Developments in Math Prior To Gödel


            Ever since the time of Euclid, mathematics has been considered the standard for objective rational though.  The reduction of geometry to a small number of self-evident propositions called axioms from which all possible theorems could be deduced set mathematics on the surest rational basis.  The simplicity of the axioms, their “obvious truthfulness” when applied to the spatial world, and the enormous number of theorems derivable from them, set geometry apart as a model of what it means to think with rigor.


            This picture of the pure logic of mathematics continued from the time of Euclid until the creation of non-Euclidean geometry in the 18th Century.[1]  Suddenly, by replacing Euclid’s parallel axiom with a contradictory axiom, an entire set of new theorems could be mathematically proven.  These alternative forms of geometry were used as evidence by Kant for his assertions that mathematics is based on pure intuition and thus is not the result of the laws of logic and strict definitions.[2]   While at the time in which Kant wrote, this assertion might not have caused a great furor, it came into direct conflict with the positivistic spirit of the last half of the Nineteenth Century. 


            Further compounding of the “difficulty” came in the work of Labachevsky and Riemann who, in the first half of the Nineteenth Century, proved that it was impossible to deduce the parallel axiom from the other axioms of Euclid’s system.  Now not only did non-Euclidean geometries exist, it was impossible to prove the superiority of Euclidean geometry without appeal to some form of intuition.


            The positivistic Zeitgeist viewed intuition with great suspicion (as well they might) and set an agenda for mathematicians of proving the consistency of mathematics without appeal to intuition no matter how sensible it seemed.[3]  In a time in which metaphysics was a dirty word, mathematicians desired to put mathematics         on the surest possible rational grounding.  The rise of the Vienna Circle of Logical Positivism, while later chronologically, is perhaps parabolically indicative of the spirit in which mathematical research was carried out.  The goal of this research was stated by Hilbert, “to remove once and for all the questions of foundations in mathematics.”[4] 


            The invention of non-Euclidean geometries that do not seem to correspond to the reality of the external world (as it was assumed that Euclidean geometry did) and this positivistic attitude resulted in the formulating of the question of the consistency of various axiomatic systems.  The question is no longer whether the mathematical formulations correlate to reality but whether they are internally and logically consistent.[5]  This question of consistency was approached in three ways, broadly speaking, to which we shall now turn.


            This first approach to the consistency of mathematics is found in Hilbert’s attempt to ground mathematics, specifically geometry, in an axiomatic fashion.  While Hilbert was able to demonstrate that certain portions of geometry needed no intuition and that the axioms of geometry was shown only by basing it on the consistency of an arithmetic model by means of analytic geometry.  Thus, if arithmetic is reduced to its axioms, geometry is only consistent if arithmetic is consistent. Thus, Hilbert proved the relative consistency of Euclidean geometry based upon the assumed absolute consistency of arithmetic.[6]  Non-Euclidean geometry had its relative consistency proved by Riemann.[7]   Hilbert temporarily left the study of foundations assuming that the consistency of arithmetic would be proven by some modification of Dedekinds methodology in number theory, for the field of theoretical physics.  In the interim; however, other developments in the field proved that Hilbert’s optimism about the consistency proof of arithmetic were unfounded.


The second approach to the consistency of mathematics lies in the efforts of Frege and Russell to reduce arithmetic to pure logic.  The formers motivation for logically formalizing proofs lay in the positivistic confidence that such proofs would need no appeal to intuition and that “there would be last be certainty that proofs were cogent and rested only on the assumptions explicitly stated.[8]   In his book, Die Grundlagen der Arithmatik, Frege critiqued previous definitions of number and arithmetical truth and attempted to prove the basic laws of arithmetic from purely logical principles.  In the companion volumes of Die Grundgesetze der Arithmatik.  Frege attempted to bolster the deficiencies in the philosophical logic of the Grundlagen by placing the proofs into a formal system.  His fatal mistake was the addition of set theory which opened his purely logical system up to paradoxes of set theory.[9] 


