There are neither logical truths, nor mathematical truths

“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” (Einstein)

Following mainly Odifreddi’s book about Gӧdel, “The God of Logic”, in this short article, I will indicate, within the EDWs perspective, that there are no logical truths, mathematical truths and physical laws at all, i.e., these truths and physical laws

• do not have any ontological status
• are just certain mental states/thoughts in each mind-EW, nothing else.

More exactly, logic and mathematics are nothing else than an old “chat-GPT”, just an “old game in an old town”. Obviously, logic and mathematics are related to language (signs), therefore, we have also to deal with language. From my viewpoint, we have to related logic, mathematics and language with perception and thinking. From my viewpoint, all the mental perceptual states and the mental thoughts are the mind-EW. More exactly, we have to use the very important distinction between the explicit/declarative/conscious knowledge and the implicit/procedural/unconscious knowledge which it is strong related to the syntax and the semantics of language. (see my works 2007, 2008) Finally, if we talk about the logical statements and language, we have to find an answer to the main problem: “truths”. I will make a clear distinction between “truths” and “real things”. Truths involves, with necessity, logical propositions (formal systems) or sentences from language. A word cannot express truth. For instance, “tree” cannot express a “truth”.[1] However, I want to emphasize, the main distinction between “truth” and “a real thing”: a statement/sentence is true or false; a real thing exists independent of our thinking/perception and logic/language. Surely, the Earth have existed before the human beings have appeared on it. (see my previous works) The last statement indicates a macro-entity (a planet, the Earth) which have existed independent of the existences of the human beings (perception/thinking/logic/language). We can extend the notion of “truth” to our “internal/implicit thoughts”: for instance, I think, without pronouncing something, that 2 + 2 = 4. I use an internal language/numbers to have this thought. Can this statement be truth or false? It depends on what I have been learning, nothing else. In the first years of school, I learned that 2 + 2 = 4, but this is not a truth statement in itself. Moreover, this statement does not even exist in itself, it is in my mind (more exactly, it is my mind). Also, I can see a tree in my garden and think: “There is a green tree in my garden”.  This thought refers to the existence of a green tree in the garden. I can remark, not being aware, a dog in my garden, I have a perceptual visual state about that dog, and later, maybe I could remember (I can have a thought or express a statement to somebody) that I saw that dog.[2] These thoughts/statements refer to something which really exist (the tree, the garden), but according to my EDWs perspective, I perceive these entities indirectly. Therefore, I cannot claim there are true statements (see my previous works or below).

Let me introduce, from Odifreddi’s book, very few notions regarding a logical system[3]: “correctitude” or “consistence” means there are no contradictions in the system; “completeness” means that if a statement is true, there has to be a proof following the axioms and the rules of the system. (Odifereddi 2020, p. 20)[4] I will investigate all these notions from my EDWs perspective and following the difference between truths and real things. I will move from one topic to another without specifying this process. However, later, I will try to relate some of these topics.

I will investigate few statements and concepts from Odifreddi’s book, but this investigation has to be extended by a specialist to the entire logic and mathematics. I have not studied logic until I have written this article; I have read just two books and very few articles on logic and I wrote this article in less than one month. Therefore, this section is not for logicians and/or mathematicians but mainly for philosophers. It refers, however, to the logical and mathematical statements related to the notions of “truth” and “reality” (i.e., the EDWs). In fact, in this section, I want to indicate there are no “truths” at all.

Obviously, logic and mathematics are related to language (signs), therefore, I have to relate, inevitable, logic and mathematics with natural language. A language of any person is the mind-EW. So, from my viewpoint, we have to related logic, mathematics and language with human perception and more generally, with human thinking (the mind-EW). From my viewpoint, all the mental perceptual states and the mental thoughts are the mind-EW. So, any logical, mathematical, linguistic statement is the mind-EW. If we see written “1 + 1 = 2” on the wall of a cave, obviously, it was written by a human person. In this context, we have to use the very important distinction between the “explicit knowledge” and the “implicit knowledge” (see my works 2007, 2008, etc.) which it is strong related to syntax and semantics. Finally, if we talk about the logical and the mathematical statements related to language in order to find an answer to the main problem: “truths”. In the following paragraphs, I will make a clear distinction between “truths” and “real things”, i.e., between logical, mathematical propositions, linguistic statements/concepts, on one side, and the real entities (which belong to the EDWs), on the other side.

Odifreddi’s book is about Gӧdel and other great logicians. According to Odifreddi[5], Gӧdel’s some essential achievements are the following: the theorem of completeness for the predicate-logic in model theory, the first theorem of incompleteness in the theory of calculability, the second theorem of incompleteness in theory of demonstrability. (p. 225) Let me introduce very few notions regarding a logical system:

• “correctitude” or “consistence” means there are no contradictions in the system
• “completeness” means that if a statement is true, there has to be a proof following the axioms and the rules of the system. (p. 17)[6]
• Gӧdel writes that “any proof of consistency has to be derived from the axioms of mathematics and logical laws”. (p. 18)[7] However, he believes that certain axioms (of arithmetics) are “true” even if those axioms cannot be proved to be true.[8]
• Gӧdel’s incompleteness theory (in his words) is that, using only the axioms and its methods of proof, we cannot demonstrate a system is consistent; there are statements which cannot be decidable through a formal proof. (p. 19)[9] Again, mathematicians believed that there have to be certain statements that are true: for instance: “1 + 1 = 2” is a true statement. Nevertheless, from my viewpoint, there are no proof that this statement is true. It is assumed to be true because the mathematicians have been learned to be true.
• “What then is Gödel’s theorem? Here is an initial informal statement of it: No axiomatization can determine the whole truth and nothing but the truth concerning arithmetic.” (Moore 2022, p. 28)[10] “The upshot at this stage, then, is that, according to Gödel’s theorem, no finite stock of basic principles and rules enables us to prove every arithmetical truth unless it also enables us to prove some arithmetical falsehoods.” (Moore, p. 33)

Granted these definitions, Gödel’s theorem can be restated as follows: No theory can be sufficiently strong, consistent, complete, and axiomatizable… being sufficiently strong means containing some core arithmetical truths; being consistent is a precondition of containing nothing but the truth; and being complete is a precondition of containing the whole truth (in the theory’s language). (Moore, p. 75)[11]

The same problem: the axioms of arithmetic are presupposed to be true. I do not claim that “1 + 1” is not “2”. I claim that there are no proof that the statement “1 + 1 = 2” to be true. There is a linguistic statement which have been learned to be true.

• “(un)decidability”: “In computability theoryand computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime.” (Wikipedia)[12] Decidability depends on computation, computation depends on discrete and static entities like numbers. However, numbers could not even exist, therefore, decidability is a notion which depends on a “formal system”, but any formal system does not have any ontology. Computation is not an “algorithm” using discrete symbols; in a computer does not exist symbols. (see below)
• Arithmetic is the study of natural numbers 0, 1, 2, 3, … and the mathematical functions like addition and multiplication. (Moore, p. 29) Again, if natural numbers exist, then we can claim that the real numbers also exist. Between 1 and 2, there are an infinite irrational numbers. Both sets of numbers (natural and real) are infinite, but the infinite cannot even exist. (see my previous works) Therefore, “between and 2 there are infinite irrational numbers” or “there are infinite natural numbers” are pseudo-notions created in arithmetic since neither the numbers nor the infinite could even exist.
• Algorithm is a purely mechanical finite step-by-step process (like in computer). (idem) Basic principles and rules belong to logic in the sense that it do not involves a subject. (Moore p. 31) Algorithm is not a “purely mechanical finite step-by-step process”: an algorithm is realized by certain formal statement by a mathematician or an IT member, but any algorithm does not have any ontological status. Algorithm does not exist in a computer. (see below)
• “Any given axiomatization will provide canons of proof. A proof of a statement, relative to the axiomatization, will be a derivation of that statement from those basic principles, using those basic rules.” (Moore, p. 31) It is about the fact that “any sound axiomatization fails” to grasp certain arithmetical truths but it does not mean that “some arithmetical truths fail” to be grasped by any sound axiomatization. (Moore, p. 34) There are no arithmetical truths. (see below)
• Gödel’s theorem does not exclude axiomatization the truth concerning other subjects. (Moore, p. 34) Essentially, even if the axiomatization of analysis involves truths about specific natural numbers like “7 + 5 = 12”. Again, there are no mathematical truths.

