Kant on Space
Pinhas Ben-Zvi thinks Kant was inconsistent in his revolutionary ideas about the nature of space and time.
In the first and second editions of his Critique of Pure Reason (A&B) Immanuel Kant asks: “What, then, are space and time? Are they real existences? Are they only determinations or relations of things, yet such as would belong to things even if they were not intuited?” (A23; B37).
At the time when he wrote that, conflicting theories of space dominated the scientific and philosophical world.
Sir Isaac Newton's averred an absolute and real space in the sense of Euclidean geometry. According to Newton, space was a self-subsistent reality, a container inside which all objects are placed; it was “God's boundless uniform sensorium.”
The expression ‘God's sensorium' added an unnecessary theological aspect to Newton's comprehensive theory and consequently attracted some criticisms like the one from Leibniz in his debate with Clarke. In an exchange of letters during 1715-1716, in which Clark defended Newton's view, Leibniz wrote that God does not need a ‘sense organ' to perceive objects. Leibniz argued that space is merely relations between objects and is not a self-subsistent reality. He rejected: “… the fancy of those who take space to be a substance, or at least an absolute being,” and added ironically that: “…real and absolute space (is) an idol of some modern Englishmen.” The ‘modern Englishmen' are of course Newton and his adherents.
Kant, in the quotation with which I began this article, refers to the Newtonian concept as the ‘real existences' view, and to the Leibnizian concept as the view according to which space is: “only determinations or relations of things.”
In his early writings Kant sided with Leibniz and his relational space. Later on he had a change of heart and in his dissertation ‘Regions of Space' (1768), he embraced the Newtonian view. This dissertation was Kant's last before he dissociated himself from both views and formulated his own perspective on space (and time). This was a result of what he called a ‘Copernican revolution' in his thinking. The revolution is described by Kant as follows:
“Up to now it has been assumed that all our cognition must conform to the objects; but all attempts to find out something about them a priori through concepts that would extend our cognition have, on this pre-supposition, come to nothing. Hence let us once try whether we do not get further with the problem of metaphysics by assuming that the objects must conform to our cognition, which would agree better with the requested possibility of an a priori cognition of them, which is to establish something about objects before they are given to us. This would be just like the first thought of Copernicus, who, when he did not make good progress in the explanation of the celestial motions if he assumed that the entire celestial host revolves around the observer, tried to see if it might not have greater success if he made the the observer revolve and left the stars at rest” (Preface to Critique of Pure Reason B/XVI)
In consequence of this revolutionary assertion Kant states that: “Space is not an empirical concept which has been derived from outer experiences.” (B/38) On the contrary: “…it is the subjective condition of sensibility, under which alone outer intuition is possible for us.” (A/26; B/42)
In other words, Kant asserts that space (and time) are not objective, self-subsisting realities, but subjective requirements of our human sensory-cognitive faculties to which all things must conform. Space and time serve as indispensable tools that arrange and systemize the images of the objects imported by our sensory organs. The raw data supplied by our eyes and ears would be useless if our minds didn't have space and time to make sense of it all.
He further argues that this sensory-spatiotemporal process requires a supreme mediator that will synthesize the sensory input within our cognition so as to turn it into meaningful knowledge.
Kant called this supreme synthesizer the Categories. There are twelve Categories such as Unity (This A is B), Limitation (A tree is not a human being), Relation (If A then B), Possibility/Impossibility (A may be B), and eight others. Kant divides the twelve into four groups of three Categories each.
He characterised the Categories as being: “…concepts of an object in general, by means of which the intuition of an object is regarded as determined in respect of one of the logical functions of judgement.” (B/128)
One commentator likened Kant's view of space and time to a scale used by a land surveyor. After he has drawn the scale on a board, the surveyor adds the results of his measurements which, without the pre-drawn scale, would lack any meaning.
Kant further asserts that since we are dependent on our limited senses, all we can know is the way things appear as they are represented to us through our senses and cognition. The world of ‘things-in-themselves' is beyond the reach of our sensory-cognitive faculties and hence can not be known to us.
Appearances in Kantian language are called ‘Phenomena' and ‘things-in-themselves,' are called ‘Noumena'.
To support his theory, Kant gave several arguments. The fourth is based on the admitted validity of Geometry which forms the bedrock for his proof of the properties of space.
This can be inferred from his statement that: “The apodeictic certainty of all geometrical propositions and the possibility of their a priori construction is grounded in this a priori necessity of space.” (B/39), and: “Geometry is a science which determines the properties of space synthetically, and yet a priori,”(B/40).
The Trouble with Geometry
Kant's view of space (and time) is the groundwork of his Critique, However the inseparable bond he claimed between geometry and the nature of space serves to undermine his case rather than support it. The following arguments question the validity of Kant's linkage between geometry and space; I will try to show that it is inconsistent with his assertions about space.
When Kant refers to geometry, he must mean Euclidean geometry, since Non-Euclidean geometry, the brainchild of the 19th Century, was unknown to him. Hence space, in Kant's philosophical system must conform to Euclidean geometry. Norman Kemp Smith, in his Commentary on the Critique, remarked that for Kant “…space in order to be space at all, must be Euclidean.”
