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. In other words, there is an injec- tary subtree (inference rule) of t that was used to obtain j tive mapping from nodes of s to nodes of s that pre- α. Then a tree can be denoted by all its branches. For in- i j serves the lexicographic and descendant ordering. Then a
M M T T reduction of (ai1 ,...,aim)to(ai1,...,ain) would mean that of Section 1, if we define all the last particulars of the Torah aM ,...,aM aT ,...,aT aM ,...,aM apart,...,apart ( i1 im)isapieceof( i1 in), i.e. ( i1 im) as axioms of the Talmud: ( i1 in ). T T (ai1,...,ain). Now, let us consider an example from Exodus 22 and let In deep theorems like FLT we have ever a forest sj ,which us try to build up a proof tree for the notion ‘responsibility M M is a substantial part of (ai1 ,...,aim), such that sj is not a for the property of his neighbour’, just basing on the text of T T piece for any forest of (ai1,...,ain). So, we can distinguish the Torah. We have the following statements mentioned in M M two kinds of forests from (ai1 ,...,aim): this chapter of the Holy Book:
T G “He is responsible for the property of his neighbour.” 1. a forest sj which is a piece of some forest from (ai1, aT ..., in), these pieces are called logical fragments; G1 “The neighbour’s property is given for safekeeping for free of charge.” 2. a forest sj which is not a piece of any forest from aT ,...,aT s ( i1 in), this j is called a new construction. G2 “The neighbour’s property is given for safekeeping for money.” We assume that logical fragments are well studied in the foundations of mathematics, but new constructions are not G3 “The neighbour’s property is borrowed.” yet. The true mathematical task is to give non-trivial es- sential theorems containing new constructions. The math- G1P1 “The safekeeping for free of charge is stolen. He ematician starts his work with logical fragments to obtain should swear that he did not lay his hand upon his new constructions later. neighbor’s property.” Thus, the higher mathematics is eternally to extend forests If a man shall deliver unto his neighbour of (aM ,...,aM ) by proposing new constructions. Therefore, i1 im moneyorstufftokeep,anditbestolenout the mathematician deals not only with proofs/trees (what of the man’s house; if the thief be found, let can be reduced completely to pieces from (aT ,...,aT )), but i1 in him pay double. If the thief be not found, more often with forests which are irreducible to pieces of then the master of the house shall be brought (aT ,...,aT ). Mathematicians know that sometimes rea- i1 in unto the judges, to see whether he have put soning by analogy allows them to propose really something M his hand unto his neighbour’s goods (KJV, new. They take a piece from one theory Ti =(ai1 ,..., M Exodus, 22:7-8). aim) to combine it with forests of another theory Tj = aM ,...,aM T ( j1 jk) and then to obtain the next theory f = G2P1 “The safekeeping for money is stolen. He should pay aM ,...,aM ( f1 fl). the loss.” In the Talmud there was proposed a metareasoning for M M G2P2 “The safekeeping for money is destroyed for natural extending the dataset (forests) of the Torah, (ai1 ,...,aik ), by analogies for the first time (see Section 2). It is a way reasons. He should swear that he did not lay his hand M M upon his neighbor’s property.” how we can add new axioms to (ai1 ,...,aik ) just looking at existing forests. In Section 3 we will show how we can use these methods in the recent foundations of mathemat- If a man deliver unto his neighbour an ass, ics. Hence, the Talmudic metareasoning in the foundations or an ox, or a sheep, or any beast, to keep; of mathematics