The philosophy of numbers(Pictured: The multitalented AdaLovelace (b. 1815) was a mathematician and
The philosophy of numbers(Pictured: The multitalented AdaLovelace (b. 1815) was a mathematician and is often regarded as the world’sfirst computer programmer.)What are numbers? You can’t think theminto existence mentally. You can’t bump into them physically. And you’d be wildto follow Plato in believing they exist as changeless abstract objects outside of spacetime. So what are they?!Let’s start with 2.‘“2” denotes a pair of things,’ youmight say, pointing two fingers from a clutched hand.But that’s cheating! Here ‘pair’ signifieswhat 2 is, ensuring your definition is circular, like a serpent eating its tail. The challenge applies to all numbers, notjust 2, all the way to infinity.[Cantor to the rescue!]Mathematician Georg Cantor offered aninteresting solution with his theory of infinite sets.Sets are definable collections of things,not just numbers. For example, take teacups, black cats, and books. Eachcollection is a set because its members share something in common: they are allkinds of teacups, black cats, or books. In set-theoretic notation we mightgroup together a set M of mathematicians, {Euclid, Lovelace, Turing . .. }, in which Lovelace, l, is a member. Thus l ∈ M (‘l is a member of M’).We use the same syntax for numbers.Let’s start with absolutely nothing (zero)and symbolise it with ‘Ø’.Thenlet’s say ‘1’ is the set of nothing, {Ø}.Nice.Then let’s say ‘2’ is the set of ‘1’, {{Ø}} …Woo! Can you see what’s happening? We’reactually defining 2 without reference to itself! And we can go on infinitely(in theory)!Numbers, Cantor thought, are expressionsof sets of sets of sets … each a distinct object.His philosophy on the nature of numberswas a precursor to further work, in which he claimed mathematicians were freeto posit the existence of abstract things so long as they were devoid ofinternal contradiction.However, Cantor’s view is controversial:if you think about it, he argued that Ø—nothing!—founds all other numbers like1, 2, and 3. This is a bit absurd, don’t you think? -- source link
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