The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[f] Euclid, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2).

https://en.wikipedia.org/wiki/Natural_number

Thus naturals was created is not by GOD but this is the pathetic snot of the ancient Greeks

its true sir................

its ok........

https://en.wikipedia.org/wiki/Pythagoras

Dear Mulyadi Rusli

Kronecker obviously wanted to say that the snot of the ancient Greeks is an ideal design and nothing else is possible.

it's possible true as I think

Dear Dr. Foukzon!

You raised a very important topic to discuss. I am a member of the Roman Catholic Church. The Bible is based on science, so the stories included in this religious book of the Catholic Church are scientifically true:

1. A video: "How Genetic Science CONFIRMS the Bible | Dr. Nathaniel Jeanson, Dec. 30, 2022, available at:

https://www.youtube.com/watch?v=lfuCdS5DqGs

Dr. Nathaniel Jeanson is a scientist of Cell and Developmental Biology (he had earned his doctoral degree at Harvard University): https://www.linkedin.com/in/nathanieljeanson

All the people in the world are representations of God:

Bhattacharyya, S. (2022). Zero—a Tangible Representation of Nonexistence: Implications for Modern Science and the Fundamental. In: Chakraborty, S., Mukhopadhyay, A. (eds) Living without God: A Multicultural Spectrum of Atheism. Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures, vol 37. Springer, Singapore. https://doi.org/10.1007/978-981-19-7249-2_11, a quote: "We insist that zero and its basic operations were likely conceived in India based on a philosophy of nothing, and classify nothing into four categories—balance, absence, emptiness and nonexistence." available at:Chapter Zero—a Tangible Representation of Nonexistence: Implications...

May I argue that there is no "nothing" - there is always something, but we just do not see it. God sees everything. We also can extend our ability to see and notice. When I close my eyes, I see "nothing", but my brain senses the environment around me. My brain is an "eye" that I can develop. Mr. Leopold Kronecker was wrong.

Yours sincerely, Bulcsu Szekely

In contemporary mathematics ancient snot has been formalized by Peano axioms

https://en.wikipedia.org/wiki/Peano_axioms

When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number". Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything. In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twenty-three problems] In 1931, Kurt Gödel proved his second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself.

According to Gödel's incompleteness theorems, the theory of PA (if consistent) is incomplete. Consequently, there are sentences of first-order logic (FOL) that are true in the standard model of PA but are not a consequence of the FOL axiomatization

Hilbert was crazy and urged everyone to believe in consistency PA (Con(PA)). Thanks to crazy Hilbert, the community fell into insanity, which has been firmly entrenched from Hilbert's time to nowadays .

@Jaykov Foukzon Reconsiderations are needed for the numbers. Kronecker is not right, I think.

First Incompleteness Theorem: "Any consistent formal system F based on FOL within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2020)

The unprovable statement GF referred to by the theorem is often referred to as "the Gödel sentence" for the system F. The proof constructs a particular Gödel sentence for the system F, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and any logically valid sentence.

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

This Gödel theorem does not make any sense

since there is no consistent formal system F based on FOL within which a certain amount of elementary arithmetic can be carried out

If God created natural numbers and 4 arithmetic operations - thereby He created rational numbers, Cantor's diagonal procedure, transfinite cardinalities, ...

That depends on God's existence... Because if there is no God, where do the natural numbers come from? Perhaps natural numbers exist per se whenever living beings require counting objects of interest (it seems that even bees can count up to some figure). If God exists, then there is no problem, and perhaps Kronecker was right, as it seems that one can attribute anything to Him (to God, not Kronecker).

Gödel incompleteness theorem does not make any sense since the following statement holds

Theorem. There is no consistent formal system F based on FOL within which a certain amount of elementary arithmetic can be carried out

http://www.ams.org/meetings/sectional/2210_program_ss17.html

http://www.ams.org/amsmtgs/2210_abstracts/1089-03-60.pdf

Set-theoretic definition

Main article: Set-theoretic definition of natural numbers

Intuitively, the natural number n is the common property of all sets that have n elements. So, its seems natural to define n as an equivalence class under the relation "can be made in one to one correspondence". Unfortunately, this does not work in set theory, as such an equivalence class would not be a set (because of Russell's paradox). The standard solution is to define a particular set with n elements that will be called the natural number n.

