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MatheRealism

 

MatheRealism as a philosophical foundation of mathematics = is based on the fact that mathematics is not independent of physical constrain= ts of reality. MatheRealism denies the existence of entities which, in principle, can never be observed= . MatheRealism distinguishes between numbers w= hich can be determined exactly and ideas= which can not. MatheRealism necessarily leads to the eliminatio= n of any actual infinity from mathematics. Please do not mix up MatheRealism with so called realism in the current philosophy of mathematics which, in f= act, is merely an idealism without any roots in the reality.

 

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= Material Considering or Supporting <= span lang=3DEN-GB style=3D'font-size:22.0pt;font-family:Arial;font-variant:small= -caps; mso-ansi-language:EN-GB'>MatheRealistic<= span lang=3DEN-GB style=3D'font-size:22.0pt;font-family:Arial;font-variant:small= -caps; mso-ansi-language:EN-GB'> Attitudes

 

Numbers are a free creation of the human mind.

- R= ichard Dedekind Was sind und was sollen die Zahlen?

 

There are at least two different ways of looking a= t the numbers: as a completed infinity and as an incomplete infinity.<= /span>

- Edward Nelson Completed versus incomplete infinity in arithmetic

 

A construction does not exist until it is made; when something new is made, i= t is something new and not a selection from a pre-existing collection.

- Edward Nelson Predicative arithmetic

&nbs= p;

When the objects of discussion are linguistic entities [...] then that collection of entities m= ay vary as a result of discussion about them. A consequence of this is that the "natural numbers" of today are not the same as the "natural numbers" of yesterday.

- David Isles What evidence is there that 2^65536 is a natural numbe= r?

 

Sequences generated by algorithms can be specified by those algorith= ms, but what possibly could it mean to discuss a 'sequence' which is not genera= ted by such a finite rule? Such an object would contain an 'infinite amount' of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics.

- Norman Wildberger<= /span>  = Set Theory: Should= You Believe?

 

Everything is finite. Give me one valid counter-example: a piece of evidence that the Infinite is actually useful in= physics, in a theoretical or practical sense.

- Han de Bruijn Nat= ural philosophy

 

We demonstrate here that the observed acceleration= of the Universe can produce a universal limit on the total amount of informati= on that can be stored and processed in the future, putting an ultimate limit on future technology for any civilization, including a time-limit on MooreR= 17;s Law.

- L.M. Kraus, G.D. Starkman Universal = limits on computation

<= o:p>

 

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Material Questioning the Concept of Act= ual Infinity

 

Die endlichen Weltmodel= le der gegenwärtigen Naturwissenschaft zeigen deutlich, wie diese Herrschaft eines Gedankens einer aktualen Unendlichkeit mit der klassischen (neuzeitlichen) Physik zu Ende gegangen ist. Befremdlich wirkt dem gegenüber die Einbeziehung des Aktual-Unendlichen in die Mathematik, d= ie explizit erst gegen Ende des vorigen Jahrhunderts mit G. Cantor begann. Im geistigen Gesamtbilde unseres Jahrhunderts [...] wirkt das Aktual Unendliche geradezu anachronistisch.

- Paul Lorenzen Das Aktual-Unendliche in der Mathematik

 

 

§ 124.  It isn't just impossible "for us men" to run through the natural numbers one by one; it's impossible, it means nothing.

…= ; you can’t talk about all numbers, because there'= s no such thing as all numbers.

§= 144  The = infinite number series is only the infinite possibility of finite series of numbers.= It is senseless to speak of the whole infinite number series, as if it, too, w= ere an extension.

§ 174 Set theory is wrong because it apparently presupposes a symbolism which doesn't exist instead of one that does exist = (is alone possible). It builds on a fictitious symbolism, therefore on nonsense= .

- Ludw= ig Wittgenstein Philosophical Remarks

&= nbsp;

Herren Geheimrat Hilbert und Prof. Dr. Cantor, I'd like to b= e Excused from your "Paradis= e": It is a Paradise of Fools, and besides feels more like Hell. [...] For many years I was sitting on the fence. I knew that it was a paradise of fools, b= ut so what? We humans are silly creatures, and it does not harm anyone if we m= ake believe that À= 0, À= 1, etc. have independent existence. Granted, some of the greatest minds, like Gödel, were fanatical platoni= sts and believed that infinite sets existed independently of us. But if one uses the name-dropping rhetorics, then one would hav= e to accept the veracity of Astrology and Alchemy, on the grounds that Newton and Kepler endorsed them.

