Talmudic Foundations of Mathematics Andrew Schumann Alexander V. Kuznetsov University of Information Technology and Voronezh State University Management in Rzeszow Universitetskaya pl. 1 35-225 Rzeszow 394018 Voronezh Poland Russia [email protected] [email protected] ABSTRACT i.e. in fact it was the first attempt to make explicitly math- In this paper, we assume that the mathematicians in prov- ematics from the point of view of symbolic logic, that is an ing new significant theorem, such as Fermat’s Last Theorem, attempt to consider mathematical theorems as logical state- deal with combining proof trees on tree forests by using the ments which are automatically inferred from axioms by log- analogy as an inference metarule. In other words, the real ical inference rules. To continue and enhance this approach, mathematical proofs cannot be formalized as discrete se- David Hilbert, the German mathematician (1862–1943), put quences, but they are concurrent and can by formalized as forward a new proposal for the foundation of mathematics analog processes within a space with some topological prop- called the Finite Program (or Hilbert’s Program). In this erties. For the first time, inference metarules in a topological proposal all of mathematics should have been formalized in space were proposed in the Talmud within a general Judaic axiomatic form, together with a proof by ‘finitary’ methods approach to concurrent or even massive-parallel conclusions. proposed by Hilbert that this axiomatization is consistent. The mathematician does not think sequentially like a logi- Now there are some basic formal theories which are re- cal automaton, but concurrently, also. Hence, we suppose garded as start points in the foundations of mathematics. that the proof technique of real mathematics cannot be for- This means that these theories, in the way how it seems malized by discrete methods. It is just a hypothesis of the to mathematicians, can cover big fragments of mathemat- foundations of mathematics that we can use discrete tools so ics by their extensions. For instance, it is assumed that in that mathematics can be reduced to logic. We show in the the foundations for number theory we should start from the paper how the mathematical proof can be formalized just five Peano’s axioms, introduced by Giuseppe Peano in 1889 by analog computations, not discrete ones. and now called the Peano arithmetic PA. Also, it is sup- posed that any set-theoretic reasoning in mathematics (like reasoning in topology) can be reduced to statements for- CCS Concepts malized in the Zermelo-Fraenkel set theory, constructed by •Computing methodologies → Parallel computing me- mathematicians Ernst Zermelo and Abraham Fraenkel and thodologies; Artificial intelligence; •Parallel algorithms denoted by the abbreviation ZFC, where C means axiom of → Massively parallel algorithms; •Artificial intelligence choice. → Philosophical/theoretical foundations of artificial intelli- To sum up, mathematicians believe still that the founda- gence; tions of mathematics in the meaning of Principia Mathe- matica are possible and any correct well-done mathematical Keywords reasoning can be rewritten in a logical theory such as ZFC. So, they believe that all the mathematics can be reduced to Hilbert’s program; foundations of mathematics; Fermat’s a logic. Is it true indeed? Is it possible? last theorem; Cauchy criterion; topological Cauchy-Cantor In mathematics there are really non-trivial theorems which intersection theorem; proof trees are so deep that they cannot be inferred without introduc- ing absolutely new mathematical constructions. For exam- 1. INTRODUCTION ple, Fermat’s Last Theorem (FLT)iswellformulatedinPA. The Principia Mathematica, a three-volume work written Therefore this statement seems to be so simple. For the first jointly by Alfred North Whitehead and Bertrand Russell and time, FLT was put forward by Pierre de Fermat in 1637 in published in 1910, 1912, and 1913, was the first book the margin of a copy of Arithmetica. However, this state- mentwasprovenformallyonlyafter358yearsofeffortby mathematicians, namely by Andrew Wiles in 1995 [6]. The most dramatic problem of FLT recently is that this theo- Permission to make digital or hard copies of all or part of this work for personal rem is proved mathematically and this proof was accepted or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice by mathematical communities, but this statement was not and the full citation on the first page. To copy otherwise, to republish, to post on checked by logicians at all. It is unknown still whether there servers or to