Kant claims that geometric knowledge is of a synthetic a priori nature. Geometric knowledge can be encapsulated by the propositions it postulates, which are the set of all statements it declares to be true. This includes such things as the Pythagorean Theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two sides. Under Kant’s theory of knowledge, propositions can fall under either one of two categories: analytic or synthetic. Analytic propositions are true simply due to the meaning of their words, whereas synthetic propositions are not. Therefore, the proposition, “all husbands are married,” would be considered analytic, as embedded in the very meaning of the word ‘husband’ is the fact that one must be married, and it would lose its specific meaning were this not the case. An example of a synthetic proposition might be that, “three plus four equals seven,” in that this is a similar sort of factive statement, but the meaning of ‘seven’ is seemingly not at all contained in any part of the subject ‘three, ‘plus’, or ‘four’. There must be some sort of active process taking place in the mind which allows us to unify such distinct concepts, in contrast with analytic knowledge which consists of a simple clarification of the meaning contained within the subject of the proposition.
There is a separate distinction between affirmative propositions which is founded on the justification offered for belief in the truth of each of their statements. A posteriori propositions are those truths which are justified by empirical evidence, whereas a priori propositions are justified independently of experience. In this way, ‘three plus four equals seven,’ is an a priori proposition, because knowledge of such a thing does not depend in the least on empirical fact. Its truth is not justified by any physical experience such as the act of counting two groups of objects together, but rather by the rules governing mathematics which offer meaning to its words. Conversely, ‘all bachelors are unhappy,” is an example of an a posteriori proposition, because its truth would be justified empirically by a polling of bachelors.
It seems likely that human consciousness is partly structured around the concepts of space and time, in that we can not help but experience the world in such terms. If this is the case, then they must be taken as a priori truths from which all other determined empirical truths later arise. Under Kant’s view, our knowledge of geometry is a reflection of the physical nature of space, which is one of a three-dimensional Euclidean geometry as dictated by Newton, such that our knowledge of it must be synthetic. We can not construct geometry analytically, through the act of rewording meaning, because we think in terms of a space contained outside of our own experience. In this way it can be seen why Kant claims that geometric knowledge is synthetic a priori, because it involves complex concepts which are not contained within the meanings of one another and which arise out of a reflection of the nature of physical space, but can not be justified by the experience of that physical space, as this same geometric knowledge is presupposed within the structure of such experience.
Russell agrees that the components of geometric knowledge can be synthetic or of a priori justification, but differs in his claim that each part must instead constitute either synthetic a posteriori or analytic a priori knowledge alone. He supports this argument by drawing a distinction between what he calls the pure and applied parts of geometry, which are associated with logical and physical spaces respectively. Kant himself realized that it must at first seem strange how we come to possess a priori knowledge of the actual geometry of physical space, but offered his theory of intuitive knowledge as a solution. Russell instead solves this dilemma by arguing that pure geometry is justified a priori, but consists of all equivalent spaces founded on formal logic and is therefore analytic, whereas applied geometry is empirical and justified a posteriori, and an approximation of physical space and therefore synthetic. In this way, we can claim that the Euclidean geometry is true of physical space with a high degree of certainty, but can never be assured of this fact because of the inherent error in all empirical observation. Conversely, pure geometry consists of a set of spaces defined by systems of axioms founded on formal logic and is therefore analytic a priori knowledge. One such system is Euclidean geometry, although it has been found that there are many more equivalent systems. In this way, Russell has shown that there is no as synthetic a priori geometric knowledge, but that it actually constitutes the separate parts of pure and applied geometry.
Although Russell stipulated that proof of physical space being Euclidean was impossible, Gauss wanted to show with careful measurement that the physical space of applied geometry was instead of some sort non-Euclidean. He did this by sending light-rays between the peaks of three distant mountaintops which formed a triangle, and present in a sufficiently non-Euclidean geometry the sum of whose interior angles would not equal the sum of two right angles. Gauss did not measure any noticeable difference from the sum of two right angles between the peaks of the three distant mountaintops. This leads to the conclusion that physical space must be Euclidean with an incredibly high degree of accuracy. Schlick points out that Gauss misses the conventionality of the geometry of physical space and the physical systems therein. This phenomenon could either be explained by light-rays traveling by straight shortest paths in a Euclidean space, or by curved shortest paths in a non-Euclidean space. Russell believed that the geometry of physical space was strictly a matter of convention and that Euclidean geometry was chosen for simplicity sake, but Shlick’s analysis of Gauss’ experiment ensured that the most that could ever be proven of physical space was that it deviates only slightly from long-standing experimental evidence in favor of the Euclidean geometry of physical space.