Even if Gӧdel was a member of Vienna Circle, he was against “positivism” just because he believed that the natural numbers really exist in their own world. From my viewpoint, the main problem would be if the logical and the mathematical concepts/functions/statements are parts of the human language or not. I consider that any logical and mathematical concept/ statement /function is just a part of the human language. I emphasize that some mathematical statements describe correctly (obviously, always with approximations) certain real facts/events. It means that, during evolution being always in indirect contact to their environment, the human beings developed certain words/numbers which fitted certain entities external to their bodies. But these numbers/equations/statements are not “true”, they just fit (with approximations) in describing certain natural facts, events, processes. Many other mathematical concepts/equations/statements have nothing to do with reality. Moreover, any language is part of the mind-EW. We admit that the animals (for instance, monkeys and dogs) do not have language, but they have perceptual representations and bodily “signs”. These perceptual representations and bodily signs are the mind/life-EW and there are, somehow, similar to the human language. But any human language is the mind-EW.

If a single dog has a quarrel with other two dogs, the single dog is somehow aware (no language, no numbers) that it is single and the enemies are more (two). However, there are no “numbers” in the mind of a dog, there is a mental perception which it includes those two dogs (any mental perception is the mind-EW). The single dog is aware there are two enemy dogs in front of its body, therefore, he prefers to avoid a fight with them. This ability is the result of species evolution, but the dogs and the monkeys do not have any numbers or languages in their minds. Only the human species has developed “language”. From my viewpoint (see my book 2024), there are no innate knowledge, therefore, there are no innate words/concepts. Numbers are not innate concepts/words, there are parts of the mind-EW initially created by some human beings (long time ago) and then the other human beings (in general) have learned numbers at school. “Numbers” do not exist in themselves (as Gӧdel believed), numbers do not exist for humans or for dogs. A dog (for instance) has certain innate abilities to distinguish between one dog and two dogs, but these abilities are firstly certain “perceptual abilities” and then certain “thoughts”, but the dog has no language, therefore, it does not have numbers in its mind. Anyway, from my viewpoint, all the perceptual states and the mental thoughts are the mind (the mind-EW is an immaterial EW). (see my previous works)

Gӧdel (in Menger) believes that “more I think about language, more I inquire how people succeed to understand one another.” (p. 45)[13] This is quite an interesting statement, therefore, I will furnish a viewpoint about it within my EDWs perspective. Let me take the example of two adult persons (a woman and a man) who, as children, learned the same language; both have diplomas at different universities in the same city, etc.; both are in the same room (with windows) of a house. The woman tells to the man: “It is raining (outside).”[14] The man understands, without any problem, her sentence and his answer is: “Yes, indeed”. How then is communication (any communication) possible between these two persons (for persons in general) using their language? The “communication” though “language” (in this case) depends on different factors. There are two bodies (a woman and a man) within the same macro-EW which cannot communicate words (having syntax and semantics) directly. The bodies do not “communicate” but interact, directly or indirectly. In this case, they indirectly interact through the air from that room/Earth. Each person has a body (both bodies are macro-entities in the macro-EW) and each body corresponds to a mind-EW (I recall the main rule of my EDWs: one EW does not exist for any EDW, that is, the mind does not even exist for anybody; all the bodies are macro-entities within the macro-EW). The eyes, brain/bodies interact with the external environment, while the corresponding minds perceives, indirectly, the “rain”. Following, partially, Kant’s perspective, “the external world is the mind” (more precisely, the mental representations of the external world are not “my representations”, they are mind-EW – see my PhD thesis and my book 2008). So, the woman-mind-EW perceives indirectly that it is “raining outside the house”. “Outside the house” is a real phenomenon in that moment: outside, it is indeed raining. The phenomena of raining is really happening in the garden in those moments. Then, it is her sentence “true”? Meaningless question since, as we will see below, “truths” do not even exist.

The brains/bodies, the house and the “outside the house” are parts of the macro-EW. The woman-mind-EW perceives that it is “raining” (the mental perceptions of the house, outside and raining), through correspondence, moving her tongue and lips and with the help of the air from the room, she transmits certain sounds (not words!) to the ears of the man. The man’s brain processes certain neuronal inputs, but there are no words or pictures in the brain; words (syntax and semantics) are only in each mind-EW. The woman-mind-EW (an immaterial EW, without “extensions” in Cartesian meaning) has a mental perceptual state of the room and “inside” of her visual perception (no size, anyway, since any perception is a multimodal process and it is the immaterial mind-EW) is the mental sub-representation of the man. For the human beings and the animals, in general, there is an innate ability to identify certain “certain elements” (in motion or static, being one essential criteria for this identifications) of any perceptual representation.

In the woman-mind-EW, there is this thought: “It is raining outside” which corresponds to the motion of her tongue/lips (her body in the macro-EW). Because of the motion of tongue and lips, the air between those two bodies moves with certain frequencies. The air in motion reach the ears of the man and his timpani of both ears move in function of the frequencies of the air. The timpani send certain inputs to the nervous system (parts of the brain which send inputs – less and less powerful – to the rest of the brain). Exactly as there are no “images” in the brain, there are no “words” within the brain, but only certain neuronal activations (electrical and chemical reactions) and certain corresponding electromagnetic waves which “surround” the brain (more exactly, the brain/body –  a macro-entity in the macro-EW – corresponds to large concentrated electromagnetic field – the field-EW). These neuronal activations in the brain/body correspond to the words/sentences that are pronounced by the woman-mind-EW and understood by the man-mind-EW. The perceptual representation (visual and auditory, for instance) of that man and that room is the woman-mind-EW (and vice-versa). Their bodies (macro-entities) are both in the same room/house with its garden in the same city/country/Earth/galaxy as macro-entities (the macro-EW). However, any mind-EW does not even exist (more exactly, is not) for the body; any EW does not even exist for any EDW. The woman-mind-EW does not exist for her body, it does not exist for the man-mind-EW. However, they indirectly communicate (through “correspondence” with no ontology) through their bodies which are placed within the same EW, the macro-EW. Their bodies use certain “signals” (noise created by tongues and the motions of the air) which just correspond to their words/language (with statements having “syntax” and “semantics”), but the language of each person is her mind. Each mind has acquired certain knowledge during learning to speak and understand a particular language (English, for instance) exactly as the same person acquired knowledge about mathematics.

Essentially, a person cannot even be aware about the sentence which she pronounces it at one moment. For instance, you can ask somebody to tell you an arithmetical statement, and that person would tell you “5 + 5 = 10” (for instance). She was not aware/“conscious” about this statement before pronouncing it. You and that person become aware about that arithmetical statement at the same time! In the previous case, it is possible, the woman becomes aware/conscious about her thought/sentence when she just pronounces it for the man. Hearing the sentence, the man becomes aware/conscious about the woman’s sentence at the same time when she pronounces it. It means that the unconscious part of the woman-mind-EW constructs this sentence/thought before she pronounces it. The woman and the man become both aware/conscious about this sentence/thought at the same time.

Any formal system has certain axioms and rules. Any language has a syntax and a semantics. The syntax means there are different (distinct) words which form the sentences/statements. How does the man understand the sentence explicitly pronounced by the woman (that sentence being an explicit knowledge of that woman in that moment)? He understands each word in context with the other words, but each word has its particular meaning in relationship with other words from his memory (the implicit knowledge). As children, both adults were thought about the meaning of the word “rain”. It means, for instance, “drops of water falling from the sky”. As children, these two persons have also learned the meaning of “water”: everybody understands when somebody else asks: “Please, give me a glass of water”. In reality, in each mind-EW, there is a “pyramid of meanings” for each word (which compose all the sentences/statements). The problem is that each mind-EW has her “own pyramid of meanings” for words, perceptual images, etc., and each “pyramid of meanings” is the language of a particular person but any “language” is the mind-EW (of that person). Then, how the man understand the sentence pronounced by the woman? Between these pyramids of meanings there is a corresponding “interval of similarity” (about it, see my previous works) for words which belong to those two minds-EDWs. Those two persons can understand each other and can communicate (within this interval of similarity) without great problems. In general, there are no problems in understanding the “common words” exactly because of this interval of similarity. However, if one person wants to “check” what the other person understands from a sentence (each word in a sentence of the learned language), they can start to have a discussion. Obviously, passing a “threshold” (about “threshold”, in other sense, see my previous works), they will reach a point in which they do not understand each other regarding certain particular words which refer to previous words which refer to the first words (the pyramid of meanings). Experience, learning, memory, reasoning and other mental functions are involved in understanding (giving, implicitly, a meaning) each sentence (its particular words). I emphasize that each mind understands the words of a sentence pronounced explicitly by other person at the explicit level but this explicit level involves different implicit knowledge (experience, learning, memory, etc.) for these persons. The implicit knowledge is not only that “pyramid of meanings” (this being available for words), but it is also the entire knowledge of that person which is the mind-EW. So, the implicit knowledge (large “parts”) and the explicit knowledge (in a context) is just the top of the iceberg of the implicit knowledge acquired during the entire life by that particular mind. “It is raining outside” is only the top of the iceberg (the explicit/conscious/declarative/pronounced knowledge); the rest of the iceberg (much larger than the top) is the implicit knowledge (the entire knowledge of a mind-EW) which furnishes the meaning of that sentence, of any sentence/word. Because of that interval of similarity, two person can understand each other in certain limits and these limits depend on their education, experience, acquiring “information” during their lives, etc., i.e., on the meanings each word has for each mind in that iceberg of implicit knowledge.

