renewing philosophy by better understanding of philosophy (see previous lectures)
Our ability to perform abstract reasoning and to contemplate universals (as for example, to contemplate a mathematical proof and its deductive logical structure) has been taken to be the feature of our current lives in which we live divinely - the one thing we do a perfect god would also have reason to do, just out of joy and love, with no pressure of need (Nussbaum 1990, 385)
Mathematics was elevated to the status of an ‘arche beyond discourse’ within the 20th century; the importance of mathematics therewithin cannot be overestimated.
Even for Plato if you want to be a ruler you must
[…] enter upon the study of calculations, take hold of it, not as amateurs, but to follow it up until they attain to the contemplation of the nature of number, by pure thought, not for the purpose [of anything practical], but for the uses of war and for facilitating the conversion of the soul itself from the world of generation to essence and truth. (Plato, R., VII, 525b-c)
This is still, to some extent, present in our time; mathematics is the privileged discourse par excellence, and science and technology are mathematical through and through.
That is interesting because it poses the fundamental question of ‘how does the study of abstract, non-physical, non-visual structures help create a knowledge that is so secure and so useful?’ ← what Mary Tiles calls the Basic Problem.
If there is one kind of discourse where we can have results that are secure, that is mathematics.
[Subtle foreshadowing.]
What is special about mathematics?
Mathematical/logical language is unambiguous as a result of its transparency - whatever symbols it does have are wholly stipulated. They have the exact meaning you put into them.
This is obviously not the case with stuff like proper names, flags, and religious symbols.
Gottlob Frege
We don’t need to memorize dates of birth, but obviously need to trace the conversation through time.
[paraphrased] “ask a mathematician what (the symbol) 1 is and he won’t have a good answer; we should study that because it is very important for math.” → “let’s renew math by a better understanding of math”
according to the professor, there might not be a more influential philosopher (of mathematics or not) than Frege.
If you take mathematics to simply be the expansion of a set of axioms, you will fail to question the fundamental stipulations. Such as, for instance, the concept of ‘number’ or ‘0’. What Frege asks is how do we understand the terms used in axioms without definitions? How do we know the axioms are true without proofs of them? What is to prove theorems/justify claims in mathematics?
Peano-Dedekind axioms of arithmetic
the entire theory of natural numbers can be derived from these axioms:
(A1) 0 is a number
(A2) The successor of any number is a number
(A3) No two numbers have the same successor
(A4) 0 is not the successor of any number
(A5) (Mathematical induction) Any property which belongs to 0, and also belongs to the successor of every number which has the property, belong to all numbers.
But this axiomatic view never asks what ‘a number’ is. This is what Frege asks about; what are the terms used in the axioms of arithmetic. And arithmetic is the basis of everything else.
Digression
Logicism is a reduction of mathematics to logic; this may be an abstract approach (in the current phrasing), but is similar between Whitehead & Russell’s Principia Mathematica and Frege, though they approach it from different angles.
One of the inspirations for Russell was solving a contradiction in Frege. We’ll focus on Frege.
What Frege is doing is abstract; the argument is in the handout.
Frege on number
“I don’t claim that my power-points are useful in studying the course, but this one might be.”
1. Numbers belong to concept.
Take a big book on the table: it has 411 pages, 5 parts, and 17 chapters.
What is its number? How many is it?
It’s one book, but there are different ways to quantify it (the pages, and so on). Per Frege, if you want to take its number, you have to connect it to the concepts we use in talking about the book.
1 is the number that belongs to ‘is a book on the table’
411 is the number that belongs to ‘is a page of the book on the table’
5 is the number that belongs to ‘is a part of the book on the table’
If we want to understand what number is, we have to think about concepts.
2. A concept applies to things.
The concept ‘horse’ applies to all things that are horses.
All horses are instances of the concept ‘horse.’
The members of the set of horses are all the things that are horses.
Cf. Moore: The good is the set of all things to which the concept ‘good’ applies.
The set is more in the dimension of Russell; Frege thinks about concepts, but he does think along these lines as well.
3. Concepts instantiated by concepts
‘Transcendental’ is a concept.
‘Difficult to understand’ is a concept.
‘Difficult to understand’ applies to the concept ‘transcendental.’
‘Difficult to understand’ is a concept instantiated by the concept ‘transcendental.’
A 7-instanced concept: a concept with 7 instances/that applies to 7 things;
‘is a day of the week’ → Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday
3.1 Different concepts have an equal number of instances: equal(-numbered) concepts.
F.e. the number of concepts instantiated by ‘is a day of the week’ is equal to those of ‘is a continent of the Earth’
4. Numbers ‘belong’ to concepts
7 belongs to the concept ‘is a day of the week.‘
A number is the name of a class of concepts; for instance, all concepts which match one-to-one and have this number of members.
the Number which belongs to the concept F
= the class of concepts that apply to things which are one to one with the things that F applies to
= the class of concepts the instances of which are one to one with the instances of F
= the class of concepts equal to F
5. Empirical vs. logical concepts
The concept ‘is not identical with itself’ is empty; thus, 0 = the class of concepts the instances of which are not one to one with the instances of ‘is not identical with itself’
1 = the number which belongs to ‘is identical with 0’
vs.
1 = the number which belongs to ‘people identical with me’
Russell’s paradox
the notion of a class that has classes as its members:
BA-program = class with stage 1, stage 2, stage 3 as members
stage 1, 2, and 3 are themselves classes with courses as members
Some classes are members of themselves: the class of big things will itself be big.
Some classes are not members of themselves: the class of small things, will not itself be small.
Take the class of classes that are not a member of itself: either it is or is not a member of itself. → if it is a member of itself, it is not a member of itself; if it is not a member of itself, it becomes a member of itself.