The problem of paradoxical contradictions in Frege’s system was pointed out by Bertrand Russell while Grundgesetzen was being printed.  The antinomy known as “Russell’s Paradox” involves sets, which are of themselves and those, which are not.  Let “A” be the class of all mathematicians.  Since “A” is not a mathematician “A” does not contain itself and is a natural set.  Let “B” be the set of all thinkable things.  Since “all thinkable things” is thinkable, the set “B” contains itself as one of its members and is therefore non-natural.  Let “C” be the set of all natural sets.  Is “C” a natural or non-natural set?  If “C” is natural, then it contains itself for all natural sets including “C” are members of “C”.   But if “C” contains itself as a member, it is non-natural.  If “C” is non-natural, it contains itself but then “C” is natural because the members of “C” are natural classes.  Ergo, “C” is natural if and only if “C” is non-natural.  Russell’s discovery of this paradox based on paradoxes of Burali-forti and Cantor meant the Frege’s system lacked an adequate logical foundation.[10]   While Russell sought to escape the paradox by various means, finally adopting the theory of types,[11] Frege ultimately abandoned the entire logicist program.[12]    Russell’s theory of types as a solution to the paradox was incorporated into the famous Principia Mathematica in 1910.[13]   In this work he discusses the various paradoxes and attempts to skirt them by differentiating levels in statement in which self reference is possible.  Thus the liars’ paradox is solved by differentiating between orders of statements:


If we regard the statement “I am lying” as a compact way of simultaneously making all the following statements:  “I am asserting a false proposition of the first order.”   “I am asserting a false proposition of the second order,” and so on we find the following curious state of things:  as no proposition of the first order is being asserted, the statement “I am asserting a false proposition of the first order” is false.  This statement is of the second order, hence the statement.  “I am making a false statement of the second order” is true.[14]



The ambiguity of words such as true, false, class name, definition, etc., produces these paradoxes by failing to distinguish orders or levels in these terms themselves.  Russell assumed that he had solved the paradoxes and that in the Principia he had reduced mathematics to logic.[15]  We shall see later that after Gödel, such was not the case.


            The third approach to grounding[16] came in Hilbert’s later years when he returned to the questions of mathematical foundations in his scholarly work.  This approach, called “formalism” is in some sense the reverse of Russell’s approach in which theorems are produced by deductive reasoning.  Hilbert sought to empty the calculus of all meaning so that they are meaningless formulae and to prove deductive theorems by pure symbolic manipulation without resource to deductive reasoning.  Thus, the problem of proof is reduced to a consideration of whether symbolic manipulations of the original formulae can produce the desired theorem.  The system returns to mathematics (rather than to mere symbols on the page) when the calculus is interpreted as representing the axioms of a mathematical system and the rules of symbolic manipulation as representing the logical rules of inference.  Thus, Hilbert provided a way to prove entire classes of theorems by means of formal symbolic representation.[17]   Hilbert made it his goal to completely formalize arithmetic in this fashion and thus to prove its absolute consistency.  This proof; however, must be by finitistic means and it must involve meta-mathematical reasoning including substitution, induction, etc.  Subsequent developments however, did not lead in this direction and while parts of arithmetic were formalized and proven to be consistent, broad systems could not be finitistically proven in this way.[18] 


            In this situation, with an absolute finitistic consistency proof as a goal with the logicism of Russell’s Principia assumed to be on target, Gödel’s paper dropped like a bombshell, what Goedel actually proved, we shall now examine. 




            In 1931, Kurt Gödel, a 25 year ole mathematician and member of the Vienna Circle[19] published his famous paper.  His motivation was evidently to show that Russell’s attempts to bypass the paradoxes by the theory of types do not solve all of the possible paradoxes. Gödel  begins by building a formal system called “P” which is essentially imposing the Peano axioms for whole number arithmetic upon the logical system of the Principia Mathematica[20].   Since Gödel is dealing with a formal system, his basic approach is that of Hilbert which is then applied to the logic of Principia Mathematica.[21] 


Gödel constructs his arithmetical system by careful definition of the signs, variables, axioms, and formulae.  In particular, Gödel defines recursive functions in such a fashion that any element of an infinite sequence can be defined by its rule of construction.   This Definition of recursiveness is the subject of the first four propositions in the proof proper.  