Odifreddi: Frege (1879) realized a syntax interpretation of logical rules, while Wittgenstein a semantic interpretation of logical rules. (p. 55) The problem was that the logical truth for the predicate-logic[15] (quantified logic) was much more complex than the truth from the propositional logic. Interestingly, for Frege, Russell and Whitehead, the problem of completeness was not even conceivable: they adopted a “syntactical method” and this they totally avoided the semantics. (p. 63) From my viewpoint: the syntax interpretation of logical rules needs “discrete and static” symbols. These symbols can be “numbers”, “words”, “codes” or even mental perceptions. How was possible for these (and many other) very famous thinkers to reduce “truths” to syntax? This step is strong related to the analogy between the human mind and the computer/machine, but before the computers/machines appeared, there were logic and mathematics. The statements of logic and mathematics use “symbols” apparently without “meanings”; from my viewpoint, any symbol (word, number, picture, mental representation) in the mind of a human being has a particular meaning (given by a context) since all of these “symbols” are the mind-EW.

We have to recall certain elements related to mathematics:

• the slogan written at the entrance of Plato’s school of philosophy: “Nobody enters here without knowing geometry”
• Copernic mathematical model of Solar System (the Sun in the center)
• the movement realized by Galileo Galilei of introducing experiments and mathematics
• Newton’s equations (based on mathematics) describing the motions of planets
• mathematical equations from Einstein both relativities
• mathematics from quantum mechanics
• the (super)string theory, etc.

Just in few great historical steps, I want to grasp the huge role of mathematics in describing the external “world/Universe”. Starting mainly with Newton, the natural laws have been expressed using mathematics. It does not mean mathematical statements are real; they are just statements involving symbols/language and any language is the mind-EW, nothing else.

Related to syntax and computer (machine), we have to recall Fodor’s LOT and his computationalism. Being student of that famous linguist, Chomsky (who was dealing with language and his “universal grammar” – against this “innate” UG, see my book 2024) and being aware about the powerful of computers, Fodor tries to apply “syntax” in explaining the human thinking: in fact, he tries to explain “thinking” as certain “rules” manipulating some “symbols” in a syntactical manner (like a computer, he thought). As somebody wrote (long time ago) in describing computationalism: “semantics is in syntax”. (see my previous works)

Searle introduces his famous attacks against computationalism in two steps (the first step being earlier than the second step):

• any computer/machine does not have semantics (the “Chinese Room” thought experiment): inside a computer there are no meaning (semantics)
• any computer does not have even syntax: there are not words, not even computer program (with 0 and 1) inside my laptop. Indeed, inside my laptop there are only electronic pieces (transistors, diodes, etc.) and electrical impulses, nothing else. Therefore, there are not even these numbers “0” and “1”. When I “ask” my laptop to give an answer to this formula, “1+1”, the computer will give a “correct” answer: “2”. The problem is that the computer has no idea about 1, 2, and + at all! It works based only on transistors (diodes, etc.) and their electrical impulses, nothing else.

I see a “picture” on the screen of my computer: a “green tree”. In reality, there is no “picture” on the screen at all: there are certain pixels (electronic pieces of light) having different colors (depending on the wavelengths and frequencies of light). If I increase the size of the screen with the picture of that green tree, that picture starts to disappear from the screen and increasing more and more the size, I would see only pixels “turn on” (lights) and “black spaces” among them, nothing else.[16] From my viewpoint, the images, numbers, words and ideas are neither in computer, nor in my brain, but they are the mind/self. In the brain, there are certain neuronal activations (more or less, the entire brain is activated) which correspond to certain mental states like the meaning of a word/sentence and having a mental picture (which all of them are the self/mind).

Again, Fodor believes in the analogy between the human minds and computers. As some people noticed (see Searle’s “Chinese Room” in 1980), his program failed since the human mind has nothing to do with a computer. However, there are the mental representations (as he claimed) but for Fodor these mental representation were just “epistemological” entities since he accepted the identity theory (at least the “token identity”) and rejected any dualism. In reality, the entire Fodor’s computationalism failed. In 1992, Searle indicates a computer does not work with syntax; since in a computer, there is no semantics and no syntax, there is no analogy between computer and human mind.

Connectionism and the dynamical system approaches were the main rivals of computationalism. (see my previous works) Both approaches refer to the brain/body, not the mind (since all researchers/philosophers were accepting the identity theory until 2005 when I have published my article at Synthese journal). My EDWs perspective indicated the failure of all these approaches in explaining the “human mind”. Interestingly, if connectionism consider the representations as being distributed (within the neuronal nets/brain), the dynamical system approach rejected completely the existence of representations. Within my EDWs, I indicated that there are certain mental representations in the mind, but all the mental representation are not “my representations”, they are the mind-EW and any mind-EW/self-EW is an immaterial EW which it does not have any “parts”. The parts-whole relationship is meaningless regarding the mind (as a whole) and the mental representations (as “parts”), the distinction “whole-parts” is meaningless regarding the immaterial EW, the mind-EW). Again, these mental representations are the mind and the mind-EW is an immaterial entity and EW. On the contrary, the computer is a physical entity (a sum of many physical/electronic sub-entities) which functions based on certain discrete and continuous/in motion entities (mainly, the electrical currents). My laptop solves “2 + 2” without any problem, but it has no meaning for this equation, for any number; moreover, this equation (its “syntactic symbols”) does not even exist inside the laptop. There is no number “2” inside the laptop; there are certain electrical inputs in certain electronic pieces, no more. Therefore, we cannot make an analogy between the mind and computer/machine since any machines has no life/mind, it works with certain physical entities/elements/processes (discrete and continuously), so there is no equivalence between the mental representations (the mind as an immaterial EW) and the computer (material entity composed of many material elements).

Now, let me return to Odifreddi’s book. Beltrami passes the consistent problem of the non-Euclidean geometry to the Euclidean geometry: if the Euclidean geometry is consistent, the non-Euclidean geometry is also consistent. (p. 52) Gӧdel: The consistence of axioms of geometry are reduced to the consistence of axioms of arithmetic which has to be directly demonstrated (there are involved the real numbers and the axioms of continuity and the irrational numbers. (pp. 53-4)[17] As I indicated in my previous works, there is no space, no time, no spacetime. Therefore, I rejected not only the existences of Euclidean space or non-Euclidean space, but also Poincare’s “conventionalism”. The geometries are not even “conventions”[18] for describing “space” (spacetime, according to the special relativity) since space/spacetime could not have any ontology.

Odifreddi: The step of passing the responsibility from one garden to another it was also realized for the consistence of “analysis” in relationship to the consistency of “arithmetic”. (p. 80) From my perspective, this passing is not something surprisingly since both the analysis and the arithmetic are different “parts” of mathematics and the entire mathematics is just statements created by different human minds during millennium;  some of these statements fit (with great approximations) with some parts of reality, but no more. These fits are not completely accidentally: for instance, numbers started from the correspondences between certain thoughts and certain perceptual images.[19] Being in a forest, a human being perceives, indirectly, two real wolves. His mind adds to those parts of his entire visual image, the concept/thought “two” and later, based on different experiences, he invented “number two”. Therefore, “number 2”, for instance, is just a human mind creation associated, indirectly, with two real wolves.

Gӧdel: Representing the real numbers through arithmetical formula implies the use of “arithmetic truth” for verifying the axioms of analysis. Gӧdel discovered that this truth cannot be defined within the arithmetic, otherwise, there would be something similar to the “liar paradox” and the system would be inconsistent. (p. 81) Gӧdel was correct, but he had to talk about the explicit and the implicit knowledge within the EDWs perspective. Unfortunately for him, he worked within the unicorn world, as all thinkers during the history of humanity until I discovered the EDWs and I published my first article in English at a Romanian journal (2002) and posted free on Internet in the same year.

