2 A brief excursion into philosophy
Relying on facts that appear self-evident can lead us astray. To demonstrate this, take a look at the work of one of the great philosophers, Immanuel Kant. His Kritik der reinen Vernunft (‘Critique of Pure Reason’) was first published in 1781.
There are many different translations of the text – the quotes here are from the 1855 translation by John Miller Dow Meiklejohn (who remarkably translated this momentous work as a teenager).
Being a work of philosophy from two and half centuries ago, it’s written in a style that may be challenging to understand out of context. Don’t worry if you find the quotes a little hard to follow.
This work contains extensive discussion of the relationship between mathematical theory and philosophy. Kant discusses several ‘antinomies’ or contradictions which, in his view, arise necessarily from our attempts to conceive the nature of reality. In his work, these contradictions take the form of a ‘thesis’ and ‘antithesis’ expressing propositions that are mutually exclusive and collectively exhaustive. In other words, both of them cannot be true, and both of them cannot be false, and yet it seems both of them can be proved! Here’s an example of one such antinomy:
Thesis: The world has a beginning in time, and is also limited in regard to space.
Antithesis: The world has no beginning, and no limits in space, but is, in relation both to time and space, infinite.
Kant proceeds to prove both of these statements, following a few basic assumptions to their logical conclusions. As an example, let us look at the ‘time’ part. Kant’s proof for the thesis above is:
Granted, that the world has no beginning in time; up to every given moment of time, an eternity must have elapsed, and therewith passed away an infinite series of successive conditions or states of things in the world. Now the infinity of a series consists in the fact, that it never can be completed by means of a successive synthesis. It follows that an infinite series already elapsed is impossible, and that consequently a beginning of the world is a necessary condition of its existence.
He then proves the antithesis (i.e., that the world has no beginning) as follows:
For let it be granted, that it has a beginning. A beginning is an existence which is preceded by a time in which the thing does not exist. On the above supposition, it follows that there must have been a time in which the world did not exist, that is, a void time. But in a void time the origination of a thing is impossible; because no part of any such time contains a distinctive condition of being, in preference to that of non-being (whether the supposed thing originate of itself, or by means of some other cause). Consequently, many series of things may have a beginning in the world, but the world itself cannot have a beginning, and is, therefore, in relation to past time, infinite.
From a modern perspective, it’s fairly clear to see that the issue causing the contradiction lies with the concepts of space and time that Kant employed. Our understanding of space and time has moved on with the development of Einstein’s theory of relativity, which links space and time in an intricate manner. The simple view of space and time as static entities in which everything develops is no longer appropriate – space and time interact with each other in a non-trivial way. We can now contemplate the possibility of the universe being a closed space that is finite without having a boundary – much like the surface of the Earth does not have a boundary, despite being evidently finite. However, this is something we have learned from science; we cannot directly experience this link between space and time. Our brains are trained by experience, and in our everyday experience space and time are separate entities. As a result, we cannot visualise the ‘true’ structure of space-time.
Similarly, we base mathematics on what we perceive as self-evident assumptions. However, what we see as self-evident is arguably influenced by the way our thinking has developed, based on our experience of the world around us. We might naively assume that any alien intelligence would use the same mathematics as we do, and that mathematics is in this sense ‘universal’, but this is far from assured.
Question 2
Can you imagine what Kant’s argument for the ‘space’ part of this antinomy looks like? Consider this, then reveal the discussion below.
You might want to look closer at the source if this interests you. It’s freely available online from Project Gutenberg: The Critique of Pure Reason [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] (open in a new tab or window so you easily return here). These particular arguments are located in ‘Section II. Antithetic of Pure Reason’.
Discussion
Here are the relevant bits of text. It’s important to note Kant’s use of proof by contradiction here, beginning by taking the opposite for granted.
Proof for the thesis:
… let us take the opposite for granted. In this case, the world must be an infinite given total of coexistent things. Now we cannot cogitate the dimensions of a quantity, which is not given within certain limits of an intuition, in any other way than by means of the synthesis of its parts, and the total of such a quantity only by means of a completed synthesis, or the repeated addition of unity to itself. Accordingly, to cogitate the world, which fills all spaces, as a whole, the successive synthesis of the parts of an infinite world must be looked upon as completed, that is to say, an infinite time must be regarded as having elapsed in the enumeration of all co-existing things; which is impossible. For this reason an infinite aggregate of actual things cannot be considered as a given whole, consequently, not as a contemporaneously given whole. The world is consequently, as regards extension in space, not infinite, but enclosed in limits.
Proof for the antithesis:
… let us first take the opposite for granted—that the world is finite and limited in space; it follows that it must exist in a void space, which is not limited. We should therefore meet not only with a relation of things in space, but also a relation of things to space. Now, as the world is an absolute whole, out of and beyond which no object of intuition, and consequently no correlate to which can be discovered, this relation of the world to a void space is merely a relation to no object. But such a relation, and consequently the limitation of the world by void space, is nothing. Consequently, the world, as regards space, is not limited, that is, it is infinite in regard to extension.