The other significant introductory matter in the proof is Gödel’s famed mapping of all propositions in his system onto numbers, which serve as signs of logical symbols and functions.  This mapping involves using primes to signify various symbols, variables, etc., by raising these primes to powers for sentential or predicate variables every possible formula in the system has a unique Gödel number.  Thus in a formal system, a proof that an arithmetic relationship exists between the Gödel numbers of two formulae also indicates a deductive relationship between the two formulae.   Perhaps more significantly meta-mathematical statements can be reduced to unique formulae within the system, which are given unique Gödel numbers.[22]   The relationship between mega-mathematical statements and their corresponding arithmetical formulae can thus be analyzed in terms of the relationship of their Gödel numbers.  Meta-mathematical concepts such as “proof-schema”, “formula”, and “provable formula” are definable in Gödel’s system.


            Gödel then uses his system to construct a formula “G” which states that a certain formula is not provable in Gödel’s system.  It turns out that this unprovable formula has the Gödel number of “G” itself.  Thus “G” affirms that “G” is not provable. Gödel acknowledges the similarity between his proof and both the Liar’s paradox and Richard paradox.[23]   His paradox; however, skirts the falsity of the others without falling into it.  The difference lies in asserting provability rather than truth or falsity.  Gödel’s “G” is false and unprovable.[24]   All of this is based on the assumed consistency of Gödel’s system.  If the system in inconsistent, every contradictory theorem is provable.  But if it is consistent (and if it is not, no full arithmetical system is consistent), “G” cannot be proven within “P”, Gödel’s arithmetical system.  Meta-mathematical considerations are necessary to determine “G” provability.  


            A corollary of proposition VI that is of such monumental importance (Gödel’s proposition XI) states that the consistency of the entire arithmetical system “P” is not provable in

“P” but only by meta-mathematical considerations.  In other words, any know arithmetical system adequate for number theory cannot be formally proven to be consistent within the system.

Thus, not only is a specific proposition not provable though correct in “P”,  “P” itself is not probable in “P”.  Thus both Russell’s and Hilbert’s programs are defeated.  The theory of types does not do away with Gödel’s theorem and arithmetic (adequate for number theory) cannot be formally proven to be consistent.




            Gödel’s theorem, although originally of interest to the handful who could understand it, is now considered to be ‘revolutionary in its broad philosophical import.”[25]  Gödel’s theorem has implications in at least three areas:  In mathematical proof-theory, in epistemology and in the analysis of human thinking processes, i.e.; in the mind, brain, computer hypothesis.  Each of these will be discussed in this chapter.


            Gödel’s theorem is most immediately relevant to various schools of mathematical foundations.  Prior to Gödel foundations, study was divided into the logicist, formalist, and the intuitionist approaches.  The logicist program, lead by Russell and Frege, attempted to base mathematics in pure logic by deductively formulating all propositions from self-evident axioms.  Gödel’s proof showed that propositions exist, which can only be proven by meta-mathematical reasoning and not by pure logic.[26]   Thus, the logicist program, while not dropped, had to be radically redefined into “pluralistic logicism.” [27] 


            Hilbert’s formalist program attempted to: 1) prove the consistency of classical mathematics by using “finitary” methods, and 2) to solve the decision/problem of classical mathematics.[28]  Gödel’s theorem proved that many propositions could not be proven by finitist means and Church’s extention of the theorem proved that a decision procedure for lower predicate calculus was in principle unrealizable.[29]   While formalist proofs have since been made for many portions of mathematics and the approach has been very fruitful for limited arithmetical systems.  Hilbert’s original program had to be abandoned.[30]  


            With logicism and formalism redefined, intuitionism is given a foothold, which cannot be relinquished.  Thus, the foundations of mathematics must now admit reliance on intuition or metaphysics or transcendence.  Thus, parts of mathematics are based in intuition.  This conclusion leads to the second domain affected by Gödel’s proof, that of epistemology.  Mehlberg states this well:


If it were the case that we shall never be able to prove the consistency of the basic mathematical theories. Then, by the same token, we would be incapable of knowing that these theories are consistent, let alone that they are true.  Our ignorance of the consistency and of the truth of these theories would, in turn, entail the impossibility of knowing whether, any other theory, which presupposes the basic mathematical theories, is true. Practically speaking, any and every scientific theory would be involved. Gödel’s result would imply that virtually no scientific theory would be classified as knowledge.  The only alternative we would be left with in regard to science would be belief:  Since knowing the truth of these theories would transcend the scope of man’s potentialities, he would have to believe blindly in the validity of his general outlook, which is presently based on scientific information; he would also have to believe blindly in the dependability of the individual and social actions which he is presently compelled to take on the basis of scientific information.  Thus, belief rather than knowledge, would have to be admitted as the basis of our theoretical outlook and practical activity in view of the persuasive part which science has come to play in man’s life.[31]


