Philosophy and History of Mathematics A Brief Introduction: Week 2

Try This… Write down any two numbers. Try to bring the smaller one up to the size of the bigger one in the following way: Calculate HALF the distance between small and big Add this to the small one Repeat as many times are you can! What do you discover? NB: A basic calculator is very useful for this – you may have one on your phone.

Recap Plato’s “Allegory of the Cave” from The Republic Claims that the physical world we experience with our senses is an illusion. It can only offer conjectures and tentative theories. Absolute truth is to be found in the “realm of ideas”, which are unchanging and eternal. Euclid’s Elements A compendium of geometrical knowledge. Claims to derive all geometry from a small number of “definitions”, “postulates” and “common notions”, together called axioms. Both of these texts have had a huge influence on almost every part of Western culture.

This Week: Overview Zeno’s paradoxes of change and motion. There are four paradoxes, all quite similar. We will need some preliminary ideas abut atomism to understand these. Aristotle’s solution to the paradoxes. If time, we will look at the proof that some numbers are “irrational” (in a special sense).

Context: Geometry & Mechanics For the ancient Greeks, “mathematics” was almost entirely “geometry”. Think of this as the abstract or idealised study of space. Geometry studies, e.g., “perfect triangles” that can’t exist in the world of the senses. But “mathematics” also included a certain amount of what we could now call “physics” or “mechanics” – the study of movement and change. This is still the case today.

What’s At Stake This session leads us further into the field of metaphysics. We’ll be asking (among other things) about the fundamental nature of space and time. But we will also be asking about the objects that geometry and mechanics deal with. Look again at the definitions in Euclid’s Elements – he describes the basic objects as points, lines and planes. These are “pieces of space”. But what are they like? Do they even make sense as ideas? We will also ask analogous questions about time, which will apply to any study of movement or change.

Time and Space Think of space as the “container” for matter, time as the “container” for events. We will ask: how is this container constituted? In particular, what parts does it have?

Atomism 1: Space Is space really made of tiny pixels, much too small to see? Or is it “smooth”, with an infinite number of points between any two points?

Atomism 2: Time Is time really like a sequence of still frames in a film? Does time “jump” between instants so quickly that it seems to us to be continuous? Or does every second contain an infinity of instants?

Zeno’s Paradoxes Against Atomism The Stadium: against atomism of space If you’re an atomist about space, it seems that fundamental intuitions about “betweenness” and order are broken. The Arrow: against atomism of time If you’re an atomist about time, you have to explain where “movement” comes from. It seems it can’t come from time itself!

Infinitism This is an appealing answer to atomism. I have sort of invented the term “infinitism”, at least in this context. The parts of space, says an infinitist, are points. But in any region of space, however small, there are an infinite number of points. The parts of time, says an infinitist, are very similar. In any interval of time, however short, there are an infinite number of instantaneous “moments” or “times”.

Infinitism with Decimals B Suppose A and B are precisely 10cm apart. We can never measure this with absolute precision – but suppose it is a fact that they’re this far about. Now we move B but keep A fixed Suppose we move B 1cm towards A. Now they’re 9cm apart. Suppose we move it back by half as much: 9.5cm. Repeat by half as much again, so it’s at 9.25cm. Repeat again, so it’s now at 9.275cm

Infinitism with Decimals The point is that this list of distances can go on forever: 9, 9.5, 9.25, 9.375, 9.3125, 9.34375, 9.328125, 9.3359375, 9.33203125, 9.333984375, … The atomist says that at some stage we will be able to find a number representing the “halfway point” but there will be no space there. We will be trying to fit “between two pixels”. The infinitist says this is absurd: space must be infinitely divisible just as numbers are.

Zeno’s Paradoxes Against Infinitism The Dichotomy If you’re an infinitist about space, it seems every motion involves an infinite number of things happening in a finite time. Many of the ancient Greeks found this absurd. Achilles and the Tortoise If you’re an infinitist about time, it seems that something we commonly observe (overtaking) can’t happen.

Summary of Zeno’s Paradoxes Whatever position you take up – even if you mix infinitism and atomism between space and time – Zeno has a paradox to defeat you! Atomist Infinitist Space Stadium Dichotomy Time Arrow Achilles

Zeno’s Paradoxes Are the paradoxes just intellectual games? No – they point to contradictions that arise when you adopt either the infinitist or atomist position. A contradiction is a Very Bad Thing, as we will see next week. Where does that leave us? Is seems that space and time can’t be infinitely divided into points (infinitism), and also can’t not be (atomism). Zeno probably intended for us to conclude that that are not made of parts at all – he may have been a monist.

Platonism For anyone with a vaguely Platonic view of things, space and time as ideas (forms) are either made of parts or not. Since we can “find” their parts (e.g., by cutting a line in half), these parts must already exist (as ideas). Nothing is created or destroyed in the realm of ideas, it’s just that we discover it. So the idea of a 10cm line includes all the little sub- lines we can consider as its parts. Zeno’s arguments then are very disturbing!

Aristotle’s Reply Aristotle makes a distinction between the actual and potential infinite. He says a line is a continuum – a single thing with no parts. When we “cut” the line, we produce something new: two shorter lines. It’s correct to say that these two shorter lines are parts of the original line. It’s wrong to say that they were already there before we made the cut. Hence a line contains the potential to be divided in an infinite number of ways, but it isn’t actually divided up in advance. It’s easy to see how this applies to space and time.

Aristotle vs Plato Plato’s world of forms is unchanging. Truth is eternal: change is a sort of illusion. Aristotle posits two kinds of reality: substance and form. Think of substance as “unformed matter” – a kind of grey goop that everything is ultimately made of. Forms are ideal objects that make a particular piece of matter what it is – a table, a horse etc. Knowledge always concerns forms, not matter. Change occurs when a piece of matter comes under the influence of a different form.

Substance + Form = Things Substance is just “stuff” – it offers the potential to be something we can have knowledge of. When it takes the “impression” of a form, such as “the form of the cup”, it becomes a fully-fledged thing.

Can Physics Help Us? A modern assumption might be that it’s the job of Physics, not Philosophy, to determine the fundamental nature of space and time. But how could an experiment or observation, carried out by finite human beings, determine whether space and time are infinitely divided? I’m not saying this is impossible, just that at present it’s hard to see how it could be done. Modern physics almost universally uses an infinitist model of the continuum – see our course on calculus for more details. But a good model can be very different from reality.