Gӧdel’s movement is also partially related to Wittgenstein’s Tractatus: language indicates its proper logical structure, but it cannot speak about it. So, for Wittgenstein, there is a correspondence between what it can be expressed through language, but not in language, with certain mystical elements which show in the world, without being from the world. (p. 81) Discovering the impossibility of defining truth was an example of mystic in the truth itself, in accord to the traditional metaphysics. (p. 81) In this sense, Gӧdel (a letter toward Balas): the impossibility of defining truth furnish the “correct solution for the semantic paradoxes” like the liar paradox. (p. 81) Without furnishing details about this paradox, I mention that any such paradox uses some statements of “common language” which refer to “truth” or “false”. Therefore, there are contradictions not only in the liar paradox, but also in common language. (p. 82) Using “diagonal proof”, Gӧdel translates the liar paradox in “arithmetic language” and proves that there is no definition for “arithmetic truth”. (pp. 83-4) The association between formulas and numbers leads Gӧdel to his theory regarding the “impossibility of defining truth”; in this sense, there is a translation of the metamathematical about truth of these formulas in the corresponding mathematical affirmations about numbers. (p. 89) Gӧdel writes that these metamathematical concepts imply only combinatorial relationships between formulas, relationships which directly reflect on the associated numbers. (p. 90) Anyway, without more details, Gӧdel indicates certain ideas exactly in opposition to those from Tratactus regarding the impossibility of expressing metalanguage through language. (p. 91)[20] In 1933, Tarski shows that there is no formal language (which it contains the arithmetic language) can admit (in its inside) a definition for truth.[21]

Let me investigate these statements from Odifreddi’s book. Exactly as the non-Euclidean geometry was reduced to Euclidean geometry, the analysis was reduced to arithmetic. However, any formal system (arithmetic included) presupposes certain axioms, rules and “truths”. Believing in mathematical “truths”, Gӧdel needed to believe in the existence of natural numbers. Again, from my viewpoint, the “numbers” do not have any ontology, the numbers are just words/thoughts in the human minds or animal minds. There are no (mathematical) “truths” at all. Moreover, the “combinatorial relationships between formulas, relationships which directly reflect on the associated numbers” send us directly to Fodor’s “language of thought” and his analogy of the human mind with a computer/machine. Apparently, any computer works with discrete symbols: however, we have the illusion that we press certain static and discrete symbols (syntax) like number “1”, “+” and “=” and the machine furnishes us the correct answer: “2”. Nevertheless, the computer does not work with “symbols” but only with certain physical entities and processes, without syntax and meaning. Otherwise, somebody can extrapolate this analogy to the analogy between the mental functions and the rotation of the Earth around the Sun (see below). Obviously, knowing about Gӧdel’s work (1931), Tarki elaborated his approach: no formal language (in arithmetic language) can admit (inside) a definition for truth. From my viewpoint, truths cannot even exist.

I return to my thought experiment: a dog fights with two other dogs. Perceiving, indirectly, the bodies of those two dogs, the single dog is aware that it has to fight with two dogs. Maybe if there were three dogs, the single dog would not even try to fight with them. The dogs does not have any language. Therefore, there are no “numbers”/words in their minds. However, the single dog has a thought about the other two dogs. It does not mean the numbers really exist; for animals and human beings, there are certain perceptual representations and thoughts associated to “two” dogs or “three” dogs, but such associations do not furnish ontology for “numbers”. During their evolutions, the human beings were able to associate “words” (number “two”, for instance) to these thoughts/mental perceptual representations which just represented, in their mind, the “external world”. A human person pronounces (or thinks): “There are two dogs (in front of me)”. This is an explicit knowledge which it involves a lot of implicit knowledge in a pyramidal structure. For instance, the meaning of the word “dog” (and other words, of course) is implicitly involves in this explicit statement. What does “dog” implicitly means? An animal with four legs, etc. What does “legs” means? The questions can go further and further but not in an infinitely way (maybe in an “un-definitely” way) since the “infinite” cannot even exist (see Aristotle’s argument against the existence of “infinite”). These thoughts and words are just the explicit and the implicit knowledge: any explicit knowledge involves the entire implicit knowledge since the entire knowledge is the mind/self.[22]

About Tarki’s approach: in any formal/arithmetic language, there is no definition of truth. However, from my viewpoint, “truth” cannot even exist; “truth” is a simple a word (human creation) inside the mind of a person. Anyway, we can talk about “truths” only when there are propositional/statements of language. Then, is the sentence “There are two dogs in front of me” true? Everybody would claim this statement is “true”. I would claim “truth” does not even exist, therefore, “This statement is true” is not “true” just because “truth” does not even exist. “Truth” has no ontology, exactly as the logical propositions, mathematical numbers and statement, linguistic sentences/statements have no ontological status; all these elements are thoughts in a mind-EW. Each explicit word/sentence/statement involves the “iceberg of implicit knowledge” which is the entire mind/self. One mind-EW is not for any mind-EDW, therefore, the notion of “truth” has no “common meaning” for both minds. Two persons speak to each other, a woman and a man. The woman pronounces: “There are two dogs here (in front of us)”. The woman perceives, indirectly, those two dogs. In reality, her mind is not in the same EW with the dogs. Those two human bodies and those two dogs are all certain macro-entities which really exist in the same EW, the macro-EW. The woman’s mind is neither for her body and dogs’ bodies, nor for the man’s mind. Each mind is in itself and for itself; it corresponds to the brain/body (a macro-entity within the macro-EW). The mind-EW does not exist (is not) for the brain/body. The man’s mind is neither for his body and dogs bodies, nor for the woman mind. Then, how can we consider that statement to be “true”?

We have to make the distinction between “truth” (a pseudo-notion) and “real”. All the bodies exist in the macro-EW; all the minds/lives are mind-EDWs. However, “truth” cannot even exist at all since my mind does not exist for your mind and my language is not exactly as your language. The top (an explicit statement, for instance) of the implicit language (knowledge) is quite similar to both the woman and the man and this is the reason those two persons are able to partially understand each other. In fact, based on these icebergs of language/knowledge, the people can use language for communication among them. However, “truths” do not even exist, we can talk only about certain implicit/mutual agreements among people which refer, in some cases, to certain real entities/processes.

Mathematics (arithmetic) is like logic: old or new “GTP” or “old games in town”. The mathematical statements do not contain “truths” since truth cannot even exist. Maybe if God existed, we could talk about “truth”, but God could not even exist (see my article/chapter). Probable, this was, implicitly, the reason Gӧdel needed “God” in his personal framework of thinking (not in his theorems). Since “God” cannot even exist, then there are neither “absolute truth” nor “relative truths” at all. There are just “mutual agreements”, nothing else.[23]

Odiffreddi: In arithmetic, the definition of demonstrability is a metamathematical notion. In a letter to Zermelo (1931), Gӧdel writes that the property of a formula to be demonstrable is pure combinatorial and formal, it does not depend of the meaning of signs, that is, the set of numbers of demonstrable formulas is reduced to simple arithmetical concepts. (Gӧdel in Odifreddi, p. 92) It means that the demonstrability in a formal system for arithmetic is definable within the system, but not the truth, so the demonstrability and truth are not the same thing. Then, there are demonstrable formulas which are not true (the system is not correct), or there are truth formulas which cannot be demonstrable (the system is not complete since there are truths which it cannot demonstrate). It means any formal correct system for arithmetic is incomplete if it permit the definition of demonstrability inside it. (Odifreddi, p. 92) Gӧdel needed certain metamathematical notions. (p. 93) In a letter to Burks, Gӧdel writes that the impossibility of the definition of truth is the reason there are undecidable propositions in formal systems which contain mathematics. Tarki indicated the same result in 1933.[24] In this context, there is an essential statement in Odifreddi’s book: “A reason for a discontent regarding the original demonstrability of the incompleteness theory was the fact that there was necessary the explicit use of truth.” (p. 94)[25] In fact, the original demonstration derived the incompleteness from the difference between truth and demonstrability. (p. 94)

I do not furnish more detail, but these statements can receive the same judgments as above. In a formal system, the “true axioms” assumed by a logician as being “truths” need a meta-formal system to be proof as “truths”. But any meta-formal system need a meta-meta-formal system, and so on… but we cannot accept a regress ad infinitum (the infinite could not even exist), therefore, some great thinkers (Descartes, Newton, Leibniz, Gӧdel, etc.) appealed to “God” (who, from my viewpoint, it could not even exist). So, all these debates regarding “truth(s)”, logic, mathematics, natural laws are “in the air”, i.e., these areas of knowledge:

– do not even exist in themselves

– do not have any ontology

– do not refer to something real (“real” means the ED entities).