The following definition was first published by John von Neumann,[36] although Levy attributes the idea to unpublished work of Zermelo in 1916.[37] As this definition extends to infinite set as a definition of ordinal number, the sets considered below are sometimes called von Neumann ordinals.

The definition proceeds as follows:

• Call 0 = { }, the empty set.
• Define the successor S(a) of any set a by S(a) = a ∪ {a}.
• By the axiom of infinity, there exist sets which contain 0 and are closed under the successor function. Such sets are said to be inductive. The intersection of all inductive sets is still an inductive set.
• This intersection is the set of the natural numbers.

It follows that the natural numbers are defined iteratively as follows:

• 0 = { },
• 1 = 0 ∪ {0} = {0} = {{ }},
• 2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},
• 3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},
• n = n−1 ∪ {n−1} = {0, 1, ..., n−1} = {{ }, {{ }}, ..., {{ }, {{ }}, ...}},
• etc.

It can be checked that the natural numbers satisfies the Peano axioms

Set-theoretic definition gives nothing, since based on wrong hypothesis that ZFC is consistent.

Arno Gorgels added an answer

February 28

>Kronecker was (only) incomplete

No at all. Kronecker's meaning about naturals was mistaken since is based on stupid antic logic.

ON THE DOGMA OF THE NATURAL NUMBERS

P K Rashevskii

1973 The London Mathematical Society Russian Mathematical Surveys, Volume 28, Number 4 Citation P K Rashevskii 1973 Russ. Math. Surv. 28 143 DOI 10.1070/RM1973v028n04ABEH001602

ON THE DOGMA OF THE NATURAL NUMBERS1

P. K. Rashevskii

God gave us the integers, all else is

man's handiwork

L. Kronecker

Of course, no one nowadays takes Kronecker's words literally, indeed,

hardly anyone ever meant them literally, including Kronecker himself. But

if we read them in an appropriate transcription, then they do, in some

sense, express the prevailing frame of mind of mathematicians right up to

the present time.

On this point I should like to say that even today the natural numbers

are the unique mathematical idealization of the processes of real calcula-

tion.2 This monopoly position illuminates them with the halo of some

truth in the highest instance, absolute, the only one possible, to which the

mathematician must have recourse in every case when he is working on the

counting of his objects. Moreover, since the physicist uses only the appar-

atus proffered to him by the mathematician, the absolute power of the

natural numbers spreads to physics as well, and by means of the real axis

predetermines to a considerable extent the possibilities of physical theories.

It may be meaningful to compare the present position regarding the

natural numbers with the position of Euclidean geometry in the eighteenth

century, when it was the only geometrical theory, and hence was considered

as some absolute truth, as binding for the mathematician as for the physicist.

It was considered quite reasonable that physical space must ideally obey

Euclidean geometry exactly (what else?). Similarly, today we consider that

the counting of sets, as large as we please, of material objects, the measure-

ment of distances as large as we please in physical space, etc. must obey

existing schemes of the natural numbers and the real axis (what else?).

The difference lies only in the fact that the first question in brackets

https://www.mathnet.ru/links/c865ed0228094622b3208e0f39a3bb0d/rm4944_eng.pdf

Can someone prove that 2 neutrons are totally identical? I think not. Having different locations, they have different fields acting upon them, thus having different energies. Therefore, even if number 2 exists, it is almost impossible to prove that it exists.

So for all practical purposes, only 1 (presence of an object) and 0 (absence of it) can be argued exist. The remaining numbers are all nothing but abstract generalizations, whose logic breaks down when taken beyond a certain limit.

Numbers themselves do not exist. They are mere abstractions of Patterns found in nature. In that sence neither do shapes, equations or formulas exist.

For an example of what happens when mathematics is taken beyond a certain limit, look no further than the Foundational Crisis in Mathematics.:-

https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis

As many modern philosophers and mathematicians said:

It is the merest truism,

evident at once to unsophisticated observation,

that mathematics is a human invention.

Percy W. Bridgman; (1882-1961); The Logic of Modern Physics; 1927/1951; p60

Is geometry derived from experience? Careful discussion will give the answer- no! We therefore conclude that the principles of geometry are only conventions … Whence are the first principles of geometry derived? Are they imposed on us by logic? Lobatschevsky, by inventing non-Euclidean geometries, has shown that this is not the case.

