phil of math

Instantiation of Numbers
- The term "set" refers to an instantiation of a number as a plurality.
- Example: "12 doughnuts" as an instantiation of the number "12".
- In this language, one would formally express: "12 doughnuts is a plurality".
- General concept of numbers leads to specific numbers (e.g., the number 12), which further leads to instantiations in the real world (e.g., plurality of doughnuts).
- Examples of instantiations:
  - 12 doughnuts
  - 2 chairs
General Concepts vs. Specific Instances
- A general numerical concept is exceptionally abstract and should exist independent of the physical world or distinct points in time.
- The account of what constitutes a number should clarify why mathematical entities are abstract rather than concrete.

Frege's Example
- A hypothetical where Julius Caesar is equated to the number two highlights why such identifications are problematic.
- Identification fails since Julius Caesar is a historical figure and not a number.
Definition of Numbers
- A proper definition of numerals must prevent confusion that makes concrete entities (like Julius Caesar) represent numbers.
- Need to explain why mathematical equations involving numbers (e.g., "Julius Caesar + Julius Caesar = 4") do not hold.

Kantian Influences
- Philosophers following Immanuel Kant, including Russell, have attempted to differentiate between two types of propositions.
- Analytic Propositions
- Truth based on the arrangement and meanings of the component terms.
- Example: "All bachelors are unmarried".
  - Analytic because knowing the meaning of "bachelor" reveals its truth.
- Synthetic Propositions
- Truth contingent upon the represented states of affairs.
- Example: "The solar system has nine planets".
  - Requires empirical investigation to ascertain its truth.

Analytic Statements
- Truth derived simply from the meanings and arrangement.
- Can often be assessed using truth tables or logical evaluation (e.g., tautologies).
Synthetic Statements
- Require more investigation and evidence-based research.
- Defining features include potential changes in truth value based on empirical developments (as with the number of known planets).

Belief in Mathematics as Analytic
- Russell posited that truth in mathematics exclusively stems from the meanings of relevant terms, not from concrete world relations.
- Potential to derive all mathematical truths through deep understanding of term meanings, albeit the challenge lies in rigorous comprehension.

Core Issues to Address in Mathematical Philosophy
- Explanation needed on how mathematics can be deemed abstract, focusing on the dissimilarity between concrete objects and numbers.
- Understanding of why truths of number theory and propositions are analytic rather than synthetic.

Logicism Defined
- Logicism proposes that all mathematics can be understood purely through logic.
- Frege, a pivotal figure in this argument, claimed that mathematical truths equate to logical truths.
- Shares a close relationship with deductive logic as referenced in prior studies.
Defining Mathematics Through Logic
- Frege’s work "Foundations of Arithmetic" sought to place mathematics on a logical foundation.
- Most notably revolved around first-order logic, showcasing how mathematics functions as a logical extension.

Extensional vs Intentional Definitions
- Extensional Definitions: Defined by listing all examples covered by the definition.
  - Inefficient for large or infinite sets (e.g., list of humans in Manhattan).
- Intentional Definitions: Defined by identifying properties that encompass all elements in the definition.
  - More practical and widely used when discussing infinite sets like numbers.

Set Definition
- A set is a collection of distinct objects referred to as elements or members.
- Not affected by ordering (e.g., set of apples, set of natural numbers).
- Bijection
- A relation showing a one-to-one correspondence between two sets.
  - Each element in one set corresponds to exactly one element in another set, and vice versa.
- Example: Each person's unique fingerprint represents a bijective relationship to the person.
Equinumerous Sets
- Sets that can relate one-to-one are said to be equinumerous, sharing the same size.
Defining Numbers Through Sets
- Number defined as a set corresponding to objects that exhibit one-to-one relations with another set.
  - Example: The number two is defined as the set of all sets containing exactly two elements.
  - This highlights the abstraction involved in the definition of numbers, highlighting their nature as collections rather than concrete individual items.

Philosophical Implications of Logicism
- Logicism strives to elucidate the truth of mathematics through logical deductions.
Challenges to Address
- The analyticity problem challenges the assumption that all logical propositions are analytic.
- Conceptualization problems about humanity’s historical understanding of numbers amidst later developments.
- Issues related to set theory could complicate definitions of numbers and ultimately logicism itself.

Platonism
- A significant alternative emphasizes the existence of a mathematical reality that accounts for all mathematical truths.
Conclusion
- These philosophical explorations highlight the complexities of defining mathematics, numbers, and their existence, invoking profound questions about the nature of reality and abstraction in mathematics.