While Mehlberg further develops this conclusion with an argument that amounts to “my scientifically based faith is superior in reliability to your non-scientifically based faith,” his initial point is more accurate as to the epistemological import of Gödel’s proof.[32]  Gödel in fact treated foundations as parallel to explanatory hypotheses in the physical sciences and ended up adopting Platonic realism in which mathematical objects are not created but discovered by man.[33]    Thus even mathematics, the standard of rigor for all rational thought must adopt some form of transcendence in order to maintain its consistency and usability.  No area of intellectual inquiry can survive without some form of transcendence.  Modern man just cannot get the ghost out of the machine!


            The third area in which Gödel’s theorem is of significance is the questions of whether a machine can conceivably be constructed that would match the human brain in mathematical intelligence.  This is related to, though not identical with the broader question of whether the human mind can be reduced to a machine model.  While logical empiricists chaff against rejecting such assertions,[34] Gödel’s proof does seem to ten in such a direction.  If a machine is by definition a formal system with finite limitations, no machine can conceivably be constructed that can solve all Gödel -type propositions, for every formal system will have such propositions.

Since a machine is a finite formal system, the last formal system will contain undecidable but true propositions, which the machine could not solve.  While man is not infinite, he is able to conceptualize an infinite method of proving every Gödel -type proposition by meta-mathematical reasoning.  While specific machines can perform certain mathematical functions better than man, a machine’s potential is not equal to the mind’s potential because of Gödel’s proof.  While this reasoning has been attacked vehemently,[35] these attacks fail to recognize that the human mind intuitively transcends formal systems and thus the mind transcends the formal system of any conceivable machine.  I see no other conclusion than that of Lucas[36] -- i.e. Gödel l proves the falsity of mechanism.




            In this paper I have attempted to show the impact of Gödel’s theorem on the logical empiricist mentality.  While empiricism is not a dead theory of knowledge, Gödel proclaims its demise. Gödel left the Vienna Circle of Logical Positivism because it’s program was unrealizable in principle.[37]  Intuition cannot be removed from mathematics even if self-evident

propositions are excluded.  Mathematics looks like it is purely objective, but it too, has its basis in transcendence.  This is a long way from demonstrating that this transcendence comes from a transcendent God who has given this gift in a limited form to man who bears his image, but it does open the door (even if only a crack) to discussion of why mathematics is as Goedel shows that it is.  We must be wary of thinking that Goedel’s theorem solves all of the difficulties of witness in a positivistic mentality, but it does give us a tool with which to confront the presuppositions of the contemporary mind.  No one is objective.  The question is not whether we have faith; this is inevitable even in highly complex symbolic logic, but which faith accounts for man in his totality.






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[1] Encyclopedia of Philosophy, 1972 ed., s.v. “Gödel’s Theorem” by J. Van Heijendort, p. 349.

[2]  Immanuel Kant, Critique of Pure Reason, translated by Norman Kemp Smith (New York: St. Martin’s Press, 1929), p. 52 where Kant says, “All mathematical judgments, without exception, are synthetic.”  See also pp. 85-86 where he alludes to non-Euclidean geometry implicitly.  (See my Progress of Science and the Foundations of Mathematics: from Euclid to Gödel)

[3] Encyclopedia of Philosophy, 1972 ed., s.v. “Mathematics, Foundations of,” by Charles Parsons, pg. 197.   “One of the purposes that Frege, Russell and many later proponents had in mind in seeking to reduce arithmetic to logic was to show that no appeal to sensible intuition was necessary in arithmetic, as had been claimed by such empiricists as John Stuart Mill, and by Kant in his theory of a prior intuition.”

[4] Encyclopedia of Philosophy, 1972 ed., s.v. “Hilbert, David,”

[5] Ernest Nagel and James Newman, Gödel’s Proof  (New York: New York University Press, 1958), p. 12.

