1. Philosophy and History of Mathematics
A Brief Introduction: Week 4

2. Recap The Ancient World Logic
Plato is an idealist: he believes what is real is eternal ideas, and that the changeable physical world is an illusion.
Aristotle is… not quite an idealist. But he inherits Plato’s idea of forms, and the notion that knowledge is concerned with ideas.
Logic
We saw some examples of logic as a “method for deduction” – that is, a method for getting reliable knowledge in the world of ideas.

3. This Week: Overview Descartes (1596-1650) Locke (1632-1704)
The method of universal doubt
Rationalism
Locke ( )
Empiricism
Rationalism and Empiricism are (still) the two basic pillars of epistemology – the branch of philosophy concerned with knowledge. (Of course we will be asking, as ever, about the status of mathematics as a branch of knowledge.)

4. A Priori and A Posteriori
Knowledge is a priori if we can know it from first principles.
If you know what the words “batchelor” and “married” mean, you know that “no batchelor is married” is true a priori. It would be true even if there were no batchelors!
If you don’t believe it, no amount of evidence could possibly convince you.
This kind of knowledge (when it’s true!) has a kind of certainty, but it’s quite rare.
Knowledge is a posteriori if we learn it from experience.
I know that when I throw a pen in the air, it will fall down again; I learned this from experience.
This kind of knowledge relies on something else and so seems less certain; my observations might have been mistaken (as in a dream) or incomplete (“all swans are white”).

5. The Scientific Context
Both Descartes and Locke were deeply influenced by the science of their time.
Descartes lived in the time of Galileo and Francis Bacon.
At least three sources of knowledge seemed to be put into question: ancient authorities, our senses and everyday “common sense”.
Science seemed to present us with new and alien ideas. In our own time, think of quantum mechanics.
Locke lived in the time of Newton, Hooke, Boyle and the rest of the Royal Society.
This seemed to represent a triumph of the study of the material world through observation and experiment.

6. Descartes: Method of Doubt
What can we be certain of?
For a modern version, consider Nick Bostrom’s 2003 essay “Are you living in a computer simulation?”
“This paper argues that at least one of the following propositions is true:
(1) the human species is very likely to go extinct before reaching a “posthuman” stage;”
(Defined as) “humankind has acquired most of the technological capabilities that one can currently show to be consistent with physical laws and with material and energy constraints.”
“(2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof);
(3) we are almost certainly living in a computer simulation.”

7. Descartes: Method of Doubt
“Cogito ergo sum” – I think, therefore I am.
I can’t doubt my own existence.
This provides an “Archimedean point” – something to grasp hold of.
From here, Descartes claims to reconstruct a large amount of our common-sense knowledge of the world, dispelling the doubt he has invoked.
He does this not by scientific methods – that would be hopeless – but by pure rational thought.
Notice: only rational thought is capable of dispelling radical doubt.

8. The “Light of Reason”
Descartes only trusts things he sees in his mind’s eye “clearly and distinctly”.
We see that = 4 and cannot be unsure about it.
Being unsure is something like seeing something in dim light or through fog; we think we make it out but we might be wrong.
With = 4, we see clearly: that is, we cannot be mistaken about it.
In this metaphor, our reason that what “lights up” the world of ideas, allowing us to see them clearly.
Remember the role played by light in Plato’s cave!

9. Rationalism
Rationalists believe that deduction is the only route to true knowledge (crudely speaking).
They point to the failings of our senses and our scientific theories.
They also point to maths – while every other part of ancient Greek science has been found to be false, Euclid’s Elements has stood the test of time.
Rationalists in philosophy of maths have:
an easy time explaining why maths is universally true, and why its conclusions seem impossible to doubt; but
a hard time explaining why maths is useful – that is, why it should have any application to reality.

10. BREAK

11. A Problem for Rationalists
Euclid’s Elements begins with a list of axioms; I’ve sometimes called these “the rules of the game of geometry”.
But how is this different from writing out the rules of chess, say, and then playing a game?
Is a game of chess “knowledge”, then?
Doesn’t this trivialise mathematics, and turn it into meaningless symbol-shuffling?
Doesn’t knowledge have to be about something?

12. Locke Locke believes that all ideas have their origins in the senses.
Hence he believes that all knowledge begins with what our senses tell us.
This makes him an empiricist.
Empiricists agree that reason is useful, they just don’t believe it’s enough on its own.
Logical deduction can only clarify what we already know (but might be implicit).
Consider again the computer, which can make amazing logical deductions but seems incapable of even the most basic knowledge.

13. Empiricism
Empiricists believe that deduction only yields empty “knowledge”.
It can show us what we already knew, but not give us anything new.
To get started, we need hypotheses, and these come from our senses, observations and so on.
Empiricists in philosophy of maths have:
an easy time explaining why maths is useful – that is, why it should have any application to reality; but
a hard time explaining why maths is universally true, and why its conclusions seem impossible to doubt.

14. What is a Number?
Remember Aristotle’s distinction between substance and form.
The substance can be thought of as “the stuff a body is made of”, but it has many properties attributable to the actions of forms (colour, texture, shape etc).
Locke thinks “number” is a quality of bodies like “colour” or “weight”.
For example, my body has two arms, and a chair has four legs.
The two-ness of my arms comes into our senses just as their colour and other properties do.

15. What is a Number?
Can Locke’s theory account for our knowledge that = 4?
Is it something we learn from experience by seeing two things alongside another two things?
Does it have the same status as “All animals with kidneys also have spinal cords”?
Can Locke’s theory explain numbers like -4 or 2 ?
What about more modern mathematical objects like 7- dimensional spheres or fractals? Or, indeed, Newton’s infinitesimals?

16. Do Aliens Have The Same Maths?
When we imagine an alien civilization, we can imagine many things different from ours.
Their bodies and environments; their social structures; how they communicate; their art, music, literature and architecture (if they have such things); their scientific understanding.
But can we imagine them having different mathematics, without just imagining them being wrong?
For example, what would it mean to have a species for whom = 4 is true?
Is such a thing imaginable, or must every intelligent species somehow arrive at the same maths?
Tune in next week to find out… (sort of)