- Doron Zeilberger http://www.math.rutgers.edu/~zeilberg/Opinion68.html

 

Cantor= 's 'paradise' as well as all modern axiomatic set theory is based on the (self-contradictory) concept of actual infinity. Cantor emphasized plainly = and constantly that all transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to persuade us to believe that the A= ST does not use actual infinity. It is an intentional and blatant lie, since if infinite sets, X and N, are potential, then the uncoun= tability of the continuum becomes unprovable, but withou= t the notorious uncountablity of continuum the modern= AST as a whole transforms into a long twaddle about nothing and really is "= ;a pathological incident in history of mathematics from which future generatio= ns will be horrified" (Luitzen Egbertus Jan Brouwer).

- Alexander Zenkin http://www.math.rutgers= .edu/~zeilberg/fb68.html=


If you= have an elaborate theory of 'hierarchies upon hierarchies of infinite sets', in which you cannot even in principle decide whether there is anything between= the first and second 'infinity' on your list, then it's time to admit that you = are no longer doing mathematics.

- Norm= an Wildberger Set Theory: Should= You Believe?

&= nbsp;

Why not?<= o:p>

&= nbsp;

A super task: Suppose you have a giant vase and a bunch of ping pong balls with an integer written on each one, e.g. just like the lottery, so the balls are numbered 1, 2, 3, ... and so on. At one minute t= o noon you put balls 1 to 10 in the vase and take out number 1. At half a minute to noon you put balls 11 - 20 in the vase and take out number 2. At one quarter minute to noon you put balls 21 - 30 in the vase and take out number 3. Continue in this fashion. Obviously this is physically impossible, but you = get the idea. Now the question is this: At noon, how many ping pong balls are in the vase? http://groups.google.nl/group/sci.math/brow= se_frm/thread/5276b15032c5d79f?scoring=3Dd&hl=3Den

 

Here are some essays of mine:

Par= adoxes of sequences of digits

On Cantor's theorem

 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= ~~~~~~~~~~~~~~~~~~~~~~~~~~

Do you know when the expression = well-orderi= ng was coined?

Do you know the difference betwe= en potential and a= ctual infinity?

To quote opinions in favour of s= et theory and actual infinity would mean carrying coals to Newcastle

but look here ...

Philosophy and the infinite (collected by E.D. Buckner)  htt= p://uk.geocities.com/frege@btinternet.com/cantor/Phil-Infinity.htm

 

~~~~~~~~~~~

 

=

 

12 lectures and a textbook on the history of the infinite (in German)

 

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=

 

10 chapters on the mathematics of the infinite (in German)

 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= ~~~~~~~~~~~~~~~~~~~~~~~~~~

 

Some useful links

 

The Göttinger Digitalisierungs-Zentrum supplies a vast amount of literature http://gdz.sub.uni-goettingen.de/no_cache/dms/suche/?tx_goo= bit3_pi1%5Blink%5D=3D0

University of Michigan Historical Math Collection <= span lang=3DEN-GB style=3D'font-size:12.0pt;font-family:Arial;mso-ansi-language:= EN-GB'>http://quod.lib.umich.edu/cgi/t/text/text-idx?page=3Dbrowse&= amp;c=3Dumhistmath

Index of historical Bookpages http://www-= groups.dcs.st-and.ac.uk/~history/Bookpages/<= /o:p>

Directory of Classic Literature http= ://www.greektexts.com/index.html

Euclid's Elements http://aleph0.clarku.edu/~djoyce/java/elements/toc.html<= o:p>

Euclid's Elements (Greek  and English Text, 4.57 MB) http://farside.ph.utexas.edu/euclid/euclid.pdf        &= nbsp;     

Earliest known uses of some of the words of mathematics http://jeff560.tripod.com/mathword.html

Eric Weisstein's Wolfram MathWo= rld http://mathworld.wolfram.com/=

Eric Weisstein's World of Scientific Biography http://scienceworld.wolfram.com/biography/

PlanetMath http://planetmath.org<= /span>/

St. Andrew's biographies of famous mathematicians by J J O'Connor and E F Robertson http://www-= history.mcs.st-andrews.ac.uk/Indexes/A.html<= /o:p>

The works of Leonhard Euler online http://www.math.dartmouth.edu/~euler/welcome.html

The Project Euclid Collection: Literature http://projecteuclid.org/DPubS?Service= =3DUI&version=3D1.0&verb=3DDisplay&handle=3Deuclid&page=3Da= bout&aboutArea=3Dsubscribers&aboutPage=3Dabout_linking

arXiv.org, the preprint server of Cornell University http://= arxiv.org/

Wikipedia http://en.wikipedia.org/wiki/Main_Page

Material on the infinite maintained by Rudolf Spon= sel: http://www.sgipt.org/wisms/geswis/mathe/m_unend0.htm

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= ~~~~~~~~~~~~~~

 

I am grateful for any useful suggestions. <= span lang=3DEN-GB style=3D'font-size:12.0pt;mso-ansi-language:EN-GB'>=

Apr= il 26, 2009 Wolfgang Mueckenheim

wolfgang.mu= eckenheim@hs-augsburg.de

 

 

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