1. Philosophy and History of Mathematics
A Brief Introduction: Week 5

2. Recap Rationalism Empiricism Descartes
Knowledge – especially maths – primarily comes from logical deduction.
Explains why maths is universally true, and why its conclusions seem impossible to doubt; but
Fails to explain why maths is useful – that is, why it should have any application to reality.
Empiricism
Locke
Knowledge – even maths – primarily comes from induction from experience.
Explains why maths is useful.
Fails to explain what makes it true, and why our knowledge of it seems so certain.

3. This Week: Overview Immanuel Kant (1724-1804)
The apparent contradiction between Rationalism and Empiricism needs to be resolved.
Transcendental philosophy – what are the conditions of possibility for knowledge?
Kent’s conclusion is surprisingly mathematical!
Critical philosophy – there are limits to what can be known, and some debates in philosophy are about matters on the other side of those limits and should be set aside.
NOTE: Everything we do today will be simplified. Kant is a very subtle and technical thinker; our intention today is just to get an orienting sense of what he was about.

4. Rationalism: Castles in the Air
Kant was writing at the end of the 18th century, when rationalist philosophy (e.g. Leibniz, Wolff) was in fashion, especially in Germany.
These philosophers had a tendency to build “systems” out of pure logical deduction.
These systems were often supposed to explain the fundamental structure and nature of the universe, God, humanity and so on.
But they were based on very little outside the philosopher’s study.

5. Empiricism: The Risk of Solipsism
The empiricists’ vice was more negative: if rationalists tended to believe anything they liked, empiricists tended to disbelieve even the most common-sense ideas.
David Hume ( ), for example, questioned fundamental ideas about cause-and-effect and personal identity.
Kant said Hume had woken him from a “dogmatic slumber”.
But Kant is not satisfied with Hume’s skepticism, which tends towards solipsism or nihilism.

6. Critical Philosophy
Philosophy’s job is to know everything that can be known and nothing else. This is an ancient ambition.
Some Rationalists claimed to know things that cannot possibly be known, such as whether God exists or whether the future is predetermined.
Some Empiricists claimed not to know things that can be known, such as whether I am awake or dreaming.
Kant’s critical philosophy aims to draw the line between what can and can’t be known.

7. Conditions of Possibility
Pre-analytically, some things are obviously knowable and some obviously not.
But what makes the difference?
In other words, what makes knowledge possible?
In fact, what is knowledge, exactly?
Are these questions answerable or not?
This is the problem Kant’s Critique of Pure Reason (1781) sets out to solve.
It is a critique in that it seeks to define a set of conditions of possibility for knowledge.
Anything that doesn’t fulfil those conditions isn’t “real” knowledge.

8. Phenomenology Both Descartes and Locke indicated a difference between:
“how things really are” independently of me; and
“how they appear to me”.
Kant calls these noumena and phenomena.
He never really talks about noumena: they are strictly unknowable.
The “science” of how things appear to us is phenomenology.
Today we probably associate this more with psychology or even neuroscience. In fact psychology emerged later (during the C19) and the two traditions are intertwined.
You could perhaps think of phenomenology as a sort of union of psychology with logic.

9. Transcendental Idealism
Phenomenology sounds like an empiricist project, because it begins with experience (phenomena).
BUT Kant is not interested in phenomena themselves but rather their conditions of possibility.
How come we can experience anything at all?
In what way do things appear to us?
How must our faculties be arranged so that such appearances are possible?
The answers must go beyond experience.
They are universal: they apply to everyone capable of knowledge.
They are logical, not empirical: they must be about ideas that are independent of any particular experiences (that is, if they exist at all).
I find it helps to imagine Kant is talking about aliens; if we met a very alien species that seemed to have intelligence, what (if anything) must it have in common with us?

10. Break

11. The Transcendental Aesthetic
Aesthetic refers to experiences that involve the senses.
An intuition is Kant’s term for a specific experience of this kind, such as seeing a chair.
You can also close your eyes and imagine a chair: this is an intuition too.
Sensibility is his name for the faculty of forming intuitions, as the nose is the faculty of smelling.
The “Transcendental Aesthetic” is a section of the Critique of Pure Reason that tries to figure out the conditions of possibility for this kind of experience.
That is, if someone possesses sensibility, what structure(s) must their mind have?
We will take some time now to read this together.

12. Summary
Space & Time are properties of our mind, not of objects or the universe.
Specifically, they are called “forms of intuition”.
Space is the form of outer intuition: without it we would not be able to tell the difference between imagined and real objects.
Time is the form of inner intuition: without it nothing could happen, including intuitions appearing. We could never experience anything, and hence never get knowledge.
Space & Time are a priori.
Geometry studies space and number studies time. So mathematical knowledge is also a priori.