Henri Poincaré; (1854-1912); Science & Hypothesis; 1905/1952; pxxv

… number is entirely the creature of the mind.

George Berkeley; (1685-1753); Principles of Human Knowledge; 1710; s12

… mathematics … although it can be applied to an exterior world, neither in its origin nor in its methods depends on an exterior world.

L. E. J. Brouwer; (1881-1966); Brouwer’s Cambridge Lectures on Intuitionism; Dirk van Dalen, ed.; 1951/1981; p92

Mathematics is a human artefact, a human conception, in which there is no truth … there is no absolute unity; no absolute space and no absolute time, there is no mathematics.

Gerrit Mannoury; (1867-1956); Quoted in Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer;

Dirk van Dalen; 1999; p121

Non-Euclidean geometry is proof that mathematics … is man’s own handiwork, subject only to the limitations imposed by the laws of thought. … [its] creation signalized the realization that mathematics in no sense depends upon our environment.

E. Kasner; (1878-1955); & J. Newman; (1907-1966); Mathematics and the Imagination; 1940; p359,361

… the whole of Arithmetic and Algebra has been shown to require three indefinable notions and

five indemonstrable propositions.

Bertrand Russell; (1872-1970); Mysticism and Logic; 1914/1957; p73

Mathematics is the language of physical science and certainly no more marvelous language was ever created by the mind of man.

R. B. Lindsay; (1900-1985); On the Relation of Mathematics & Physics; 1963; p151

The scientist’s world

is perfectly mathematical,

but the sense world

is not.

Gordon Clark; (1902-1985);

A Christian View of Men & Things; 1952/1981; p210

Modern astronomers might agree with Kepler that the heavens declare the glory of God and the firmament showeth His handiwork; however, they now recognize that the mathematical interpretations of the works of God are their own creations …

Morris Kline; (1908-1992);

Mathematics and the Search for Knowledge; 1985; p85

All mathematics begins with a set of axioms. Any set of axioms is as valid as any other as long as it avoids contradictory assumptions.

Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189

… Science, and especially Mathematics, the ideal form of science, are creations of Intellect in its quest for Harmony.

Cassius J. Keyser; (1862-1947); The Human Worth of Rigorous Thinking; 1913/1925; p23

… the number 2 … is a metaphysical entity about which we can never feel sure that it exists…

Bertrand Russell; (1872-1970); Introduction to Mathematical Philosophy; 1919/1993; p18

… the mathematician … derives from the axioms only what he puts into them, since all conclusions that follow are logically implicit in the axioms.

Morris Kline; (1908-1992); Mathematics: Method and Art; 1963; p165

… numbers are free creations

of the human mind …

Richard Dedekind; (1831-1916);

Essays on the Theory of Numbers; 1901/1963; p31

Nature does not count nor do integers occur in nature.

Man made them all, integers and all the rest …

Percy Williams Bridgman; (1882-1961); The Way Things Are; 1959; p100

… mathematics is something that has been created over time as a means of conceptualizing the natural world. We should not be surprised by its effectiveness at doing what it is designed to do.

Why is it that so much of science can be explained mathematically? Because, so much of mathematics is speculative brainstorming.

The unreasonable effectiveness is an illusion.

Paul Cox; What is Mathematics? Part 2

The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. … he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God.

J. W. N. Sullivan; (1886-1937); Mathematics as an Art; 1963; p271

Not only are there no two identical objects,

no single unchangeable object exists in nature.

… we see that an object with identity is an abstraction corresponding exactly to nothing in nature.

P. W. Bridgman; (1882-1961); The Logic of Modern Physics; 1927/1951; p35

No two real things are precisely equal.

Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p188

… language imposes subjects and predicates on a world that does not have stable, enduring units corresponding to its terms.

F. Nietzsche; (1844-1900); Will to Power; cited in Truth in Philosophy; Barry Allen; 1993; p46

We are prepared to say that one and one are two,

but not that Socrates and Plato are two …

Bertrand Russell; (1872-1970); Introduction to Mathematical Philosophy; 1919/1993; p196

It is certain that all natural bodies, even those said to be of the same kind, differ from each other, that no two portions of gold are exactly alike, and that one drop of water is different from another drop of water.