So, “undecidable propositions” is quite a wrong concept since “truths” could not even exist; moreover, this undecidability refers to the iceberg of the implicit/tacit knowledge but each person has her own iceberg which is one mind-EW or a mind-EDW. Maybe we can consider logic and mathematics as “games” created by human minds, nothing else…

Odifreddi: In a letter toward Wang, Gӧdel: heuristically, syntactic and finite incompleteness was obtained through the consideration of a semantic and infinite concept, i.e., the truth. (p. 95) Indeed, the phantom of truth disappeared from the main formula of the theorem, but still haunted through the semantic hypothesis regarding the correctness of the system. Gӧdel could not do this, however, his student, Rosser replaced this semantic hypothesis through a syntactic hypothesis of consistence in 1936. Today, the demonstrability of correctness and consistency of a theorem is purely syntactic without appealing to truth. (p. 96) Moreover, on 2 July 1931, Gӧdel announced Vienna Circle about the undefinability of truth and from this the derivation of incompleteness of arithmetic. (p. 101) In this context, avoiding many other details, I write about the second theorem of incompleteness: a consistent system cannot demonstrate its consistency. (p. 104, Odifreddi’s italics) In a letter toward Gӧdel, von Neumann writes that in his opinion, the result indicates a negative solution to the fundamental problem: there is no rigorous justification of classical mathematics. (p. 104) The entire Hilbert’s program totally failed. Also, following Gӧdel, von Neumann writes (1930): there is no demonstrability of consistency of arithmetic, such demonstrability can be realized only with non-formal tools inside arithmetic. (p. 105)

Syntax involves static and discrete symbols (words, signs, etc.); however, each symbol has its own “meaning” (perceptual meaning or linguistic meaning, for instance). If there are two dogs in front of me, I “understand” there are two dogs in front of me. This understanding/meaning is the top of the iceberg of the entire implicit knowledge (which it is the mind). But the mind-EW1 does not exist (more exactly, “is not”) for the mind-EW2, therefore, we cannot speak about truth(s) because truths could not even exist (they could not have any ontology). Then, theorem of incompleteness has to be re-written (in a much more general view) within the EDWs perspective.

Odifreddi writes Carnap’s opinion about Gӧdel who agrees with Brouwer’s intuitionism of mathematics: in agreement with Brouwer, the mathematics cannot be completely formalized. In any formalization, there are problems surprised by the common language, but not by the language of that formalization. Conform Brouwer, the mathematics is inexhaustible and we have to appeal to “intuition”. There is neither a “characteristic universalis”, nor a process of decision for the entire mathematics. (p. 107) According to Gӧdel, the consistency of a system does not furnish its correctness. (p. 108) In a letter to Wang, Gӧdel: metamathematics is the unique part of the mathematics which has a meaning. In itself, the symbols do not have any meaning, but they receive it through the metamathematics, through the rules furnished by it for the using of symbols. Moreover, metamathematics has not been considered a science which described the objective mathematical factual states, but it represents, more, a theory of human activity of manipulation of mathematical symbols. (p. 108) From Gӧdel’s viewpoint, logic is the predicate theory (which it is complete), while mathematics starts with elementary arithmetic (which it is incomplete). (p. 108) For instance, I have the perception of a dog. I cannot say that this mental state is “true”. Moreover, the perception of a dog is part of the entire visual perception which is part of the multimodal interactions which is a mind-EW. (Maybe, the metalanguage would corresponds to the mind as an EW, I do not know…)

Obviously, the consistency cannot furnish the correctness. From my viewpoint, we have not to confuse “correctness” with “real” (the things which really exist, for instance). It is not surprisingly, apparently, Gӧdel supports Brouwer’s intuitionism since he cannot “prove” the truths of his axioms.[26] Again, related to “symbols”, there is Fodor’s LOT or connectionism’s networks; it is syntax related to semantic, so we have to recall, against this “symbolism”, the “Chinese Room”. Even the statement “2 + 2 = 4” is not a true statement, it is a formal statement created by certain minds-EDWs, exactly as there are the entire logical and mathematical statements/symbols. In my example, for the simple dog, there are “two” dogs, but in reality there are those dogs. “Two” is added by the mind of the dog. The same is available for the human mind: a human person perceives, indirectly, two dogs but the word/thought “two” are associated with those dogs (their bodies) who really exist in the macro-EW. The mental perception of those two dogs is the mind-EW. Again, an EDW, for instance a particular mind-EW cannot be considered as having a true statement (“there are two dogs in front of me”) in her mind since the words of this statement are the top of the iceberg of the entire implicit knowledge (which it is the mind-EW). Moreover, this “true” statement of the woman is indirectly transmitted to the mind-EW of that man. Indirectly, it means that there are no “truth”; somebody can claim, each statement is true in a particular mind and in its correspondence with those real dogs. However, this is not “truth”, it is just an “implicit agreement”, indirectly agreement, inside a human mind, no more. Moreover, we cannot consider a mind as being a kind of “meta-language” or metamathematics since one mind does not exist for any EDWs/any mind-EDW.

In this context, it is much easier to understand that the second incompleteness theory (the impossibility of demonstrating consistency) has an intensional and not extensional characteristic: its validity depends not on what the system demonstrates, but on how it do it. (p. 109, Odifreddi’s italics) Regarding the “truth”, we cannot give up to intensionality/semantics; the “approximate truths” (my “implicit agreements”) cannot be reduced to the extensionality/syntax exactly as the mental functions of any human mind could not be reduced to its “syntax” (see the “Chinese Room” again) just because any mental state (“number 2”, for instance) is not “a part of my mind”, it is mind-EW (the mind is an immaterial EW without any extension or an “illusory spacetime framework” associated by the human mind to certain phenomena belonging to the for the macro-EW, for instance).

Again, I am interested in the relationship between a formal systems (logic), the machines/computers and the human mind. Regarding the human mind, Gӧdel considers that either the human mind exceed infinitely more the power of any finite machine[27] (at least in the area of pure mathematics), or there are certain “Diophantine problems” which are absolute insoluble and we cannot exclude that both are true. (p. 128) Odifereddi mentions that Gӧdel, Lucas and Putnam agree that there are no difference between machines and formal systems. (idem) Related to computers, there are the notions of computability, calculability and mechanic demonstrability (pp. 132-142). Odifreddi deals with these topics in the second part of this book. For Gӧdel, there are two ways:

(a) There are some mental non-demonstrable intuitions in a mechanical way, or there are mathematical truths which cannot be mentally intuitive or both. Moreover, Gӧdel believes that the human mind is not reducible to the brain activity (which seems to be a finite machine with finite parts). From my viewpoint, Gӧdel was somehow right: mind is not only irreducible to the brain activity, the mind is an EDW than the brain/body within the macro-EW. The mind-EW does not even exist for the brain/body/macro-EW.

(b) The mathematical objects and facts exist in an objective way independently of our mental decisions, i.e., a kind of Platonic view or realism for mathematical objects and theorems which are as objective as physical world. (p. 142) From my viewpoint, the mathematical entities/objects/facts cannot even exist.

In the end, Gӧdel rejected intuitionism. (p. 152) All these ideas/concepts are strong related to the difference between syntax and semantics: a computer does not have semantics; moreover, it does not have even syntax (see above). Therefore, from my EDWs perspective, it is meaningless to work on the similarities between the human minds and machines (computers, robots, GPT, etc.).

A very important problem is the relationship between real numbers, natural numbers and the “continuum hypothesis”. I do not want to deal with these topics since I have never worked on logic and mathematics. Gӧdel believes that the natural numbers really exist. The number of the real numbers is infinite and the dichotomy infinite/transfinite-finite is quite important in Odifreddi’s book. I want to mention that “infinite” and “numbers” (natural, real, etc.) cannot even exist. This is the reason, I do not deal with this problem. About the “continuum problem” (not in logic or mathematics): we can apply the continuum problem to the physical objects. For instance, in front of my body, there is a continuous table, i.e., its extension is has a continuity. The cup of coffee sitting on the table is also a continuous object and the cup interacts with the table in a continuous way. All these macro-entities have the “continuity” of their material, physical ontology. Obviously, the “continuity” of a table (its extension) does not exist for the corresponding amalgam of microparticles since the particles do not exist for a macro-entity. However, the corresponding electromagnetic field/waves has also a continuity…

Again, the relationship between syntax and semantic: Gӧdel believes that, influenced by Wittgenstein and being positivist, Carnap and others from the Vienna Circle (Hahn, Schlick) considered that mathematics was totally reducible to the syntax of language, it was nothing else than this syntax. In other words, the mathematical theorems are valid only as consequences of certain syntactical conventions in using symbols but not as descriptions of factual states of an objective world. (p. 180) Again, we have here certain “syntactical conventions”. I rejected the syntax mentioned above; above, I explicitly rejected Poincare’s “conventions”. As I indicated in my previous works, space could not exist, therefore, space is not given by “conventions” (i.e., by the “tools” which we measure it) since “space” itself could not even exist. Moreover, there is not a convention that in front of me “there are two dogs”. In front of me, there are indeed two dogs but the sentence “There are two dogs” is not “true” since “truth” cannot even exist. This sentence is just an explanation (it is my mind) which corresponds to something which really exist in front of my body: those dogs, in this case, two dogs, but “two” does not exist in reality. I associated, in my mind, those dogs with “number two”, but the sub-representations of those two dogs are “parts” of my entire perceptual representation/image/movie and my entire perceptual representation is the mind-EW. So, even this sentence is not fully “true” since in front of me, there are not only those two dogs, but also the entire external environment. Neither the statement: “In front of me, there are two dogs” is “true”, since my visual perception does not even exist for those two dogs, and each word has a particular pyramid of meanings in my mind and this pyramid is partially different than the pyramid of any other person. Again, each mind-EW is not for any mind-EDW, so how can we establish the existence of “truth”?

