more on this theme     |     more from this text

Single Idea 23622

[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism ]

Full Idea

Numbers cannot be mental objects constructed by our own minds: there exists at most a potential infinity of mental constructions, whereas the axioms of mathematics require an actual infinity of numbers.

Gist of Idea

We can only mentally construct potential infinities, but maths needs actual infinities

Source

Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro 2)

Book Ref

Hossack, Keith: 'Knowledge and the Philosophy of Number' [Routledge 2021], p.3

A Reaction

Doubt this, but don't know enough to refute it. Actual infinities were a fairly late addition to maths, I think. I would think treating fictional complete infinities as real would be sufficient for the job. Like journeys which include imagined roads.

Related Idea

Idea 23626 Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack]

The 29 ideas from Keith Hossack

23621 Numbers are properties, not sets (because numbers are magnitudes) [Hossack]
23622 We can only mentally construct potential infinities, but maths needs actual infinities [Hossack]
23623 Predicativism says only predicated sets exist [Hossack]
23624 The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack]
23625 Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack]
23626 Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack]
23627 'Before' and 'after' are not two relations, but one relation with two orders [Hossack]
23628 The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack]
10663 A thought can refer to many things, but only predicate a universal and affirm a state of affairs [Hossack]
10664 Complex particulars are either masses, or composites, or sets [Hossack]
10665 Leibniz's Law argues against atomism - water is wet, unlike water molecules [Hossack]
10666 Plural reference will refer to complex facts without postulating complex things [Hossack]
10669 Plural reference is just an abbreviation when properties are distributive, but not otherwise [Hossack]
10668 We are committed to a 'group' of children, if they are sitting in a circle [Hossack]
10671 Plural definite descriptions pick out the largest class of things that fit the description [Hossack]
10675 A plural comprehension principle says there are some things one of which meets some condition [Hossack]
10673 Plural language can discuss without inconsistency things that are not members of themselves [Hossack]
10674 A plural language gives a single comprehensive induction axiom for arithmetic [Hossack]
10676 The Axiom of Choice is a non-logical principle of set-theory [Hossack]
10677 Extensional mereology needs two definitions and two axioms [Hossack]
10678 The relation of composition is indispensable to the part-whole relation for individuals [Hossack]
10682 The fusion of five rectangles can decompose into more than five parts that are rectangles [Hossack]
10681 In arithmetic singularists need sets as the instantiator of numeric properties [Hossack]
10680 The theory of the transfinite needs the ordinal numbers [Hossack]
10684 I take the real numbers to be just lengths [Hossack]
10683 We could ignore space, and just talk of the shape of matter [Hossack]
10685 Set theory is the science of infinity [Hossack]
10686 The Axiom of Choice guarantees a one-one correspondence from sets to ordinals [Hossack]
10687 Maybe we reduce sets to ordinals, rather than the other way round [Hossack]