Nicolas Malebranche; (1638-1715); The Search After Truth; 1674/1997; p253

A typical statement of empirical arithmetic is that 2 objects plus 2 objects makes 4 objects. This statement acquires physical meaning only in terms of physical operations, and these operations must be performed in time. Now the penumbra gets into this situation through the concept of object. If the statement of arithmetic is to be an exact statement in the mathematical sense, the “object” must be a definite clear-cut thing, which preserves its identity in time with no penumbra. But this sort of thing is never experienced, and as far as we know does not correspond exactly to anything in experience.

P. W. Bridgman; (1882-1961); The Logic of Modern Physics; 1927/1951; p34

A simple arithmetic statement like “7+5=12” is true, not because it conforms to a set of empirical facts, but because it is a theorem of arithmetic which is deducible from certain prior theorems which in turn derive from the postulates, rules and basic concepts of that system.

Joseph Gerard Brennan; The Meaning of Philosophy; 1953; p85

If we are to add at all, we must add unlikes,

in violation of all mathematical regulations.

Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189

… we must concede that no material object

is truly and simply one.

St. Augustine; (354-430); De Libero Arbitrio; p45

You cannot step into the same river twice.

Heraclitus, 500 B.C.

You cannot step into the same river

even once.

Cratylus, 400 B.C. (Also found in Buddhist Philosophy -> rsm)

Mathematics has been shorn of its truth;

it is not an independent, secure,

solidly grounded body of knowledge.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p352

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? … In my opinion the answer to this question is briefly this:- 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.

Albert Einstein; (1879-1955); Sidelights on Relativity; Dover; 1983; p28

The object of mathematical theories is not to reveal to us the real nature of things; that would be an unreasonable claim.

Henri Poincaré; (1854-1912);

Science & Hypothesis; 1902/1952; p211

… the exact validity of mathematical laws as laws of nature is out of the question.

L. E. J. Brouwer; (1881-1966); Intuitionism and Formalism; 1912

Truth to the mathematician merely means freedom from internal inconsistencies.

Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189

In fact, consistency, not truth, is the key word to mathematical thought. … The thought that the axioms underlying a mathematical system must be “obvious truths” slowly became a thing of the past.

Carroll V. Newsom; (1904-1989); An Introduction to Modern Mathematical Thought; 1963; p75

In recent years consistency replaced truth as the god of mathematicians and now there is a likelihood that this god too may not exist.

Morris Kline; (1908-1992); Mathematics: Method and Art; 1963; p161

… Mathematics is still the paradigm of the best knowledge available.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p352

Although there is at present still considerable disagreement about the ultimate foundations of mathematics, nobody can nowadays hold the opinion anymore that “arithmetical propositions” communicate any knowledge about the real world. … Their validity is that of mere tautologies; they are true because they assert nothing of any fact …

Moritz Schlick; (1882-1936); Form and Content: An Introduction to Philosophical Thinking; in v2 Philosophical Papers; H. L. Mulder; ed.; 1979; p344

Thus the absence of all mention of particular things or properties in logic or pure mathematics is a necessary result of the fact that this study is, as we say, “purely formal.”

Bertrand Russell; (1872-1970); Introduction to Mathematical Philosophy; 1919/1993; p198

Mathematics is the most abstract of all the sciences. For it makes no external observations, nor asserts anything as a real fact.

C. S. Peirce; Collected Works- The Regenerated Logic; 1896; p23

Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Bertrand Russell; (1872-1970); Recent Work on the Principles of Mathematics; International Monthly; vIV; p84; 1901. In The Monist; v22

The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true…

Cassius J. Keyser; (1862-1947); The Human Worth of Rigorous Thinking; 1916; p221

In the words of the philosopher Wittgenstein,

mathematics is just a grand tautology.

Morris Kline; (1908-1992); Mathematics: Method and Art; 1963; p165

The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter.

Carl G. Hempel; (1905-1997);

On the Nature of Mathematical Truth; 1945

The propositions of mathematics are of exactly the same kind as the propositions of logic: they are tautologous, they say nothing at all about the objects we want to talk about.

Hans Hahn; (1879-1934); Logic, Mathematics and Knowledge of Nature; 1933

… for a period of over two thousand years, mathematicians pursued truth.