Space, in Euclidean Geometry, is a concept which is independent of the attributes of our human minds and senses. The word Geometry is derived from Greek – geo “earth”, and metron “to measure”, namely “earth measurement”. With such semantic-conceptual roots its hardly conceivable that Euclid regarded Geometry as divorced from an objective independent space.
Euclid's conception of space is revealed, for instance, in Proposition 16, Book III of his Elements where he states that:
“The straight line drawn at right angles to the diameter of a circle from its end will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed, further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilinear angle.
Corollary. From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its end touches the circle.”
Following this statement Euclid proves it:
“I say next that into the space between the straight line AE and the circumference CHA another straight line cannot be interposed.”
(Euclid concludes his proof with his usual Greek phrase meaning “which was to be demonstrated”. This phrase in the Latin translation of the Elements, is given as Quod Erat Demonstrandum, better known by its abbreviation - Q.E.D.)
Euclidean geometry deals with space when describing three-dimensional figures such as spheres, cylinders and cones. If we eliminate from geometry the property of space, it becomes a concept as meaningless as the concept of an non-angular triangle.
Kant, in a different context, indeed observes that: “To posit a triangle, and yet to reject its three angles, is self-contradictory.” (A/594; B/622).
Furthermore, Kant remarks that the perceiving of space (and time) is unique to us:
“We know nothing but our mode of perceiving them – a mode which is peculiar to us, and not necessarily shared in by every being, though, certainly, by every human being.” (B/59).
If the mode of perceiving space is ‘peculiar to us' as ‘human beings', then it may be different to ‘Other Beings' such as the light dove who while: “… cleaving the air in her free flight, and feeling its resistance, might imagine that its flight would be still easier in empty space.” (A/5; B/8)
In the light of Kant's own observations it is permissible to argue that to ‘Other Beings', like Kant's ‘light dove', space may be real and not just a mode of perceiving.
The Unthinkable Absence of Space
Kant presents an additional proof of his subjective space argument:
“We can never represent to ourselves the absence of space, though we can quite well think it as empty of objects. It must therefore be regarded as the condition of the possibility of appearances, and not as a determination dependent upon them.” (B/38-39).
The statement that we are powerless to “represent to ourselves the absence of space” is however not so self-evident as to require no proof. Indeed the controversy as to whether the absence of space is conceivable, can be traced right back to Pre-Socratic philosophy.
Parmenides of Elea, in the early 5th Century BC, asserted that only the ‘Is' is, whilst to speak of the ‘Is not' is to take a “…wholly incredible course, since you cannot recognise Not Being (for this is impossible), nor could you speak of it, for thought and Being are the same thing.”
(The resemblance between Kant's statement about the inability to ‘represent' the absence of space and Parmenides' statement about the inability to ‘think' of the ‘Is Not', is quite interesting.)
Parmenides' concept, embraced by his disciples – the Eleatics – is considered to be a refutation of the teachings of his predecessor – Pythagoras – who claimed that a kind of Non-Being does indeed exist.
Other pre-Socratics, such as Democritus of Abdera, the most prominent of the atomist scholars, and one who wrote and taught some decades after Parmenides, also insisted, like Pythagoras, that Non-Being must in fact exist, in spite of Parmenides' rigorous logic.
Plato, in his dialogue The Sophist, argues that what ‘is not' in some sense also ‘is', refuting Parmenides' concept of the impossibility of the Non-Being to exist. Non-being is just a being characterised only by its difference from ‘another' being. He asserted that the antinomy between Being and Non-Being is false. The only real antinomy is that of a single object of consciousness and all other things from which it is distinguished.
St Thomas Aquinas affirms in his Summa Theologica, that from the act of Creation “…It is necessary to say that God brings things into being from nothing” (ST, Ques. 45, Art 2). Aquinas had earlier presented the same argument in his Summa Contra Gentiles (Book II Chapter 16).
As I've already noted, Leibniz saw space as being merely the relations between things; space per se doesn't exist in the Leibnizian world. Indeed in Leibniz's philosophy the absence of space, as well as the representation that absence, is obligatory.
Kant is not satisfied by just eliminating the notion of the existence of a real objective space. He even denies the possibility to ‘represent' to ourselves such absence.
In contrast to Kant's rigourous view it seems that some of the greatest minds of all times like Plato, Democritus, Aquinas and Leibniz had no trouble to ‘represent to themselves' a concept of nothingness which means absence of space as well. Their standpoint defies Kant's assertion of the inability to represent the absence of space.
Kant's views on space and time have been the subject of controversy ever since the Critique first appeared in print. For instance, there was an interesting dispute during the 1860's and 1870's between Trendelenburg and Kuno Fischer. Trendelenburg claimed that Kant had presented the dilemma between the subjectivity and objectivity of space as being exclusive. Accordingly, by refuting the objective alternative, Kant left us no other choice but to choose the subjective view of space. In contrast to Kant, Trendelenburg argued that there is indeed a third choice available. The third choice is the view that space may be both subjective and at the same time objective and real, independent of our human constitution.
In view of the aforesaid submissions it follows that Kant's assertion of a subjective space is inconsistent with his embrace of Euclidean Geometry as an indivisible and fundamental part of his philosophical system.
© Pinhas Ben-Zvi 2005
Pinhas Ben-Zvi graduated from the Law Faculty of the Hebrew University in Israel. He has also studied law in London and philosophy in Tel-Aviv. He practices commercial law in Tel-Aviv.