For a long time people wanted to be able to deduce the parallel postulate of Euclidean geometry from the other axioms, which given any line and another point in space posits the existence of a single parallel line through that point. Lobachevski decided to construct a system of geometry through accepting all axioms of Euclidean geometry, but denying the parallel postulate. From this he deduced a series of equivalent theorems to those found in Euclidean geometry with often subtle differences, such as in the case of the sum of the interior angles of a triangle being always less than the sum of two right angles, and which form equally as valid a system of formal logic. In fact, this system is consistent with Euclidean geometry in the sense that the terms found in each can be exactly translated on to one another. In this sense the choice of geometry is only convention, because it is a matter of preference to specify the axioms of the space being formulated, and so the axioms of Euclidean geometry were only chosen by the physicist for ease of use rather than any exact equivalence to physical space.
Poincare offers an example using the idea of a heated disk to explain how the geometry of physical space can be chosen by convention. He asks us to consider the interior of a heated disk with a uniform temperature such that its radius could be measured with a certain number of equally lengthed metal rods. Then consider a similar heated disk with a non-uniform temperature such that it is greatest at the center and gradually less towards the circumference. Because of heat dilation, objects in this disk will become larger as they travel from the circumference towards the center. The radius of the disk might be measured by use of metal rods, but these rods will be of unequal size in accordance with this heat dilation phenomenon and so a different number will be needed to measure its radius. Furthermore, the uniformly heated disk could otherwise be explained by the same number of non-equally sized rods found in the non-uniformly heated disk, if the geometry of the disk’s space was non-Euclidean. In the same way, the non-uniformly heated disk could be explained by the number of equally sized rods found in the uniformly heated disk, if the geometry of its space was non-Euclidean. This serves to show how the geometry of physical space can be chosen by convention, as long as its physical systems are modified accordingly.
Reichenbach’s bump example is another illustration of the conventionalism of physical geometry. It consists of a non-Euclidean geometry with a sort of hump on its surface being projected onto the surface of a linear Euclidean plane. Imagine people in the non-Euclidean geometry taking measurements of their world, who will discover that it takes a non-uniform number of measuring rods to map the length of distances on the hump, and so realize they live on such a space. However, people who exist as shadows of the projections on to the Euclidean plane will experience a similar phenomenon due to the direct mappings from the non-Euclidean geometry, and thus they will come to believe their space too has a non-Euclidean hump to it. Because the results of measurement are the same in both spaces, it is not right in the first place to say that one is necessarily Euclidean and the other is not.
It has come to be known that pure geometry is founded on a system of formal logic and therefore consists of arguments deduced directly from primary axioms. The meaning of geometric knowledge lies in the form of such arguments arrived through a series of logically consistent principles. Philosophers have long wanted to set down what parts of philosophy could not legitimately be refuted, and so similar forms of skepticism pervade many philosophical arguments. The verificationist criterion of meaning posits that the meaning of all content consists of the way in which such content comes to be verified. Conversely, if a statement has no way of being verified, then it is accordingly declared meaningless. It was suggested that many historical arguments had no method of verification, such as the distinction between realism and idealism of the world. The verificationist criterion of meaning sets out that the very fact that the content of perception is the same regardless of whether the objects of perception exist outside the mind stipulates that realism can not possibly be empirically distinguished from idealism and so the distinction has no physical meaning.
Protocol sentences are those grounded in some form of observation which all meaningful statements can be reduced to through formal logical analysis. The protocol sentence debate arose over what sort of statements the protocol sentences should consist of. This led to competing views that they must constitute of such things as reflections of the material world, objects of sensory experience, or empirical findings in natural science.
The form of the protocol sentence debate seems incredibly similar to those past dogmatic arguments which the verificationist criterion of meaning set out to correct. There is no way to verify what the protocol sentences might be, and so philosophers on each side of the argument base their claims on what the verificationist principle would refer to as meaningless conjecture. It has been suggested that there are positive universal claims which can not be verified, but still seem to possess meaning, and in a similar sense the same holds for many negative existential claims. This led to the realization that the verificationist criterion of meaning would not be able to solve all problems found in philosophy.