are called by us ‘Talmudic foundations of and it die, or be hurt, or driven away, no mathematics’, see [4], [5]. We assume that in a real mathe- man seeing it: Then shall an oath of the matical practice mathematicians use a similar metareason- Lord be between them both, that he hath ing to build up new forests by looking at existing forests (let not put his hand unto his neighbour’s goods; us notice that this metareasoning is considered often as a and the owner of it shall accept thereof, and mathematical intuition). he shall not make it good. And if it be stolen from him, he shall make restitution unto the owner thereof (KJV, Exodus, 22:10-12). 2. WHAT IS TALMUDIC LOGIC? Usually, the term ‘Talmudic logic’ means the Judaic her- G3P2 “The borrowed is destroyed for natural reasons. He ”.first formulated should pay the loss ,[4] (מידות ,meneutic rules (Hebrew: middot as a special hermeneutics by Hillel in the 1st century B.C. (he proposed the 7 rules), by Rabbi Ishmael in the 2nd cen- And if a man borrow ought of his neighbour, tury A.D. (he established the 13 rules for inferring the law, and it be hurt, or die, the owner thereof be- halakhah), and by Rabbi Eliezer ben Jose HaGelili in the ing not with it, he shall surely make it good. same 2nd century A.D. (he examined the 32 rules for infer- But if the owner thereof be with it, he shall ring the holy stories, haggadah). As a result, in the Talmud notmakeitgood:ifitbeanhiredthing,it we face a logical point of view in respect to the Torah: all the came for his hire (KJV, Exodus, 22:14-15). -or‘gen(פרט ) ’Biblical statements are considered ‘particular As we see, we deal here with the four last particulars (i.e. implying some conclusions by using hermeneutic (כלל ) ’eral Talmudic axioms): (G1P1, G2P1, G2P2, G3P2). How- rules. A Biblical statement, ϕi , is regarded particular,if part ever, the data set for ‘responsibility for the property of his |ϕi | ϕi its verification part (i.e. all denotates of part) is a subset neighbour’ is not complete – we know nothing about the |ϕj | ϕj of a verification gen for another statement, gen,namely: following two particulars (axioms) which are supposed also: i j i j |ϕpart|⊆|ϕgen|. In this case the implication ϕpart → ϕgen is true. Hence, we can draw appropriate proof trees by using G1P2 “The safekeeping for free of charge is destroyed for modus ponens and other classical inference rules in the way natural reasons. What should he do?”
G3P1 “The borrowed is stolen. What should he do?” G1P2 = min(G2P2, G1P1). In the picture form, the complete data set must be seen as follows: From this it follows that
G1P2 “The safekeeping for free of charge is destroyed for natural reasons. He should swear that he did not lay his hand upon his neighbor’s property.” G3P2 G3P1 G2P2 G2P1 G1P2 G1P1 Analogically, G3P1 is defined by qal wa-h. omer thus:
G3P1 = min(G2P1, G3P2). Then it is inferred that G3 G2 G1 G3P1 “The borrowed is stolen. He should pay the loss.”
So, the main goal of qal wa-h. omer is to add new axioms for the Torah data sets to make the proof trees more sym- G metrical: in the same tree t with only one root all subtrees must bear the same number of edges. For instance, in the tree
t = {G[{G1[{G1P1[{}]}], G2[{G2P1[{}], G2P2[{}]}], Nevertheless, in the Torah an appropriate data set is sketched G3[{G3P2[{}]}]}]} in the following manner, i.e. it is absolutely incomplete for inferring ‘responsibility for the property of his neighbour’: we had the subtree t = {G2[{G2P1[{}], G2P2[{}]}]} with the number of edges 2 that exceeds the numbers of edges of all other subtrees. Then we must add new branches to each subtree where the number of edges is less than 2.