redistribute to lists, requires prior specific permission and/or a fee. is a logical proof of FLT. In other words, FLT is not covered BICT 2017, March 15-16, Hoboken, United States by any foundations of mathematics still. *4#/ As we said, FLT is well written in a first-order sentence %0*10.4108/eai.22-3-2017.152404 of PA. However, it does not mean that it can be proved Copyright © 2017 &"* in PA. After the Paris-Harrington theorem [3], it is well stance, the notation t = {c[{a[{}],b[{}]}]} means that we known that there are ever first-order arithmetic statements have a tree consisting only of two edges/premisses a[{}]and written in PA which cannot be proved in PA, such as the b[{}] and one conclusion from them c[{a[{}],b[{}]}]: Strong Ramsey Theorem that can easily be proved in the second-order arithmetic from the infinite version of the stan- dard theorem. Also, it is known that there are many other combinatorial problems that are beyond PA. In [2], Colin McLarty supposed that FLT can be proved in some higher- order extensions of PA, but nobody has checked it still. a b Another hypothesis of McLarty [2] is that FLT is beyond ZFC. It is quite evident taking into account the fact that cohomological number theory used by Wiles [6] is based on Grothendieck’s universes which model ZFC, but the exis- c tence of a universe is not provable in ZFC. Grothendieck’s own axiom of universes, which was added to ZFC,affirms that every set is contained in some universe (there is an un- countable strongly inaccessible cardinal for sets) [1]. Hence, in cohomology we deal with ZFC+U consisting of ZFC The notation t = {c[{a[{}],b[{}]}],a[{}]} designates the with the assumption U of a universe. So, FLT can be proved following tree: at least in ZFC+U or even in higher extensions, and evi- dently not in ZFC. Nevertheless, there is no formal proof still what is set theory looks like for FLT. Thus, there are ever serious mathematical theorems such as FLT which are beyond the recent foundations of mathe- matics (for instance, beyond PA or ZFC). However, math- a b ematicians and logicians unconsciously obey the quite re- ligious faith and follow the deep-inner intuition that any mathematical theorem can be reduced to a theorem of ex- isted symbolic logic. In symbolic logic we appeal to a formal T aL theory i that possesses logical axioms/theorems ( i1,..., a c L ain) within a logical system L like the classical propositional T T logic and non-logical axioms / theorems (ai1,...,aim)for defining properties of predicates and functions introduced T L in i. Then by using inference rules of we can infer L L The formula t = {d[{c[{a[{}],b[{}]}],a[{}]}],f[{}],e[{}]} in Ti all possible provable sentences from (ai1,...,ain)+ T T satisfies the following tree: (ai1,...,aim). Surely, it does work in symbolic logic indeed, but it is unknown still whether it gives something to real mathematics. In real mathematics, i.e. in a proof of deep sentences such as FLT, we use some axioms / presuppo- M M sitions / sentences (ai1 ,...,aik )andthemaintaskofthe T foundations of mathematics is to set up a formal theory i a M M b to find out an injective mapping from (ai1 ,...,aik )into T T (ai1,...,aim). The problem is that, on the one hand, some M M sentences from (ai1 ,...,aik )inFLT can be just intuitive and not well-formulated (or even not conscious). On the other hand, their true symbolic-logical analogues in Ti can T T a c be absent from the list (ai1,...,aim) and not sketched still. M M In this paper we assume that (ai1 ,...,aik ) for serious the- orems like FLT are not non-logical axioms / theorems which T T can be represented as (ai1,...,aim). So, we assume that higher mathematics cannot be reduced to symbolic logic. f M M M M Let us suppose now that (ai1 ,...,aik )=(ai1 ,...,aik )+ e L L T T T T L d (ai1,...,ain) and (ai1,...,aim)=(ai1,...,aim)+(ai1,..., L ain), i.e. they are closed under the inference rules of L. T aT aT t t Each proof in i =(i1,..., im)isatree .Letej be an inference rule of T , i.e. it is an elementary tree i Hence, in T we deal just with finite trees labelled in – it has one parent (the inferred statement) and several i (aT ,...,aT ) in the way defined above. its direct children (axioms as premisses). Then t consists i1 im Let an ordered sequence of trees be called a forest.We of te1 ,...,tej as its subtrees. Let {} be an empty tree. say a forest si = ti,ti,ti ,... is a piece of a forest sj = Then each edge/branch of t can be labelled by a sentence t ,t ,t ,..., written s s if s can be obtained from α ∈ (aT ,...,aT )asfollows: α[t], where t is an elemen- j j j i j i i1 im s by removing nodes.