[6] “Hilbert, David,” p. 500.

[7] Goedel’s Proof, p.18.

[8] Encyclopedia of Philosophy, 1972 ed., “Frege, Gottlob,”  by Michael Dummett, P. 226.               

[9] Ibid., p. 227.

[10] Encyclopedia of Philosophy, 1972 ed., s.v. “Logical Paradoxes” by John Passmore, p. 46., notes Frege’s 1902 letter in which he stated that the discovery of the paradox had shaken the foundation of the system of logic on which he intended to build arithmetic.

[11] Ibid., p.47.

[12] An interesting sidelight is that Frege ended up basing all arithmetic on geometry which is what Kant called “synthetic a priori.” See “Frege, Gottlob,” p. 227.

[13] Alfred Worth Whitehead and Bertrand Russell, Principia Mathematica, 2nd ed., (Cambridge: University Press, 1970), pop. 37-65

[14] Ibid., p. 62

[15] William and Martiya Kneale, The Development of Logic (Oxford:  Clarendon Press, 1968), p. 724, quotes Russell, “If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point in the successive definitions of the Principia Mathematica they consider that logic ends and mathematics begins.

[16] I have not here discussed the neo-intuitionist approach of Brouwer as this was not viewed as a way of finding ultimate grounding with intuition.

[17] R.B. Braitawaite, Introduction to on Formally Undecidable Propositions of Principia Mathematica and Related Systems, translated by B. Meltzer (New York:  Basic Books, Inc., 1962), pp. 2-3

[18] For instance Braitawaite, Ibid., notes that Presurger proved a decision procedure for every proposition of a mutilated arithmetic statement in which addition was used.

[19] Encyclopedia of Philosophy, 1972 ed., “Logical Positivism,” by John Passmore, p.52.

[20] Kurt Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, trans. by B. Meltzer (New York: Basic Books, 1962), p. 41.

[21] Braithwaite, p. 5., discusses the terminological difficulties.

[22] Nagel, Proof, p. 27.

[23] Gödel, p. 40.

[24] Encyclopedia, “Goedel’s Theorem,” p. 352.

[25] Nagel, Proof, p. 4.

[26] Henryk, Mehlberg, “The Present Situation in the Philosophy of Mathematics,” in Logic and Language (Dordrecht: D. Reidel Publishing Co., 1962), p. 74.

[27] Ibid., pp. 99-103.

[28] Ibid., pp. 79-81

[29] Alonzo Church, “A Note on the Entscheidungs Problem,” The Journal of Symbolic Logic, 1 (1): 40-41.

[30] Gödel, himself, allowed for a finitist proof that cannot be stated in his arithmetical system “P” but this has proven to be unrealistic in that a finitist proof not stated “P” cannot be conceived of.

[31] Mehlberg, Ibid., p. 82.

[32] He actually says, p. 84, “yet the case of empirical sciences shows clearly that knowledge may be dependable to a high degree without being deductively established and therefore virtually infallible.”   Such a statement shows positivist arrogance and not cogent reasoning!

[33] Nagel, Proof, p. 99.

[34] It is amazing how such thinkers miss the import of Gödel’s theorem. See Encyclopedia “Gödel,” p. 357, “The bearing of Gödel’s results on epistemological problems remains uncertain.  No doubt these results and other ‘limitations’ results have revealed a new and somewhat unexpected situation insofar as formal systems are concerned.  But beyond these precise and almost technical conclusions, they do not bear an unambiguous philosophical message.”

[35] See I.J. Good, “Logic of Man and Machine,” New Scientist 26 (1965): 182-183., and Paul Beneraci, “God, the Devil, and Gödel, “Monist 51 (1967): 9-32.

[36] J. R. Lucas, “Mind, Machines and Gödel,” Philosophy 36 (1961): 112-127.

[37] See the fine discussion of this issue in Stanley L. Jaki, Brain, Mind and Computers (South Bend: Gateway Editions, 1969), pp. 214ff.  See also R. Carnap, Meaning and Necessity, 1956; B. Blanchard, Reason and Analysis, Open Court Pub., 1964; A.J. Ayer, Logic, Truth, and Language; J. R. Weinberg, an Examination of Logical Positivism, 1936 ; Karl Popper, Logik der Forshung, 1935.