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Topic: Is it possible that 3+4=8?
Replies: 21   Last Post: Jan 29, 2011 9:06 AM

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Joshua Cranmer

Posts: 487
Registered: 8/20/08
Re: Is it possible that 3+4=8?
Posted: Jan 23, 2011 8:20 PM
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On 01/23/2011 02:47 PM, Craig Feinstein wrote:
> In our real world, it is a known fact that 3+4=7. But how can we be
> sure that there are no invisible pixies running around taking balls
> away from us causing us to think that 3+4=7 when really 3+4=8?

It all depends, really, on how you define the symbols `3', `+', `4',
`7', and even `='.

One of the ways we formalize arithmetic [1] is via use of Peano's
axioms. If we take 0 to be an arbitrary entity--for example, the state
of having no balls in a set--and we add in a successor relation
S(x)--for example, the state of making a new set of balls by adding
another one two it--we can define numbers. Let our `3' be S(S(S(0)))
and `4' be S(3) or S(S(S(S(0)))), etc. We can then define this property
of addition:

a + 0 := a
a + S(b) := S(a + b)

With sufficient application of this rule, we can obtain that `3' `+' `4'
= `7', so we have a consistent set of definitions that can model our
observed phenomena. We can't necessarily rule out other models
there--what's to say that when we combine two sets of balls, they have
sex and produce an extra ball which is then immediately removed by
invisible ball pixies?

> My point is that mathematics is considered a deductive science, in
> which everything is absolutely certain. But how can mathematics prove
> that the above scenario cannot be true?

We can't exactly prove that the 3 + 4 = 8 isn't true for "reasonable"
definitions of those symbols, as "reasonable" can be very subjective,
but it is probably the case that you can't create a notion of `+' that
has the properties of addition that we're used to (most notably, x + 0 =
x) while keeping the other symbols defined the same.

[1] I am not an expert in formalism, and I'm sure I'll be criticized for
my description.

Beware of bugs in the above code; I have only proved it correct, not
tried it. -- Donald E. Knuth

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