Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

Topic: Can addition be defined in terms of multiplication?
Replies: 58   Last Post: Aug 23, 2013 3:56 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Jim Burns

Posts: 1,130
Registered: 12/6/04
Re: Can addition be defined in terms of multiplication?
Posted: Aug 16, 2013 8:05 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 8/16/2013 4:54 AM, Peter Percival wrote:
> Can addition be defined in terms of multiplication? I.e.,
> is there a formula in the language of arithmetic
>
> x + y = z <-> ...
>
> such that in '...' any of the symbols of arithmetic
> except + may occur? Or, alternatively, is there a
> formula in the language of arithmetic
>
> x + y = ...
>
> with the same requirement?
>
> The symbols of arithmetic (for the purpose of this question) are either
>
> individual variables, (classical) logical constants including =,
> S, +, *, and punctuation marks;
>
> or the above with < as an additional binary predicate symbol.


How about

x + y = z <-> 2^x * 2^y = 2^z

where 2^x is just an abbreviation for the function 2pwr: N -> N,
defined by
2pwr(0) = 1

2pwr( Sx ) = 2 * 2pwr( x )

It's true that I use the successor S, which I hear implicitly
uses addition, but without S I don't see how to do much of
anything at all.

I wonder how Nam defines * without using either + or S.

Come to think of it, if I can use S, why can't I just
go ahead and define + in the usual (so far as I know) fashion
x + 0 = x

x + Sy = S(x + y)

It seems to me that, if one is going to do arithmetic
without addition, unique prime factorization becomes central.
It might be useful to represent numbers by their prime exponents,
so that
3 * 4 = 12
becomes
( 0, 1, ...) * ( 2, 0, ...) = ( 2, 1, ...)
with special rules for 0, of course. It looks like countably
many copies of N, with only finite many copies non-zero.
Each copy has its own successor function S2(x) = 2*x,
S3(x) = 3*x, ...

However, I am daunted by the prospect of defining * in
this system. We would need to give rules that explain why
3 + 4 = 7
or, rather, why
( 0, 1, 0, 0, ...) + ( 2, 0, 0, 0, ...) = ( 0, 0, 0, 1, ...)





Date Subject Author
8/16/13
Read Can addition be defined in terms of multiplication?
Peter Percival
8/16/13
Read Re: Can addition be defined in terms of multiplication?
William Elliot
8/16/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/16/13
Read Re: Can addition be defined in terms of multiplication?
David C. Ullrich
8/16/13
Read Re: Can addition be defined in terms of multiplication?
namducnguyen
8/17/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/17/13
Read Re: Can addition be defined in terms of multiplication?
namducnguyen
8/17/13
Read Re: Can addition be defined in terms of multiplication?
fom
8/23/13
Read Re: Can addition be defined in terms of multiplication?
tommy1729_
8/16/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/16/13
Read Re: Can addition be defined in terms of multiplication?
Robin Chapman
8/16/13
Read Re: Can addition be defined in terms of multiplication?
Helmut Richter
8/16/13
Read Re: Can addition be defined in terms of multiplication?
Rotwang
8/16/13
Read Re: Can addition be defined in terms of multiplication?
Virgil
8/22/13
Read Re: Can addition be defined in terms of multiplication?
Rock Brentwood
8/16/13
Read Re: Can addition be defined in terms of multiplication?
Shmuel (Seymour J.) Metz
8/17/13
Read Re: Can addition be defined in terms of multiplication?
Helmut Richter
8/16/13
Read Re: Can addition be defined in terms of multiplication?
Jim Burns
8/16/13
Read Re: Can addition be defined in terms of multiplication?
fom
8/17/13
Read Re: Can addition be defined in terms of multiplication?
Robin Chapman
8/17/13
Read Re: Can addition be defined in terms of multiplication?
fom
8/17/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/17/13
Read Re: Can addition be defined in terms of multiplication?
fom
8/17/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/17/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/18/13
Read Re: Can addition be defined in terms of multiplication?
William Elliot
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/18/13
Read Re: Can addition be defined in terms of multiplication?
William Elliot
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/18/13
Read Re: Can addition be defined in terms of multiplication?
David C. Ullrich
8/18/13
Read Re: Can addition be defined in terms of multiplication?
David C. Ullrich
8/17/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/18/13
Read Re: Can addition be defined in terms of multiplication?
David Bernier
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Ben Bacarisse
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Jim Burns
8/18/13
Read Re: Can addition be defined in terms of multiplication?
fom
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Ben Bacarisse
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/18/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/19/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/19/13
Read Re: Can addition be defined in terms of multiplication?
Alan Smaill
8/19/13
Read Re: Can addition be defined in terms of multiplication?
fom
8/19/13
Read Re: Can addition be defined in terms of multiplication?
Alan Smaill
8/20/13
Read Re: Can addition be defined in terms of multiplication?
Alan Smaill
8/20/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/20/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/20/13
Read Re: Can addition be defined in terms of multiplication?
Graham Cooper
8/22/13
Read Re: Can addition be defined in terms of multiplication?
David Libert
8/22/13
Read Re: Can addition be defined in terms of multiplication?
Peter Percival
8/20/13
Read Re: Can addition be defined in terms of multiplication?
fom

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.