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The 'One Over Many' argument is a classical argument for the existence of abstract objects (including numbers), stating that since many different things can share a common property, there must be a single abstract entity (the 'One') corresponding to that property.

The Quine-Putnam indispensability argument claims that our best scientific theories require that we believe in the existence of mathematical entities, suggesting a form of mathematical realism based on the empirical success of science.

In the philosophy of numbers, the concept of mathematical beauty pertains to the aesthetics, elegance, or simplicity of mathematical proofs and theorems, which some argue is indicative of mathematical truth and plays a role in theory choice and discovery.

Intuitionism is the philosophy that mathematics is a creation of the human mind. Numbers and mathematical truths do not exist outside of being conceived by a reasoning being.

Formalism is the philosophy that mathematics is not about any particular existing objects, but about the manipulation of symbols according to rules and conventions, devoid of any content.

Gödel's Incompleteness Theorems show that in any consistent formal system with basic arithmetic, there are true mathematical statements that cannot be proven within the system, emphasizing the limitations of formal systems and influencing views on the nature of mathematical knowledge and truth.

In the context of mathematics, Empiricism is the view that mathematical knowledge is derived from sense experience and that mathematical entities are abstractions from the physical world, as opposed to being independently existing objects or purely logical constructs.

The debate between Constructivism and Platonism in the philosophy of mathematics revolves around whether mathematical objects are purely mental constructs (Constructivism) or whether they have an independent, nonmental existence (Platonism).

Frege's Julius Caesar problem questions how we know that the numbers we talk about are numbers, and not something else, like a person. It's a problem of criteria for identity in the philosophy of numbers to distinguish numbers from other types of objects.

Finitism is the philosophical view that only finite mathematical entities have real existence. It often rejects the actual infinite and holds that mathematical objects must have a finite procedure to be valid or meaningful.

Axiomatic systems form the basis of formal mathematics by establishing a set of foundational truths or axioms. In the philosophy of numbers, they raise questions about the nature and justification of these fundamental principles which seemingly cannot be proven within the system itself.

Mathematical Platonism is the philosophy that numbers and other mathematical objects exist independently of the human mind, in a timeless, non-physical space of mathematical truths.

Mathematical fictionalism is the view that mathematical statements do not describe an objective reality but are like fiction; they are useful fictions that allow us to express and reason about quantitative and structural aspects of the world.

The Infinite Regress Argument in philosophy of mathematics concerns the foundations of mathematics. It questions whether mathematical truths are based upon other truths ad infinitum, or if there is a foundational truth that terminates the regress.

Benacerraf's Identification Problem is the issue of how to identify numbers or mathematical objects with physical entities, without losing their mathematical properties, which seems impossible and thus challenges the plausibility of a Platonist viewpoint.

Structuralism is the philosophical approach that focuses on the relationships between mathematical objects rather than the objects themselves. It maintains that the essence of mathematics lies in the structure those objects instantiate.

The application problem challenges the philosophy of numbers by questioning how abstract mathematical theories apply so effectively to real-world empirical observations if they are separate from empirical reality.

In the philosophy of mathematics, 'number sense' refers to the intuitive understanding or grasp of numbers and numerical concepts that human beings seemingly possess, particularly evident in basic arithmetic and estimation abilities.

The question 'Do numbers exist?' addresses the ontological status of numbers: whether they are real and have a separate existence or whether they are constructs of the human mind.

Conceptualism is the view that mathematical entities are neither physical nor located in a separate Platonic realm, but rather exist within our minds as concepts, reliant on human thinking and rationality.

In the philosophy of numbers, the problem of universals is about whether numbers, which may be seen as universals, truly exist independently (as realists claim) or whether they are simply names we give to particular patterns or properties that we observe in the world (as nominalists claim).

'How?' questions in philosophy of numbers pertain to the methodology and practices of mathematical activities, including how we come to know mathematical truths. 'What?' questions, on the other hand, aim at the ontological status of numbers and mathematical entities themselves.

The Indispensability Argument claims that since mathematical entities are indispensable to our best scientific theories, we must account for them as existing entities.

Logicism is the philosophical belief that mathematics can be reduced to and based entirely upon formal logic, implying that mathematical truths are essentially logical truths.