G3P2 G2P2 G2P1 G1P1 Definition 1 (Inference Metarule qal wa-h. omer). Let a tree t have i subtrees: t1, ..., ti and let Nj be a number apart,...,apart j ,i of axioms ( j1 jN ) for each = 1 and let all ax- ioms for each subtree be lexicographically ordered in the same i way. Assume that max Nj = n. This means that there exists =1 G3 G2 G1 j l such that 1 ≤ l ≤ i and Nl = n. Let us suppose now that there exists k ∈{1,...,i} such that k = l and Nk = m proving their non-trivial sentences. In other words, they Let us denote definition 3 by Conv({xn}; R, |·|). deal not with mechanical proofs from existing axioms within the foundations of mathematics, but they combine different Definition 4. {xn}⊂R satisfies the Cauchy condition trees to expand the set of possible axioms beyond any foun- iff {xn}⊂R ∧∀(ε>0)∃(Nε ∈ N)∀(m, n ∈ N|n ≥ Nε ∧ m ≥ dations of mathematics. This way of proving is called by Nε)[|xn − xm| <ε]. us Talmudic because of the priority of the Talmudic logic in {x } R, |·| proposing some inference metarules for defining axioms. So, Definition 4 is denoted by CC( n ; ) let us generalize definition 1 as follows. Axiom 1 (Dedekind completeness). ∀(A ⊆ R|A = Definition 2 (Inference Metarule I). Let a tree t ∅)∀(B ⊆ R|B = ∅)[∀(a ∈ A)∀(b ∈ B)[a b] →∃(ξ ∈ R)∀(a ∈ A)∀(b ∈ B)[a ξ b]]. has i subtrees: t1, ..., ti and let Nj be a number of axioms M M i (aj1 ,...,ajN ) for each j = 1,i. Assume that max Nj = n. The completeness axiom is denoted by Ax3(A, B, a, b; R, |·|). j=1 This means that there exists l such that 1 ≤ l ≤ i and Nl = Definition 5. A sequence [an,bn] of closed intervals of n. Let us suppose now that there exists k ∈{1,...,i} such R is called a nested closed intervals system iff ∀(i ∈ N)[ai ∈ that k = l and Nk = m 3. METAREASONING IN MATHEMATICAL ∀(a ∈ A0)∀(b ∈ B0)[a b], Ax3(A, B, a, b; R, |·|) → ANALYSIS Ax3(A0,B0,a,b; R, |·|); Let us consider an example from mathematical analysis to show that a mathematician, extending a mathematical A ,B ,a,b R, |·| ↔ horizon, operates with forests and appeals to metarules for Ax3( 0 0 ; ) combining different trees, indeed. Great mathematicians do not prove theorems in the way like a logical automaton does ∃(ξ ∈ R)∀(a ∈ A0)∀(b ∈ B0)[a ξ b]. it. For example, Augustin-Louis Cauchy, the French mathe- The last inequality means that ξ is a common point of the matician of the 19th century who became a founder of mod- nested closed intervals system. ern mathematical analysis, put forward an axiom now called (2) The uniqueness of the common point. Dedekind Completeness for the expansion of mathematics. Assume a contrary: Due to this axiom, some properties of real numbers (non- logical axioms describing real numbers) can have transmit- ∀(n ∈ N)∃(ξ,ξ ∈ [an,bn])[ξ = ξ ] → ted to complex numbers, vectors, and even infinite sequences by a mathematical analogy of the Talmudic qal wa-h. omer ∀(n ∈ N)[|ξ − ξ | bn − an]; rule. b − a →R ∀ ε> ∀ n>N b − a <ε 3.1 Cauchy Criterion n n 0iff ( 0) ( ε)[ n n ]; Let us denote an infinite sequence (i.e. a countable set) of 1 1 objects x1,x2,...,xk,... by {xn}. Then some basic defini- ε = |ξ − ξ | > 0 →|ξ − ξ | < |ξ − ξ |. tions of the Cauchy approach are as follows: 2 2 R def It is a contradiction. Definition 3. xn → x = {xn}⊂R converges to x ∈ R what is defined thus: {xn}⊂R ∧∃(x ∈ R)∀(ε>0)∃(Nε) The statement of theorem 1 is denoted by CCP({ αn}; R, |· ∀(n ≥ Nε)[|xn − x| <ε]. |) where αn =[an,bn] is a closed interval. Definition {x } 6. Asequence n is called bounded iff 3.2 Qal Wa-H. omer for the Cauchy Criterion Let us fix the most important steps in the proofs of the ∃(C)∀(n ∈ N)[|xn|≤C]. previous subsection. {x } R, |·| Definition 6 is denoted by B( n ; ). The proof of theorem 1 can be represented as the following Theorem 