Lecture 4 Philosophy of Mathematics

Question 1

What are the three theories on existence?

Answer 1

1. Empiricism: know what exists by using our senses (seeing, hearing, touching, etc.)
   -Knowledge comes from experience -> sensory experience
2. Physicalism: only physical things exist
   - everything that exists is physical -> made of matter, energy, or governed by physical laws -> no supernatural stuff unless shown to be physical
3. The quantifier view: to exist is just to be something we can refer to in a true sentence
   - comes from formal logic

Question 2

Empiricism info

Answer 2

Empiricism leads to either:
- idealism: external physical objects do not exist -> only subjects and their mental representations
- skepticism: external physical objects, if htye exists, are not empirically knowable

Question 3

What are the motivations for the quantifier view?

Answer 3

• pluralistic: our ontology is not tied to any general metaphysics or worldview
• problem of negative existential statements: how can we talk about what doesn’t exist with contradiction?

Question 4

What is the quantification argument?

Answer 4

1. mathematical statements quantify over numbers
2. mathematical statements can be true only if what they quantify over actually exists
3. there are true mathematical statements
   therefore,
4. numbers exist

Question 5

Psychologism - Mathematical existence

Answer 5

Psychologism: Numbers are abstractions from physical quantities -> to be a number is to be a concept of a physical quantity

Leibniz’s law: A = B ⟺ A and B have exactly the same properties

Question 6

Platonism info

Answer 6

abstract things (like numbers, ideas, and forms) exist - even if we can’t see or touch them

-Real: mathematical objects are real, existing objects
- abstract: they are non-physical entities, without spatiotemporal location
- mind-independent: they are not psychological constructs and exist independently of our perceptions and beliefs

The epistemological challenge: how to know mathematical objects or true statements about them if they are abstract and mind-independent?

Question 7

What are the challenges for Platonism?

Answer 7

1. If Platonism is true, then mathematical objects are abstract and mind-independent
2. If mathematical objects are abstract and mind-independent, then they are unknowable
3. mathematical objects are knowable
   therefore,
4. Platonism is false

Question 8

what is Benacerraf’s Dilemma?

Answer 8

1. any adequate theory of mathematics should explain the truth-conditions of mathematical statements and how such statements are knowable
2. platonism can explain the former only at the expense of the latter
   therefore,
3. platonism is not an adequate theory of mathematics

Question 9

Functionalism info

Answer 9

functionalism: mathematical objects do not really exist -> are useful fictions

1. if mathematical objects exist, then abstract mind-independent objects exist
2. abstract mind-independent objects do not exist
   therefore.
3. mathematical objects do not exist (mathematical statements are not literally true)

occam’s razor: if two theories account for all the same facts, then we should endorse the more ontologically parismonious of the two
- fictionalism and platonism share same account of the truth-conditions of mathematical statements
- fictionalism denies that mathematical statements are literally true

Question 10

what is the indispensability argument?

Answer 10

1. we should believe in all and only those entities that are indispensable to our best scientific theories
2. mathematical objects are indispensable to our best scientific theories
   therefore,
3. we should believe in the existence of mathematical entities