Odifreddi continues Gӧdel’s paragraph: for the positivists, the mathematical affirmations say nothing and talk about nothing because they are necessarily true and totally abstract. I agree with positivists in the sense that mathematical statements (but not “affirmations”[28]) are totally abstract, but I disagree with them regarding “affirmations” and “true”; these statement are the explicit or the implicit “agreements” that we learned/acquired long time ago. There are not innate concepts/propositions/statements. (see my book 2024) Odifreddi: Gӧdel is against positivists, the scheme of syntactic program (the replacement of mathematical intuition with rules for using symbols) does not work: this replacement does not furnish consistence and for a demonstration of consistency asks for a mathematical intuition as strong as that for identifying the truth of axioms or acknowledging the empirical facts in relationship to the equivalent mathematical content. (pp. 180-1) Interestingly, Gӧdel continues saying that those who complain that the mathematical propositions have no content (the experiments are not involved), the same thing we can say about the laws of nature since these laws (without mathematics) do not involve experiments. In the majority of cases, it is the mathematics which adds content to the natural laws. (p. 181)[29] I consider that the natural laws do not really exist (see my article 2024 or next part of this work). Moreover, above I rejected above Gӧdel’s belief about the existence of numbers. Indeed, the natural laws, always involves mathematics, therefore, since there are no mathematical truths, there are no natural laws. More exactly, there are no laws at all. (see next section)

Rejecting Kant’s phenomenalism, Gӧdel believed in the objectivity of physical objects.[30] This position is quite similar to my EDWs perspective. The problem for me is that Gӧdel believes in the existence of natural numbers. In 1944, Gӧdel wrote that the sets and concepts could be conceived as real objects, exactly as we considered the natural objects as being real. (p. 216) Just because the axioms of numbers imposed to us as being “true” exactly as the natural objects indicate they really exist. (p. 216) The mathematical perception/intuition is no less objective than the perceptions of real objects on which we construct the physical theories. “The problem of the objective existence of mathematical world is therefore an exact replica to the problem of the objective existence of external world.” (p. 217) Gӧdel extends this Platonism from mathematics to logic and to the existence of mathematical entities and the existence of laws and real objects. (p. 217) as I showed above, such extensions have been quite wrong. Moreover, Gӧdel believed in the immaterial mind and in a spiritual non-subjective world (not the world of God![31]) related to mathematics. (p. 219) Again, we have to reject the analogy between the existence of two dogs in front of me and the existence of number “2”: the dogs really exist in front of my body (not in front of my self/mind), but number “two” does not really exist, it is just an word/thought in my mind which it is associated/corresponds to those dogs in front of my body.

Gӧdel’s theorems of incompleteness indicate the limits of the demonstrations of mathematics. (p. 226) Gӧdel writes that his theorems do not establish any limit of the powers of human reason, but the limits of the potentialities of pure formalism in mathematics. (p. 227) Also, Gӧdel declares that his philosophy is rationalist, idealism, optimism, and theological. Influenced by Leibniz, he writes about the “central monad”, God[32]. “The true philosophy has to push you to something like a religious conversion which it should push you to see the world differently.” (Gӧdel in Odifreddi, p. 241)[33] Again, Gӧdel was totally correct regarding the limits of the demonstrations of pure formalism in logic and mathematics, but these limits do not indicate the limits of human mind. Leibniz was wrong: like all the philosophers and the scientists and other great thinkers until me, he was working within the unicorn world. (see my book about those 12 philosophers and my EDWs perspective 2024) Moreover, God cannot even exist, therefore, we have to accept the correspondences between the Hypernothing (EW0 with its hyperontology) and the EDW1a-n and all the EDWs. In the end, my verdict is the following: logic and mathematics are just human minds inventions/creations, some of them fit certain “external” phenomena, but most of them have nothing to do with reality.

Gödel’s theorem ends up bearing witness to the infinite potential of meaning, then. But it is worth noting, as a parting shot, that this is not just because of what it teaches us concerning the truths of arithmetic. It is also because of what it

teaches us through the very methods employed in its proof and the ingenious use of linguistic resources that these themselves involve. (Moore, p. 121)

Indeed, there is an “infinite potential of meaning” but infinite cannot even exist. Truths of arithmetic cannot even exist, the mathematical statements are just linguistic resources and all language is the mind-EW.

Just as Gödel’s theorem can teach us something profound about meaning through its content, then, so too can it teach us something profound about meaning through the vehicle of that content. Not that this should come as any surprise. This is just what we might expect of this endlessly fascinating exercise in self-reflexivity. (p. 122)

The “content” and the “vehicle” related to meaning involves the explicit knowledge and the pyramid of the implicit knowledge, but all these knowledge are the mind-EW. Syntax of any language exists only written on a paper or on the screen of my computer, but in any mind-EW, the words/sentences do not really exist as discrete and static entities. These entities (their syntax and semantics) are the mind-EW.

Logic, mathematics[34] and physical laws are “in the air”[35]:[36] there are just “old games” in an old city/world[37]; there are “old GPT”[38]. It means, many great thinkers have lost their careers working on logic and mathematics.[39] All the physicists until me had been working within the wrong framework, the universe/world (the unicorn world).[40] Somebody can claim, because of their works, we have computers, we describe, quite exactly, the motion of the Earth around the Sun, etc. I agree, but from my viewpoint, all these results are only pragmatic/practical results. We have not to mixture philosophy and pragmatic results. Like “philosophy of language”, “pragmatism” was a totally wrong philosophical direction in the history of philosophy since these programs did not furnish any “image of the world”. The main role, the single role for a philosopher is to create, through his philosophy, a new “image of the world”.[41] All the other philosophies are “GPT” (philosophy of mind/language, philosophy of ethics (morality), etc.), nothing else.[42] With my discovery, the EDWs, I have demolished the entire human knowledge imposed, during millenniums (until 2005) by different “imperialists”.[43] No great scientist or philosopher remains in front of my EDWs…[44]

Bibliogaphy

Crocco Gabriella (2016), “Kurt Gödel’s Philosophical Remarks (Max Phil)”,

https://www.researchgate.net/publication/270565216

Holt Jim (2005), “Time bandits”, https://www.newyorker.com/magazine/2005/02/28/time-bandits-2

Holt Jim (2017), When Einstein walked with Gӧdel – Excursions to the edge of thought, Mcmillan, USA

Moore A. W. (2022), Gӧdel’s theorem – A very short introduction, Oxford University Press

Odifreddi Piergiorgio (2020/2018), Dumnezeul logicii – Viata geniala a lui Kurt Gödel, mathematicianul filosofiei, Polirom

Weinert Friedel (2009), “Einstein, science and philosophy”, Philosophia Scientiæ 13-1, Varia

Yourgrau Palle (2005), A world without time – The forgotten legacy of Gödel and Einstein, Basic Books

[1] Obviously, if a person asks another person: “What do you have in your garden?” the second person can answer: “Trees”, but this answer has the explicit word, “trees” and other implicit words: “In my garden, I have tress”.

[2] My EDWs perspective is beyond nominalism-realism distinction; however, my approach is much closer to nominalism…

[3] I mention again that, during my career, I have not worked on logic and mathematics at all. Therefore, almost sure, there are some mistakes in this article. However, I will analyze only very general concepts of these domains, more exactly, I am interested in investigating their foundations. This is the reason, I quoted many paragraphs from different books/articles in my work: I am not specialist in logic or/and mathematics and I have not time to translate the ideas from those paragraphs in my words. The reader can imagine I talk with the great thinkers mentioned in so many quoted paragraphs… The notions from logic are from Odifreddi’s book (in Romanian), so, certain notions and statements from this book are just my translation (very possible, some of them I translated incorrectly).