… Creations of the early 19th century, strange geometries & strange algebras, forced mathematicians, reluctantly and grudgingly, to realize that mathematics proper, and the mathematical laws of science were not truths.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p3,4

… arithmetic is a calculus which starts only from certain conventions but floats as freely as the solar system and

rests on nothing.

Friedrich Waismann; (1896-1959);

Introduction to Mathematical Thinking; 1951; p121

The current predicament of mathematics is that there is not one but many mathematics … It is now apparent that the concept of a universally accepted, infallible body of reasoning– the majestic mathematics of 1800 and the pride of man– is a grand illusion.

… one cannot speak of arithmetic as a body of truths that necessarily apply to physical phenomena. … Thus the sad conclusion which mathematicians were obliged to draw is that there is no truth in mathematics, that is, truth in the sense of laws about the real world.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p6, 95

… we do not experience numbers as we experience colors and sounds, numbers are nothing in and by themselves…

Hans Hahn; (1879-1934);

Empiricism, Logic and Mathematics; 1931/1980; p15

Gradually … mathematicians granted that the axioms and theorems of mathematics were not necessary truths about the physical world. … As far as the study of the physical world is concerned, mathematics offers nothing but theories or models.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p97

The philosophical aims of the three schools [Intuitionism- Brouwer, Logicism- Frege, Formalism- Hilbert] have thus not been achieved, and it seems to us that we are no nearer to a complete understanding of mathematics than the founders of these schools.

Andrzej Mostowski; (1913-1975); Thirty Years of Foundational Studies; 1966; p8

The fundamental concepts of mathematics are … empty space, empty time … points without extension, lines without breadth, surfaces without depth, spaces without content. All these concepts are contradictory fictions, mathematics being based upon an entirely imaginary foundation, indeed upon contradictions. Upon these foundations the psyche has constructed the entire edifice of this amazing science. Mathematicians have occasionally realized that they were dealing with contradictions, but seldom or never was this made the subject of any profound study. The frank acknowledgement of these fundamental contradictions has become absolutely essential for mathematical progress. The efforts made to conceal this fact have all worn threadbare.

Hans Vaihinger; (1852-1933); The Philosophy of “As If”; 1924; p51

… if mathematics is true (in the best, i.e. Platonistic, sense) then it seems we cannot know its truths.

Morton, A. & Stich, S. P.; eds.; Benacerraf and his Critics; 1996; p67

… all mathematics is a gigantic tussle

with nonexistent impossibilities.

Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189

Geometry is not true, it is advantageous.

Henri Poincaré; (1854-1912); Science & Method; 1908

… we can no more say that Einstein’s geometry is “truer” than Euclidean geometry, than we can say that

the meter is a “truer” unit of length than the yard.

Hans Reichenbach; (1891-1953); Philosophy of Space and Time; 1958; p35

A straight line has no width, no depth, no wiggles and no ends. There are no straight lines. We have ideas about these non-existent impossibilities; we even draw pictures of them. But they do not exist … A straight line hasn’t even a definition.

A point has no dimensions, no existence, and no definition. … Euclid lists twenty-three definitions which define more than twenty-three figments of the imagination. … He assumes all right angles are equal, although there are no right angles. … Lastly, Euclid introduces five “common notions” as axioms, that is, as self-evident truths, the very first of which is impossible, let alone true: “Things equal to the same thing are equal to each other.” No two things are precisely equal. …

The whole of geometry is consciously, willfully, deliberately antagonistic to reality.

Billy E. Goetz; (1904-1986); President of MIT 1958;

The Usefulness of the Impossible; 1963; p187ff

… for geometry as a mathematical science, there is no problem concerning the truth of the axioms. This apparently unsolvable problem turns out to be a pseudo-problem. The axioms are not true or false, but arbitrary statements.

Hans Reichenbach; (1891-1953); Philosophy of Space and Time; p5

… there is no reason to suppose that [a] triangle is a revelation of an eternally pre-existing truth– such as a thought in the mind of God. It is an arbitrary creation of the mathematician’s mind, and did not exist until the mathematician thought of it.