2 (Bolzano–Weierstrass). Each bounded tree: {x }⊆R {x }⊆R sequence n has a convergent subsequence nk , t1 = {CSP[{ECP[{Ax3[{{an}[{}], {bn}[{}], SES[{}]}]}], R ∃(x ∈ R)[xnk → x]. R b − a → 0[{}]}]}, Proof. The fact that {xn} is bounded means that n n the same is in the graph form: ∃(C)∀(n ∈ N)[|xn|≤C] → ∀(n ∈ N)[xn ∈ [−C, C] ⊂ R]. We apply the following algorithm notated as BWAlgo: {an} {bn} SES({[an,bn]}; R, |·|) 1. a0 = −C, b0 = C. 2. For all i ∈ N \{0} let us divide [ai,bi] into two equal segments, and choose one of them which has an infi- nite number of members of the sequence, denote it by a ,b [ i+1 i+1]. Ax3({an}, {bn},a,b; R, |·|) 3. We obtain a sequence {[an,bn]} of closed intervals, i.e. there is SES({[an,bn]}; R, |·|). All closed intervals in this sequence contain an infinite number of members of the sequence {xk}. Lengths of this closed intervals |b0−a0| 2 R |b − a | C C ECP({[an,bn]}; R, |·|) converge to zero: m m = 2m = 2m = 2m−1 , bn − an → 0 C →R 2m−1 0. 4. Choose a sequence xkm ∈ [am,bm],m∈ N with the following condition: ∀(i ∈ Z)[ki−1 R)∀(m ∈ N)[ξ ∈ [am,bm]]] →∀(m ∈ N)[|xkm − ξ| |bm − R am|]] →|xkm − ξ| → 0. Theorem 2 is denoted by BWT({xn}; R, |·|). The proof tree for theorem 2 is built up as follows: Theorem 3 (Triangle inequality). ∀(x, y ∈ R)[|x+ t { { {{x } {} } } , y|≤|x| + |y|]. 2 = SES[ BWAlgo[ n is bounded [ ] ] ] The proof of theorem 3 for R is obvious. Let us denote its ∃!(ξ ∈ R)∀(m ∈ N)[ξ ∈ [am,bm]][{CCP[t1]}]}, statement by Tr(x, y; R, |·|). which is pictured in the following manner: R Theorem 4 (Cauchy criterion). ({xn}⊂R ∧ xn → x) ↔ CC({xn}; R, |·|) Proof. Necessity. If {xn} converges to x, then ∀(ε> 0)∃(Nε ∈ N)∀(n>Nε)[|xn −x| <ε], Hence, ∀(ε>0)∃(Nε ∈ {x } is bounded t N)∀(m>Nε)[|xm − x| <ε]. Then ∀(ε>0)∃(Nε ∈ N) ∀(n, n 1 m>Nε)[|xn − x| <ε∧|xm − x| <ε]. Thus, ∀(ε>0)∃(Nε ∈ N)∀(n, m > Nε)[|xn − x| <ε∧ |xm − x| <ε] ∧ Tr(xn − x, xm − x; R, |·|)impliesthat∀(ε> 0)∃(Nε ∈ N)∀(n, m > Nε)[|xn − xm| < 2ε], then ∀(ε> ∃ N ∈ N ∀ n, m > N |x − x | <ε 0) ( ε ) ( ε)[ n m ]. BWAlgo CCP({[an,bn]}; R, |·|) Sufficiency. CC({xn}; R, |·|) →∀(ε>0)∃(Nε)∀(m> Nε)[|xm − xNε | <ε]. If C =max{|x1|, |x2|, |x3|, ..., |xNε−1|, |xNε − ε|, |xNε + ε|}, then ∀(n ∈ N)[|xn|≤C]. This means that {xn} is bounded. So, {xn} is bounded and BWT({xn}; R R, |·|), then from ∃({xn }⊆{xn})∃(x ∈ R)[xn → x]it k k SES({[an,bn]}; R, |·|) ∃!(ξ ∈ R)∀(m ∈ N)[ξ ∈ [am,bm]] follows that ∃({xnk }⊆{xn})∃(x ∈ R)[|xnk − x| <ε]. Thus, CC({xn}; R, |·|)impliesthat∀(ε>0)∃(Nε ∈ N)∀(m, k ∈ N|n ≥ Nε ∧ m ≥ Nε)[|xnk − xm| <ε]. Then ∀(ε>0)∃(Nε ∈ N)∀(m, k ∈ N|n ≥ Nε ∧ m ≥ Nε)[|x − xm|≤ |x − x | |x − x | < ε nk m + nk 2 ]. Let us define now the proof tree for theorem 4: The Cauchy criterion is denoted by CCrit({xn}; R, |·|). t4 = {CC[{∀(ε>0)∃(Nε ∈ N)∀(n, m > Nε)[|xn − x| <ε∧ |xm − x| <ε][{Conv[{}]}], Tr[{}]}], We can formulate the same generalization by using defi- R R nition 2. Assume that we have some axioms (a1 ,...,aN ) C C ∀(ε>0)∃(Nε ∈ N)∀(m, k ∈ N|n ≥ Nε∧ for real numbers, some axioms (a1 ,...,aM )forcomplex V V numbers, and some axioms (a1 ,...,aL ) for vectors. We N>M N>L m ≥ Nε)[|xnk − xm| <ε][{B[{CC[{}]}]BWT[t2]}]}, know that (respectively, ), because in the beginning, R was studied better than C (respectively, bet- with the following graph: R R ter than V). Let us take a subset of axioms (a1 ,...,an ) R for real numbers, which express the real metrics. So, an is Ax3. Now let us define a partial ordering relation on R R (a1 ,...,an ). Let be a standard implication. For instance, Conv({xn}; R, |·|) → Ax3 means that Conv({xn}; R, |· Conv({xn}; R, |·|) Tr(x, y; R, |·|) CC({xn}; R, |·|) |) Ax3. Suppose that is extended to an ordering re- C C lation on the set of axioms (a1 ,...,am), where m BWT({xn}; R, |·|) ≺ CCrit({xn}; R, |·|)]∧ of elements in any reasonable way. Immediately, a question arose: what properties should X and ρ have in order to satisfy [Tr(x, y; R, |·|) ≺ CCrit({xn}; R, |·|)]. {x } X, ρ Theorem 4 holds for real numbers. Nevertheless, this the- CCrit( n ; ) = true? orem can be extended to some other numbers by a qal wa- It is clear that in this case we have to replace all expressions h. omer: of the form |x − y| by ρ(x, y) at all occurrences in {x iy }⊂C 1. A sequence of complex numbers n + n can CCrit({x }; R, |·|) {x iy } C, |·| n be defined in the way of CCrit( n + n ; C) 2 2 where |x + iy|C = x + y , i.e. it is sufficient for us Obviously, the field of real numbers is a special case of ρ x, y |x − y| to replace all occurrences of R to C and all occurrences metric space with the following metric: R( )= . of the metrics |·| to the metrics |·|C in the proof At the same time, the Cauchy criterion holds not in every statements of the theorem. metric space, for example, it does not hold in the space of continuous functions C[0, 1] with the metric ρL1 (x, y)= s 1 2. Also, sequences of vectors from R can be formulated |x(t) − y(t)|dt. Cauchy himself, of course, knew nothing s 0 as CCrit({xn}; R , ·Rs ) where about the metric spaces and could hardly suspect that there s 1/2 will be the following definition: 1 s i 2 (v ,...,v )Rs = (v ) . Definition 7. The metric space (X, ρ) where i=1 ∀ {x }⊂X {x } X, ρ ↔∃x ∈ X x →X x , So, we can generalize theorem 4 (the Cauchy criterion) due ( n )[CCrit( n ; ) ( )[ n ]] to the fact that there is an analogue for the Dedekind com- is called complete. pleteness axiom (Ax3) (defined in real numbers) that holds for aforesaid types of numbers which can be represented as The idea was to define the property Tr(x, y; X, ρ), natural some tuples of real numbers. In this case some new in- for (R, |·|), on the elements of (X, ρ), by transforming this equalities ai ξi bi, i =1,...,n for a =(a1,...,an), property from the derived theorem to the preexisted axiom, b =(b1,...,bn), ξ =(ξ1 ...,ξn) take place instead of the also by transforming CCrit({xn}; X, ρ) from the theorem one inequality of Ax3. to the axiom and by expecting that the final object will behave like numbers with almost the same structural the- proof is considered a discrete process that can be formal- orems. Surprisingly, it became true! For example, for R ized by discrete methods. However, it is only a hypothesis there exists the compactness of the bounded and closed set that mathematics can be reduced to logic and the mathe- and for the complete metric space of functions, continuous matical thinking is discrete. We can assume that it is not on [0, 1], with the norm xC =maxt∈[0,1] |x(t)|, also there so and a mathematical proof is an operation in a space of exists the compactness of the bounded set (with an addi- proof trees with some topological properties. As a result, tional condition of equicontinuity) – it is the claim of the the mathematical proof can be formalized just by analog Arzel`a–Ascoli theorem, denoted by AAT, the analogue of computations, not discrete ones. The meaning of mathe- BWT. The Cauchy-Cantor intersection theorem, i.e. CCP, matical proofs is to transform one space of proof trees to gets its counterpart in the form of the topological Cauchy- another space with inducing new topological properties. For Cantor intersection theorem, denoted by TCCP,forthe example, in the Talmud this transformation means that each closed non-empty nested subsets of X. branch in proof trees with one root should have the same Hence, the mathematicians of the end of the 19th cen- number of subbranches and inferring allows us to construct tury proposed to replace the axiom Ax3 by the theorem additional subbranches to make