[4] Odifreddi’s book has been published in Italian in 2018. Rosser dialed with the theorem of incompleteness for consistent systems (there cannot be demonstrated contradictions; the theorem of incompleteness for correct system (there cannot be demonstrated the false) was elaborated by Gӧdel. (p. 17)

[5] In this section, I will refer to Odifreddi’s book when I indicate only the page. When I refer to other authors, I indicate their names.

[6] Odifreddi’s book has been published in Italian in 2018. Rosser dialed with the theorem of incompleteness for consistent systems (there cannot be demonstrated contradictions; the theorem of incompleteness for correct system (there cannot be demonstrated the false) was elaborated by Gӧdel. (p. 17)

[7] For instance, regarding Hilbert’s axioms, some are similar to Euclid’s axioms, some are new. Among the “primitive concepts”, there are “point”, “line”, and “plan” among others. My viewpoint refers to the notions of these axioms and concepts…

[8] “.. a formal system had to be consistent: two theorems that contradict each other should not be able to be derived from the axioms. And the system should be complete, in the sense that all true statements expressible within the system (under a suitable interpretation) should be derivable from the axioms… Hilbert’s formalism… As it core it involves the dominance of form over content, syntax over semantics, proof over truth. It is no surprise that the principal embodiment of a formal system, the computer, a pure syntax machine, would become the century’s dominant mechanical device.” (Yourgraur, pp. 53-4) Symbols, symbols, symbols, nothing else…

[9] For Gӧdel, an unsolved problem means a problem which solution is neither proved nor disproved, this being an example of incompleteness of a system in which the statement is realized. (p. 76) With his theorem, Gӧdel indicates that “in any particular formal system of sufficient strength, given the limitations imposed on such a system insofar as it is truly formal, there would always be some formula which, while intuitively true, could not be proved in or relative to that system. And the same holds for its negation. But the formula would be a perfectly ordinary, though complex, mathematical proposition, which nevertheless, because of its form, slipped through the net of the given formal system. That very formula, however, could always be proved in a more inclusive formal system; only that new formal system, in turn, would be unable to prove some new formula, which was nevertheless intuitively true. And so on. There was, then, no ‘supervirus’ that affected all formal systems. Instead, for each particular formal system, there would be some perfectly ordinary bug or virus that rendered that system incomplete.” (Yourgraur, p. 59) Believing in mathematical truths, Gӧdel needed a “superbeing”, “God”…

[10] With Moore’s book, I do the same thing as I do with Odifreddi’s book: I selected just few paragraphs which help my viewpoint. Obviously, I do not understand too much from a logical book…

[11] Writing about the “liar paradox” in relationship to Gӧdel’s theorem, Holts concludes that “no logical system can capture all the truths of mathematics-is known as the first incompleteness theorem. Gӧdel also proved that no logical system for mathematics could, by its own devices, be shown to be agree from inconsistency, a results known as the second incompleteness theorem.” (Holt 2005, p. 6 or 2017, p. 18)

[12]

https://en.wikipedia.org/wiki/Decision_problem#:~:text=8%20References-,Definition,which%20the%20answer%20is%20yes.

[13] In this section, I will investigate certain concepts and statements from Odifreddi’s book. The order of investigating these concepts/statements follow the order of the pages of Odifreddi’s book. I recall that I will investigate only some of these concepts/statements from my EDWs perspective.

[14] Obviously, it is raining “outside” and not “inside” the house. The word “outside” is implicitly in this sentence. Otherwise, that person would say: “It is raining inside of your house.”

[15] There are two types of predicate-logic: the first order logic (predicates are satisfied by individual elements like points or numbers) and the second order logic (or by sets of elements like the prime numbers or the points of a circle). (p. 77) Regarding the second order logic, there is the axiom of continuity which assures the categorical logic. Gӧdel’s incompleteness refers to the second order logic.

[16] I recall some “criminals” who tried to prove the brain (occipital lobe, for instance) have “images”. In fact, their researches were just real crimes: recall the experiment with a monkey “seeing a black circle on a screen”. The authors claimed he sees the activation of certain neurons having the form of a circle. What a liar! He was able to manipulate many other professors with the help of journals and mass-media.

[17] “In effect, Gödel was borrowing a leaf from Descartes’s book. Descartes, by assigning numbers to figures in geometrical space through what are now known as Cartesian coordinates, was able, as we would now say, to ‘arithmetize geometry.’ In this system of so-called analytic geometry, statements about geometrical figures are translated into statements about numbers, and the powerful rules of manipulation of numbers, in turn, can be exploited to make discoveries about geometry.” (Yourgraur, p. 62)

[18] Poincare considers that these conventions in describing the space are given by the tools we use to measure it: if the tool is Euclidean (a straight line), the space would be represented as Euclidean; if we use a curved tool, the space would be represented as non-Euclidean. (see my previous works)

[19] I recall  Einstein’s verdict for mathematics “As far as the laws of mathematics refer to reality, they are not certain;

and as far as they are certain, they do not refer to reality.” (Einstein) I totally agree with this slogan.

[20] Following Gӧdel (against Wittgenstein, now), Carnap developed the Logical Syntax of Language. (p. 91)

[21] There was a complementarity of Gӧdel and Tarski’s theorems (apparently in contradictions): Gӧdel with the completeness of predicate logic + the incompleteness of arithmetic, Tarski with the impossibility of defining truth in a formal language + the possibility of defining truth through a meta-language. (p. 87) However, Tarski recognizes that he got his theorems based on Gӧdel’s theorems. (p. 88)

[22] From my viewpoint, maybe we can say that Wittgenstein’s “in language” means the explicit statement/thought, while “without being from the world” means the implicit knowledge. I have not worked on Wittgenstein’s “philosophy of language”, so the above statement is just a presupposition, nothing else…

[23] Somebody could judge this notion as being a pragmatic notion. I also reject pragmatism. It is about the EDWs here, not about pragmatic elements…

[24] Anyway, Frege, Russell, and Whitehead tried to construct a universal formal system for mathematics. (p. 95) In the letter to Zermelo, Gӧdel: through Cantor’s diagonal procedure, it was showed that the entire mathematics (or even certain partial systems) could not be contained within a single formal system when the system contains at least addition and multiplication for natural numbers. (p. 94)

[25] “There simply was no safe method by which the security of formal mathematical systems powerful enough to represent the natural numbers could be ensured. And there simply was no such thing as a formal system that could adequately and completely represent the natural numbers… Never again would syntax be substituted for semantics, proof for truth.” (Yourgraur, p. 74)

[26] As I mentioned above, related to this support is that Gӧdel’s believes in God within his personal framework of thinking.

[27] “In like manner, with his incompleteness theorem, Gödel exploited the favorite tool of the mathematical positivist, formal systems of proof, to construct a proof that formal systems for number theory will always be incomplete. In essence, the theorem was a mechanical, algorithmic demonstration of the limits of mechanical, algorithmic methods, and as it turned out, of the inescapable limitations of the computer.” (Yourgraur, pp. 105-6)

[28] The mathematical statements are “true” just because we have been learned to be true, nothing else. These statements do not really exist, therefore, their truths cannot even exist.

[29] Believing in natural numbers, Gӧdel believed in God. Following Leibniz and others, he even tried to furnish an argument God’s existence. (Chapter “From Gӧdel to God” in Odifreddi’s book) Indicating that God cannot even exist, I avoid to write about this topic.

