary.
But his argument involves admitting that Euclidian geometry is among the geometries that can be used to describe actual
physical space – even considering Einstein’s discoveries
Reichenbach and Nagel also hold that the choice between geometries is a matter of convenience.
The physical laws get much more complicated if you choose the Euclidean alternative, but physics does not contradict Euclid.
Hence, choosing an Euclidean geometry would not mean letting apriori propositions override empirical observations
Choosing between geometriesChoosing between geometries
• When light from a distant star passes near a celestial body with a considerable mass (such as the sun), it seems to bend
• I say ”seems to bend”, since contemporary physics instead says that the light rays follow straight lines, but space is curved.
• But the popular way of describing the situation is not wrong, only inconvenient for the physicist (since it involves postulating a universal force acting at a distance: gravitation).
• None of the alternative geometries is the ”true” system.
• When light from a distant star passes near a celestial body with a considerable mass (such as the sun), it seems to bend
• I say ”seems to bend”, since contemporary physics instead says that the light rays follow straight lines, but space is curved.
• But the popular way of describing the situation is not wrong, only inconvenient for the physicist (since it involves postulating a universal force acting at a distance: gravitation).
• None of the alternative geometries is the ”true” system.
How to handle bent light (a.k.a. curved space)How to handle bent light (a.k.a. curved space)
Don’t bother about
Einstein, but take out a course towards where you know the star is (1), and go to warp speed!
A more mundane example: consider a
heron that must catch a fish in spite of the light being bent at the surface.
Staying with an Euclidean geometry means aiming directly at the place
where it knows that the fish is located.
Don’t bother about
Einstein, but take out a course towards where you know the star is (1), and go to warp speed!
A more mundane example: consider a
heron that must catch a fish in spite of the light being bent at the surface.
Staying with an Euclidean geometry means aiming directly at the place
where it knows that the fish is located.
What does it mean to ”use” a geometry?
(Or: what is spatial intuition?)
What does it mean to ”use” a geometry?
(Or: what is spatial intuition?)
Geometrical statements have an empirical meaning only in combination with coordinative definitions (Reichenbach)
In physics, the coordinative definition of length is in terms of rigid bodies that are assumed not to change length when transported
An empirical geometry can also be determined by general constraints such as the non-existence of universal forces ( space is curved)⇒
Or by the conditions that certain geometrical statements, for example those of Euclid, shall hold ( space is not curved)⇒
But in the latter case, are there any corresponding coordinative definitions? If not, ”Euclid’s geometry holds” is empirically empty
In my interpretation, spatial intuition offers the needed definitions, and therebye ties geometry to experience
Spatial intuition should be identified with personal action space
Spatial intuition is not specifically tied to vision, and the translation of Anschauungsform as ”visualisation” (Reichenbach) is extremely misleading. ((For a similar confusion cf. the term ”visuospatial”.))
Geometrical statements have an empirical meaning only in combination with coordinative definitions (Reichenbach)
In physics, the coordinative definition of length is in terms of rigid bodies that are assumed not to change length when transported
An empirical geometry can also be determined by general constraints such as the non-existence of universal forces ( space is curved)⇒
Or by the conditions that certain geometrical statements, for example those of Euclid, shall hold ( space is not curved)⇒
But in the latter case, are there any corresponding coordinative definitions? If not, ”Euclid’s geometry holds” is empirically empty
In my interpretation, spatial intuition offers the needed definitions, and therebye ties geometry to experience
Spatial intuition should be identified with personal action space
Spatial intuition is not specifically tied to vision, and the translation of Anschauungsform as ”visualisation” (Reichenbach) is extremely misleading. ((For a similar confusion cf. the term ”visuospatial”.))
Why Euclidean space is a necessary condition for all empirical knowledge (1)
Why Euclidean space is a necessary condition for all empirical knowledge (1)
All empirical knowledge is ultimately based on direct observation
If a direct observation gives rise to the hypothesis that the object a has property P, this hypothesis can be verified by observation only if one can re-identify a
More generally, for an observation of a to give knowledge that a is P, as distinct from that b is P, c is P etc., one must know that what one observes is a, and is not b, c etc.
If a direct observation gives rise to the hypothesis that the object a has property P, and no property of a except its location is known, this hypothesis can be verified by another observation only if one can re-identify a by means of a direct identification of its spatio- temporal location. (In the following, ”temporal” is disregarded!)
More generally, for an observation of a to give knowledge that a is P, as distinct from that b is P, c is P etc., when none of a:s properties except its location are known, one must identify a as a by means of a direct identification of its spatial location.
All empirical knowledge is ultimately based on direct observation
If a direct observation gives rise to the hypothesis that the object a has property P, this hypothesis can be verified by observation only if one can re-identify a
More generally, for an observation of a to give knowledge that a is P, as distinct from that b is P, c is P etc., one must know that what one observes is a, and is not b, c etc.
If a direct observation gives rise to the hypothesis that the object a has property P, and no property of a except its location is known, this hypothesis can be verified by another observation only if one can re-identify a by means of a direct identification of its spatio- temporal location. (In the following, ”temporal” is disregarded!)
More generally, for an observation of a to give knowledge that a is P, as distinct from that b is P, c is P etc., when none of a:s properties except its location are known, one must identify a as a by means of a direct identification of its spatial location.
The nature of spatial intuition The nature of spatial intuition
Human beings (and many other animals) have an inbuilt very reliable spatial metric for length and angle that is independent of sensory input, and that can therefore be used to re-identify objects
independently of their non-spatial properties
This metric is used when we perform so-called ”ballistic” and ”semi- ballistic” movements. Example: the fast grabbing of an object on the other side of a fragile vase, with eyes shut. (And the heron’s strike.)