the trees more symmetric. CCrit({xn}; X, ρ) in the proof trees and, as a result, they In mathematics the goal of proofs is quite similar and it is changed all proof sequences invented by the mathematicians to extend the mathematical limits to make proof trees in of the early 19th century. They did it to extend the math- forests more symmetric, too. ematical limits in building new proof trees. For instance, if Hence, we suppose that computer-assisted proofs can be we define an ordering relation among non-logical axioms, ≺, based on some analog computations involving topological as follows: A ≺ B iff an axiom A is contained in the proof properties of proof trees. three of B, then the reasoning proposed by the mathemati- cians of the late 19th century is as follows: Acknowledgment For the first time, these ideas were expressed as a joke in [CCrit({xn}; X, ρ) ≺ AAT({xn}; X, ρ)]∧ the Andrew Schumann’s Russian-speaking blog: http://minski-gaon.livejournal.com/99986.html (3 Septe- [Tr(x, y; X, ρ) ≺ AAT({xn}; X, ρ)]; mber 2016); http://minski-gaon.livejournal.com/100791.html (21 Sep- [CCrit({xn}; X, ρ) ≺ TCCP(X, ρ)]∧ tember 2016). x, y X, ρ ≺ X, ρ . [Tr( ; ) TCCP( )] 5. REFERENCES To sum up, it is important to point out that if the mathe- [1] M. Artin, A. Grothendieck, and J.-L. Verdier, maticians followed the standard foundations of mathematics Theorie des topos et cohomologie et ale des schemas, in fact, they would not change the proof sequence. But they Seminaire de Geometrie Algebrique du Bois-Marie, did it indeed, because it is significant for drawing new trees 4, Springer-Verlag, 1972. by inference rules defined in definition 2. So, the new theo- [2] Colin McLarty, What does it take to prove Fermat’s {x } X, ρ X, ρ rems AAT( n ; )andTCCP( ) allow the mathe- last theorem? Grothendieck and the logic of number maticians to apply definition 2 in their reasoning more often theory, Bulletin of Symbolic Logic 16(3):359-377, and in more cases. 2010. [3] Paris, J. and Harrington, L. A Mathematical 4. CONCLUSIONS Incompleteness in Peano Arithmetic, in: Handbook We have just tried to show that the mathematicians deal for Mathematical Logic (Ed. J. Barwise). not with a logical way of automatic proving from some ax- Amsterdam, Netherlands: North-Holland, 1977. ioms, but with combining proof trees on tree forests by us- [4] Schumann, A. Preface, History and Philosophy of ing the analogy as an inference metarule. For the first time, Logic, 32(1):1-8, 2011. such metarules were proposed in the Talmud within a gen- [5] Schumann, A. Qal wa-homer and Theory of eral Judaic approach to concurrent or even massive-parallel Massive-Parallel Proofs, History and Philosophy of conclusions, see [4], [5]. The mathematician does not think Logic, 32(1):71-83, 2011. sequentially like a logical automaton, but concurrently, also. [6] A. Wiles, Modular elliptic curves and Fermat’s Last In the logical foundations of mathematics there are two Theorem, Annals of Mathematics, vol. 141, 1995, pp. approaches in drawing computer-assisted proofs: (i) auto- 443-551. mated theorem proving (i.e. proving mathematical theorems by computer programs) and (ii) automated proof checking (i.e. using computer programs for checking proofs for cor- rectness). There are many objections for these approaches. For instance, for (i) one the main objections is that these methods do not give new and useful concepts in mathemat- ics in fact, but they present just a long gloomy calculation. For (ii) one of the main objections is that these methods can check just very simple theorems. There are no even insights how to check FLT by computer programs. In our opinion, the most significant problem of existing logical foundations of mathematics is that a mathematical