[30] “Gödel, however, was not through with Kant. In an essay written in 1961 but never published, he noted that it was “a general feature of Kant’s assertions that literally understood they are false, but in a broader sense contain deeper truths’.” (Yourgraur, p. 107) “Both Gӧdel and Einstein insisted that the world is independent of our minds, yet rationally organized and open to human understanding.” (Holt 2005, p. 2)

[31] Gӧdel believed that God has to be a person. (p. 222)

[32] “His dream of an exact theology and the role of religion (beliefs 13 and 14) echoes Leibniz’s assertion (which Gödel quotes elsewhere) that philosophy—as a systematic enterprise—has to speculate rationally about God. Human experience, knowledge and action are in need of an explanation for which the idea of God is the keystone.” (Crocco 2015, p. 5)

[33] “The fact is that natural languages represent for him a concrete example of combinatorial tools, which are supposed to be images of the infinite possibility of combinations of objective concepts. Words and concepts are so related that, as Gödel says, even the finite combinatory is an image of God.” (Crocco 2015, p. 20) “For Gödel, all numbers are ‘the

work of God’.” (Yourgrau 2005, p. 23)

[34] “In Göttingen, the dean of mathematicians, David Hilbert, declared, ‘Where else would reliability and truth be found if even mathematical thinking fails?’ Cantor’s paradise, in particular, had to be shored up. ‘The definitive clarification of the nature of the infinite,’ said Hilbert, ‘has become necessary, not merely for the special interests of the individual

sciences, but rather for the honor of human understanding itself’.” (Yourgrau 2005, p. 52) Hilbert, there is no infinite and the “honour of human understanding itself” is not its being (recall Hitler, Stalin, Ceausescu or those from my Dark list), but the fact that each mind is an EW which is in itself and for itself…

[35] “In the air: Love is in the air. in the air in American English 1. current or prevalent 2. not decided; not settled; still imaginary (https://www.collinsdictionary.com/dictionary/english/in-the-air#google_vignette)

In circulation, in people’s thoughts. For example, There’s a rumor in the air that they’re closing, or Christmas is in the air. (https://www.dictionary.com/browse/in–the–air#google_vignette) up in the air idiom If a matter is up in the air, it is uncertain, often because other matters have to be decided first: uncertain and with an unknown result:

The whole future of the project is still up in the air. https://dictionary.cambridge.org/dictionary/english/up-in-the-air#google_vignette; (up) in the ˈair (of plans, etc.) uncertain; not yet decided: Our plans for the summer are still very much up in the air. ♢ At the end of the meeting, the matter was left in the air. in the ˈair (of an idea, a feeling, a piece of information, etc.) felt by a number of people to exist or to be happening: Spring is in the air. ♢ There was a strong feeling of excitement in the air. https://idioms.thefreedictionary.com/in+the+air” “But now, with Cantor, mathematics had seemingly overreached itself. It had tried to fly too high in the thin air of infinity and was in danger of crashing down on the solid earth below, the empirical soil on which natural science is based.” Yourgraur 2005, p. 48)

[36] “One would have thought that after Gödel’s incompleteness theorems, which established the essential limitations of formalization, the very enterprise of formalizing mathematical domains would have been reconsidered. Yet nothing of the kind happened. The American logician Emil Post was one of the few to take note of this curious fact.” (Yourgraur, p. 74) Obviously, all the scientists and the philosophers working, until 2005, under the unicorn world, it is a great surprise they did not care too much about Gödel’s incompleteness theorems…

[37] “Further separating Einstein from Gödel was the fact that Einstein never fully resolved his native suspicion of mathematics. To the end, the great physicist favored his cherished physical intuitions. Even though it was precisely Minkowski’s mathematical reworking of special relativity in terms of four-dimensional geometry (which Einstein resented at the time) that led to the mathematical abstractions of general relativity, the physicist remained forever wary of being led by the nose by mathematicians. He confessed once to being suspicious of a new move in general relativity that he said he could reach only mathematically (i.e., not intuitively). Gödel, in contrast, always felt most secure when he had formulated a problem in symbolic, mathematical terms. ‘If you had a particular problem in mind, wrote Taussky-Todd, ‘he would start by writing it down in symbols.’ Yet Gödel also believed, famously, that in mathematics too there are intuitions (a doctrine for which logicians still have not forgiven him).” ( Yourgraur 2005, p. 15) From my viewpoint, Einstein was better than Gödel in their relationship to the mathematical “truths”…

[38] “Translation” of GPT in French: “J’e pété”.

[39] “The incompleteness theorems sent a shock wave through the world of mathematics. Hermann Weyl, one of the first permanent members of the mathematical faculty at the Institute for Advanced Study, spoke of the Gödel “debacle,” the Gödel “catastrophe’.” (Yourgraur, p. 59) With my discovery, the EDWs, I have shocked the entire world of knowledge (physics, cognitive science, philosophy and, in this work, logic and mathematics)…

[40] Toward the end of his life, Gödel told to one of the few close friend that “he had long awaited an epiphany that would enable him to see the world in a new light, but that it never came. Einstein, too, was unable to make a clean break with time. “To those of us who believe in physics,” he wrote to the widow of a friend who had recently died, ‘this separation between past, present, and future is only an illusion, if a stubborn one.’ When his own turn came, a couple of weeks later, he said, ‘It is time to go’.” (Holt, p. 11) “Like Einstein, Gödel had led a life of increasing isolation and reclusiveness since coming to the institute, a tendency that only increased after he received the Einstein Award and delivered the Gibbs Lecture. He too spent his final years in a lost cause, part formal and part philosophical, searching for new axioms to decide the continuum hypothesis (and thus settle the question of whether there is an infinity between

the number of points on a line and the cardinality of the natural numbers), and seeking a definitive refutation of the thesis—bolstered by Cohen’s independence result for the continuum hypothesis—that the results of mathematics are in some sense only the reflection of human convention.” (Yourgraur, p. 150) From my viewpoint, the epiphany would be that not only the “spacetime” could not even exist, but even the universe/world cannot even exist, but the EDWs are. About continuum hypothesis, see above.

[41] “When the already distinguished Carnap suggested to his young student that he write some encyclopedia entries to gain recognition, Gödel responded that he had no need for such devices to achieve renown.” (Yourgraur 2005, p. 26) I do not need to be recognized by “imperialists” who have felt so embarrass by the fact that a “gypsy” from “Somalia” (ask Sean Carroll/Caltech about this) discovered the EDWs and not one of them… Moreover, in this work, I have indicated that great thinkers, indeed, from logic and mathematics have worked, somehow, in vain. They have wasted their lives working in the air… “Yet Einstein too, as Gödel wrote to his mother, ‘was in many respects a pessimist. In particular, he didn’t have a very good opinion of humanity in general. Among other things, he based this on the fact that those who wished to do some good, like Christ, Moses, Mohammed, etc., either died a violent death or had to use violence against his followers’.” (Yourgraur, p. 95) See my article: “God cannot even exist”…

[42] Since he was probably the greatest logician, I would be not surprised about Godel’s statement: “Only fables present the world as it should be and as if it had meaning.” (Yourgrau 2005, p. 5) In logic and mathematics, there are no “truths”, then fables seem to be a better “reality” for the “world” (which does not even exist). About Einstein and mathematics: “Einstein, before fleeing Germany, had already become a refugee from mathematics. He later said that he could not find, in that garden of many paths, the one to what is fundamental. He turned to the more earthly domain of physics, where the way to the essential was, he thought, clearer. His disdain for mathematics earned him the nickname ‘lazy dog’ from his teacher Hermann Minkowski (who would soon recast the lazy dog’s special relativity into its characteristic four-dimensional form). ‘You know, once you start calculating,’ Einstein would quip, ‘you shit

yourself up before you know it’.” (idem) “‘Every boy in the streets of Göttingen,’ his countryman David Hilbert wrote, ‘understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians’.” (Yourgrau 2005, p. 6) I do not know mathematics at all, I am much “lazy” then Einstein… However, I am a philosopher: “As Einstein realized, when the foundations of science become problematic, the man of science becomes a philosopher. [Einstein 1936, § 1]… As Einstein realized himself, science makes philosophical presuppositions. The scientist needs philosophical ideas, simply because amongst the experimental and mathematical tools in the toolbox of the scientist there are conceptual tools, like the fundamental notions.” (Weinert 2009, p. 17)

[43] “Gödel’s desire was to become a great philosopher in the tradition of Plato, Leibniz, and Kant, but he discovered that he had set this goal too late in his life, having devoted his best years to logic, mathematics and physics.” (Yourgraur, p. 104) Gödel believes that concepts “have an objective existence” and the “materialism is false.” “The world is rational.” “This puts one in mind of philosophical theism, according to which the order of the world reflects the order of the supreme mind governing it. Plato, a philosopher Gödel greatly admired, held similarly that all order is a reflection of rationality.” (idem, pp. 104-5) I have been lucky than Gödel: even if, in my career as student, I studied four years computer science, but then I gave up to these studies and I switched  to study philosophy: I got a diploma of bachelor, a Master diploma and two PhD diplomas in Philosophy (one at my university in 2006, another at UNSW, Australia in 2007).

[44] Probably, in 200 years, another approach will replace my EDWs perspective. However, some ideas of my perspective will remain as “truths” for at least two millenniums: “Their most famous discoveries behind them, Einstein and Gödel led increasingly quiet lives in the backwaters of their respective fields. In domains they had once ruled as titans they were now but part of the furniture, albeit, as Einstein cracked, ‘museum pieces.’ ‘In Princeton,’ he told friends, ‘I am known as the village idiot’.”(Yourgraur, p. 95) Unfortunately, both Einstein and Gödel worked within the wrong paradigm of thinking, the unicorn world.