It would be ridiculous in a work of this sort, which cannot claim to do more than break the ground for further research, to attempt to set forth anything like an inventory of 'final phenomenal truths': or, again, to attempt to deal in detail with more than a very small number of them. Most of the 'truths' which seem to me to have a reasonable chance of being thus established could not be treated in a manner calculated to carry any conviction without embarking upon a lengthy specialised inquiry into the respective fields of experience to which they refer. I propose to conclude this chapter, nevertheless, by saying something about the reaction of the general doctrine I am maintaining upon one group of propositions which are of peculiar interest for the theory of Truth - the group, namely, pertaining to the science of geometry. The apparent certainty of geometrical 'truths' has always been a sore stumbling-block for the Coherence theory of the test of Truth. To insist that propositions of the nature 'the three angles of a triangle are equal to two right angles' express only 'partial truth,' has been a paradox which it has proved hard for the Idealist to sustain. On the presupposition as to the nature of the mind's apprehension of space which the Idealist customarily adopts, I believe that it is impossible to sustain it. On the other hand, the Idealist doctrine does, I think, emphasise one side of the truth with just as much propriety as the critics of it emphasise the other. The correct solution is not visible until we make the distinction which has been the main theme of the present chapter, the distinction between Phenomenal Truth and Ideal or Noumenal Truth. In the following pages I shall try to show that, if we accept the Idealist presuppositions above referred to, our proper course is to deny, with Idealism, that geometry can attain Noumenal Truth, but to admit with the critics of Idealism that it can attain intellectual incorrigibility. Final Phenomenal Truth is open to geometry.

The presuppositions in question are that the apprehension of space is the logical (though certainly not the psychological) prius of our apprehension of objects, and that the space so apprehended is Euclidean space, a homogeneous, tridimensional continuum. I do not propose to offer a defence of these assumptions, highly disputable though they be. Those who are unable to accept them may be requested to neglect this brief discussion, or else to regard it merely in the light of an illustration of the principle upon which Phenomenal is distinguished from Noumenal Truth.

Let us suppose, then, that these presuppositions are conceded. What I wish to maintain is that such a judgment as'two straight lines cannot enclose a space' enjoys final Phenomenal Truth: and that in general, with respect to the science to which this judgment belongs, final Phenomenal Truth is everywhere possible.

It is easiest to demonstrate this if (as in the analogous argument above) we contrast the judgment here in question with the ordinary judgment of particular objective fact, whose capacity for attaining final truth is denied. What is the relevant difference? In both cases we have the connection of differences through a 'ground.' The apprehended nature of space is the ground of the judgment 'two straight lines cannot enclose a space,' just as much as the apprehended nature of material objects is the ground of judgments in physical science. But - and here is the point - in the spatial judgment the ground is present in its final form from the beginning. What we are 'given,' as simple brute fact which we cannot question (this is the obvious consequence of its a priori character), is space as a homogeneous, tridimensional continuum, and a judgment such as that which we are considering simply reads off such implications of the construction we make as are determined by the known character of this 'given.' It is the ground (here the apprehended nature of space) which prescribes the specific connections which we affirm in all judgments. But in this instance, since the ground is not liable to revision, or even question, by the intellect, the connection it prescribes is (if we have read the implication aright) not liable to question either. In judgments in physical science the grounds obviously possess a quite different status. They are, and must ever remain, hypotheses, not datal facts. They are susceptible of indefinite modification from the increase of our experience of the physical world, both intensive and extensive. And with these alterations in the apprehended grounds there must alter concurrently the meaning of the connections which they warrant. But the child who sees - not says but sees - that 'two straight lines cannot enclose a space' has precisely the same logical warrant for his assertion as has the most accomplished mathematician. And that warrant is infallible, not a questionable hypothesis, but a self-complete datum. Here, accordingly, there is no infinite process imposed upon the mind before it may justly claim finality. We have the ground in what must remain for us its final form from the beginning. Neither advancing spatial knowledge nor developing objective knowledge can - if space, as defined, is indeed the logical prius of our apprehension of objects - impose upon it the slightest modification. We can make our assertion, therefore, knowing that no accretion of experience attainable by the human mind will have any power to overturn it.

It is the failure to distinguish the intellectual incorrigibility of the ground in geometrical reasoning from the admittedly developing character of the grounds in our reasonings upon physical things which vitiates the treatment of geometrical truth by most Idealist writers. These writers are accustomed to treat a proposition like 'two straight lines cannot enclose a space' as though it fluctuated in logical stability according to the more or less developed nature of the ground. The proposition is supposed to be in this respect precisely the same in principle as such a proposition as 'water freezes at 32° Fahrenheit.' The meaning of the latter proposition, it is argued (as I think, justly), is conditioned and even constituted by the 'appercipient background' of the judging mind, and this background is of a very much higher calibre in the case of the scientist than in the case of the plain man. What the scientist asserts and what the plain man asserts in the verbally identical propositions are very different things, and the scientist's more systematic assertion must be adjudged, by the Coherence test of Truth, the more true. Now this is probably very right and proper where the 'backgrounds' are physical hypotheses. It is mere perversion of the facts where the background is the given nature of space. The proposition 'two straight lines cannot enclose a space' does without doubt depend vitally upon the nature of the system which prescribes the connection. But the point is that this system in so far as relevant is precisely the same for the tyro as for the most accomplished geometrician. The development in our knowledge of spatial inter-relations does not add one iota of cogency to the ground of this proposition. It would be interesting to know if the Idealist can produce any geometrician who has felt it to be otherwise.

The fundamental point is, in a word, that in geometrical judgments we have all the relevant data before us from the start; in so far at least as such data are procurable under the conditions of finite mind. The qualification denoted in the last clause, however, is essential. We know how we have to think space. And we know how, thinking space thus, we have to think the mutual connection of constructions made within it. But we do not understand space. We have not the least conception of its place in the economy of the Whole, and there are even excellent reasons for believing that in the manner that we think it, and must think it, it cannot ultimately be. Our 'ground,' therefore, is once more (as in the case of 'self-awareness') a sheer mystery, even though it be intellectually incorrigible. And if this is so, it becomes nonsense to suppose that in spatial judgments we are asserting ultimate truth. A connection of differences cannot be self-consistent, intrinsically satisfying, where the ground is related to the rest of our experienced world in a way that is not intelligible to us, as is the case with space. The Idealist is right, therefore, in denying Truth in the fullest sense to these judgments. But since they may reasonably be pronounced 'intellectually incorrigible,' it only promotes misunderstanding to treat them as though they possessed no higher status than patently modifiable judgments in other spheres. They should be recognised as 'ultimate' for finite mind: or, to use the terminology of this chapter, 'final phenomenal truths'.