J. W. N. Sullivan; (1886-1937); The Limitations of Science; 1933; p152

Geometry predicates nothing

about the relations of real things …

Albert Einstein; (1879-1955); Sidelights on Relativity; Dover; 1983; p35

The theory contends that an innate property of the human mind, the ability of visualization, demands that we adhere to Euclidean geometry. In the same way as a certain self-evidence compels us to believe the laws of arithmetic, a visual self-evidence compels us to believe the validity of Euclidean geometry. It can be shown that this self-evidence is not based on logical grounds.

Hans Reichenbach; (1891-1953); Philosophy of Space and Time; p32

It was thought till recently that geometry dealt with space and spatial points, and laymen probably thought that, even if they themselves did not succeed in grasping what a point of space was, mathematicians at least had a perfect grasp of it. But this was an enormous error: mathematicians had no better grasp of it, and they were no more capable of saying what a point really was than the nearest layman.

Hans Hahn; (1879-1934); Empiricism, Logic and Mathematics; 1931/1980; p13

The ideas expressed in the preceding considerations attempted to establish Euclidean geometry as epistemologically a priori; we found that this a priori cannot be maintained and that Euclidean geometry is not an indispensable presupposition of knowledge.

Hans Reichenbach; (1891-1953); Philosophy of Space and Time; p31

I shall not attempt to prove that mathematics is useful. I will admit it and so save myself the trouble that here is a great and respected discipline where all is impossible yet much is useful. The usefulness largely flows from the impossibility. Mathematical concepts have been simplified and generalized until they describe an imaginative world no part of which could possibly exist outside men’s minds.

Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189

Though the axioms of non-Euclidean geometry appeared to be contrary to ordinary human experience, they yielded theorems applicable to the physical world.

Morris Kline; (1908-1992); Mathematics: Method and Art; p160

Every mathematical system contains undefined terms: for example, the words ‘point’ and ‘line’ in a geometric system. In deductive proof from explicitly stated axioms the meaning of the undefined terms is irrelevant. … pure mathematics is not … concerned with … meanings (of) undefined terms. … it is concerned with deductions that can be made from the axioms …

Morris Kline; (1908-1992); Mathematics: Method and Art; p166

Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world.

Morris Kline; (1908-1992); Mathematics: Method and Art; p167

Thus, the analysis outlined on these pages exhibits the system of mathematics as a vast and ingenious conceptual structure without empirical content and yet an indispensable and powerful theoretical instrument for the scientific understanding and mastery of the world of our experience.

Carl G. Hempel; (1905-1997); On the Nature of Mathematical Truth; 1945

We modify the mathematics when applications reveal misrepresentation or downright errors in the mathematics.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; p344

Freeman Dyson agrees: “we are probably not close yet to understanding the relation between the physical and the mathematical worlds.” … it is important to realize that nature and the mathematical representation of nature are not the same. The difference is not merely that mathematics is an idealization, the mathematical triangle is assuredly not a physical triangle.

These … “explanations” … say rather little … in impressive language that tempers the admission that they have no answer to why mathematics is effective.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; p349

It is sometimes assumed that the effectiveness of mathematics … shows that mathematics itself exists in the structure of the physical universe. This, of course, is not a scientific argument with any empirical evidence.

George Lakoff; (1941-); Where Mathematics Comes From; 2000; p3

How then under this view can mathematics apply to the physical world and especially to physical phenomena? There are several answers. One is that mathematical axioms use undefined terms and these can be differently interpreted to suit the physical situation.

Should we reject mathematics because we don’t understand its unreasonable effectiveness? … Should I refuse my dinner because I do not understand the process of digestion? … mathematics deals with the simplest concepts and phenomena of the physical world. It does not deal with man but with inanimate matter.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; p342, 350

The question of the existence of a Platonic mathematics cannot be addressed scientifically. At best, it can only be a matter of faith, much like faith in God. … The burden of scientific proof is on those who claim an external Platonic mathematics does exist … At present there is no known way to carry out such a scientific proof … as far as we can tell, there can be no such evidence, one way or the other. There is no way to tell empirically whether proofs proved by human mathematicians are objectively true, external to the existence of human beings or any other beings.

George Lakoff; (1941-); Where Mathematics Comes From; 2000; p2, 342

Why then should the deductions still apply? Poincaré’s answer is that we modify the physical laws to make the mathematics fit.

Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; p343

Let us grant that the pursuit of mathematics is a

divine madness of the human spirit.

A. N. Whitehead; (1861-1947); Science and the Modern World; /1958; p22