# Week 5: Mathematical approaches

## What is the role of mathematics in philosophy?

Philosophy as theory:

• A priori: Euclidean model of deductive certainty.
• Naturalistic: use math as sciences do, to build models.

Other views of philosophy: therapy, analysis, politics, history, poetry.

## Bayesian Epistemology

### Basic Assumptions

Beliefs are graded, with levels of confidence expressed by probabilities between 0 and 1.

To be rational, beliefs must be consistent with the laws of probability. Otherwise, the Dutch book argument shows that they will lose.

Bayes theorem: the probability of a hypothesis given the evidence is equal to the probability of the hypothesis times the probability of the evidence given the hypothesis, all divided by the probability of the evidence.

Bayesian confirmation: evidence E supports hypothesis H if P(H/E) > P(H).

Learning is changing probabilities in accord with Bayes theorem.

### Criticisms

Beliefs do not have probabiliies: few grades.

Probability is useful for statistical phenomena (frequencies), but not for subjective degrees of belief.

Hence the Dutch book argument is irrelevant.

Bayes' theorem is a mathematical truth, but hard to apply in real life.

Explanatory coherence provides a more plausible account of scientific inference.

Human learning is non-probabilistic.

### Response

Bayesian epistemology is normative, not descriptive.

## Decision theory

### Basic assumptions

Agents do and should make decisions that maximize expected utility.

Expected utility (action) = SUM Probability (outcome) * Utility (outcome).

### Criticisms

People rarely have information about probabilties and utilities.

People's preferences do not conform to the axioms of utility theory.

Decision making uses multiple criteria, not just utility.

Decision theory promotes immoral decisions: Kant vs. utilitarianism.

## Game theory

### Basic assumptions

Well-behaved preferences can be represented by utility functions, and rationality is maximizing utility.

Games have players, strategies, and payoff functions. E.g. prisoner's dilemma.

Game players jointly maximize expected utility, seeking equilibrium.

Game theory can explain morals by agreement.

### Criticisms

Rationality isn't maximizing utility.

The games described in game theory are highly artificial.

Social interactions are appropriately emotional, e.g. the ultimatum game.

The social contract is historically and psychologically irrelevant.

## Discussion Questions for Week 5

1. Are emotions contrary to rationality? Would you want to be emotion-free?
2. Are emotions evaluative?
3. Are emotions propositional attitudes?
4. Are emotions biological states?
5. Are emotions feelings?
6. Can emotions contribute to theoretical rationality?
7. Can emotions contribute to practical rationality?

PHIL 680

Computational Epistemology Laboratory.

Paul Thagard