Suppose, for simplicity’s sake, that in any situation we can either take a step forwards, or turn 90 degrees either clockwise or anti- clockwise. The step length is changeable at will, but fixed in the sense of being pre-determined for each occurrence of a movement.
Further, we keep track of our movements for a long time.
Using the supposed capacities, we can knowingly re-visit all places that we have visited lately (= not too late for our memory span)
Hence we can re-identify all objects that we have observed during this time, without relying on any non-spatial properties
Human beings (and many other animals) have an inbuilt very reliable spatial metric for length and angle that is independent of sensory input, and that can therefore be used to re-identify objects
independently of their non-spatial properties
This metric is used when we perform so-called ”ballistic” and ”semi- ballistic” movements. Example: the fast grabbing of an object on the other side of a fragile vase, with eyes shut. (And the heron’s strike.)
Suppose, for simplicity’s sake, that in any situation we can either take a step forwards, or turn 90 degrees either clockwise or anti- clockwise. The step length is changeable at will, but fixed in the sense of being pre-determined for each occurrence of a movement.
Further, we keep track of our movements for a long time.
Using the supposed capacities, we can knowingly re-visit all places that we have visited lately (= not too late for our memory span)
Hence we can re-identify all objects that we have observed during this time, without relying on any non-spatial properties
Why Euclidean space is a necessary condition for all empirical knowledge (3)
Why Euclidean space is a necessary condition for all empirical knowledge (3)
Re-visiting a point via a detour in non-Euclidean space using a fixed angle of the turns (such as 90 degrees) will not work, because the angles of the composite path that you walk will not sum up to the intended result.
In a spherical geometry with constant curvature, the angles will be too small.
The walk will be even more problematic in a geometry with irregular ”bumps” and ”sinks”.
Re-visiting a point via a detour in non-Euclidean space using a fixed angle of the turns (such as 90 degrees) will not work, because the angles of the composite path that you walk will not sum up to the intended result.
In a spherical geometry with constant curvature, the angles will be too small.
The walk will be even more problematic in a geometry with irregular ”bumps” and ”sinks”.
Source: Wikimedia Commons
What I have not said What I have not said
My argument is not the common one that we can use Euclidean geometry for object re-identification in everyday life, because the actual curvature of close space is negligible and the
Euclidean system therefore a good approximation of the truth.
This is not my argument because
1) there is no true geometrical system and therefore no
”actual” curvature, and
2) the Euclidean system can be used even in regions of space that the non-Euclidean would describe as having a strong
curvature. All points in such a region can, in principle, be visited and re-visited using the proposed inbuilt ability.
Another mundane example. Suppose that you want to leave a hall of mirrors through its only door as fast as possible, in spite of a lot of confusing mirror images suggesting an irregularly curved space. If you remember how you walked from the door to where you are, you can simply close your eyes and go back to the door without caring about the non-Euclidic suggestions.
My argument is not the common one that we can use Euclidean geometry for object re-identification in everyday life, because the actual curvature of close space is negligible and the
Euclidean system therefore a good approximation of the truth.
This is not my argument because
1) there is no true geometrical system and therefore no
”actual” curvature, and
2) the Euclidean system can be used even in regions of space that the non-Euclidean would describe as having a strong
curvature. All points in such a region can, in principle, be visited and re-visited using the proposed inbuilt ability.
Another mundane example. Suppose that you want to leave a hall of mirrors through its only door as fast as possible, in spite of a lot of confusing mirror images suggesting an irregularly curved space. If you remember how you walked from the door to where you are, you can simply close your eyes and go back to the door without caring about the non-Euclidic suggestions.
The nature of the synthetic a prioriThe nature of the synthetic a priori
Euclid’s geometry is not a theory that is true or false because the world is the way it is, but a framework for describing the world that we use because it fits our epistemic purposes
Neither are the propositions of Euclid analytical statements about the relations between axioms and theorems
So they do not describe anything factual, nor anything logical
Instead they express
1) our fundamental choice of framework and
2) logical consequences of that choice that we must, consequentially, adapt to
In other words, they do not state any Laws of Nature, but formulate our own Laws for describing Nature.
The statements of Euclid’s geometry are normative in nature.
In this, they are like all other statements of applied mathematics.
Euclid’s geometry is not a theory that is true or false because the world is the way it is, but a framework for describing the world that we use because it fits our epistemic purposes
Neither are the propositions of Euclid analytical statements about the relations between axioms and theorems
So they do not describe anything factual, nor anything logical
Instead they express
1) our fundamental choice of framework and
2) logical consequences of that choice that we must, consequentially, adapt to
In other words, they do not state any Laws of Nature, but formulate our own Laws for describing Nature.
The statements of Euclid’s geometry are normative in nature.
In this, they are like all other statements of applied mathematics.
Thank you for your attentionThank you for your attention1,2,3,41,2,3,4!!
1) Note that attention is an internalised version of a prototypical ballistic reach for an object.
2) Note that Euclid held an emission theory of vision.
3) Many thanks also to all who commented on earlier versions of this text. You will be given full credit in the definitive version of the paper.
4) (Note added in March 2020) A few very minor edits have been performed on pp.
8 and 12 to fix two copyright issues and to improve clarity.
1) Note that attention is an internalised version of a prototypical ballistic reach for an object.
2) Note that Euclid held an emission theory of vision.
3) Many thanks also to all who commented on earlier versions of this text. You will be given full credit in the definitive version of the paper.
4) (Note added in March 2020) A few very minor edits have been performed on pp.
8 and 12 to fix two copyright issues and to improve clarity.
