Philosophy of mathematics

These next several outlines deal with philosophy of certain specialized topics, starting with this one on the philosophy of mathematics. Here we dig into issues of what is abstraction.

First we survey the following branches of mathematics: algebra, analysis, numbers theory, logic, model theory, and category theory. Then we discuss unification programs across branches, followed by sections on positions in philosophy of mathematics, in particular dealing with the realism/antirealism or platonism/nominalism debate.

Contents

  1. Algebra
    1. Introduction
    2. History
    3. Finite groups
    4. Lie groups
    5. More
  2. Analysis
    1. Introduction
    2. History
    3. Development of calculus
    4. Differential geometry
  3. Number theory
    1. Introduction
    2. Set theory
    3. Transfinite numbers
  4. Logic
    1. Introduction
    2. History
    3. Propositional logic
    4. First-order logic
    5. Second-order logic
    6. Modal logic
    7. Alternative logics
    8. Proof theory
  5. Model theory
    1. Introduction
    2. Incompleteness
    3. Complexity theory
  6. Category theory
    1. Introduction
    2. History
    3. Homotopy type theory
    4. Cobordism hypothesis
    5. Topos theory
  7. Unification programs
    1. Introduction
    2. Erlangen program
    3. Langlands program
    4. Univalent foundations
  8. Platonism
    1. Introduction
    2. Pythagoreanism
    3. Plato
    4. Contemporary platonism
    5. Visual proofs
  9. Nominalism
    1. Introduction
    2. History
    3. Science Without Numbers
  10. Logicism
    1. Introduction
    2. Principia Mathematica
    3. ZFC
    4. Neologicism
  11. Formalism
    1. Introduction
    2. Hilbert’s program
  12. Intuitionism
    1. Introduction
    2. Criticism
  13. Fictionalism
  14. Structuralism
  15. Naturalism
    1. Unreasonable effectiveness
    2. Quine-Putnam indispensability thesis
    3. Mathematical naturalism
  16. My thoughts
  17. Annotated bibliography
    1. Wigner, E.P. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
    2. Field, H. (1980). Science Without Numbers.
    3. Snapper, E. (1979). The Three Crises in Mathematics: Logicism, Intuitionism, Formalism.
    4. More articles to do
  18. Links and encyclopedia articles
    1. SEP
    2. IEP
    3. Wikipedia
    4. Others
    5. Videos
  19. References

Algebra

Introduction

TODO

History

Finite groups

Lie groups

More

See also:

Analysis

Introduction

TODO

History

Development of calculus

Figure 1: Leibniz’s notation of integration: \int (summa) and differentiation: d (differentia) summarized in the margin of his notes in 1675. Note that \Pi was Leibniz’s notation for equality. Photo by S. Wolfram (2013) of notes in the Leibniz-Archiv in Hanover, Germany.

Differential geometry

See also:

Number theory

Introduction

Set theory

Transfinite numbers

Logic

Introduction

Pedagogy:

More:

History

Propositional logic

Syntactic consequence:

\[ A \vdash B \]

means that \(B\) is logically derivable/provable from \(A\).

Semantic consequence:

\[ A \models B \]

means that in all possible valuations in which \(A\) is true, \(B\) is also true. One says that \(A\) “entails” \(B\), or \(A\) “models” \(B\).

Note that \(\vdash\), \(\models\), and \(\equiv\) are all metalogical symbols, not part of the rules of logic; they are shorthands.

\(\vdash\) means “from which, it is derivable that.”
\(\models\) means “entails that, i.e. in every case this is true, that is true (regardless of provability).”
\(\equiv\) means “is notationally identical to.”

Lecture notes on soundness and completeness:

A formal system is sound if everything that is provable is in fact true, i.e. if \(A_1, A_2, \ldots A_n \vdash B\), then \(A_1, A_2, \ldots A_n \models B\).

A formal system is complete if everything that is true has a proof, i.e. if \(A_1, A_2, \ldots A_n \models B\), then \(A_1, A_2, \ldots A_n \vdash B\).

Propositional logic was proven to be sound (\(\vdash\) implies \(\models\)) and complete (\(\models\) implies \(\vdash\)) by Emil Post in 1921.16

Material implication:

\[ P \rightarrow Q \equiv \lnot P \lor Q \]

Modus ponens:

\[ P \rightarrow Q, P \vdash Q \]

Modus tollens:

\[ P \rightarrow Q, \lnot Q \vdash \lnot P \]

Peirce’s law:

\[ ((P \rightarrow Q) \rightarrow P) \rightarrow P \]

See also:

First-order logic

See also:

Second-order logic

See also:

\(\Box\) means “necessarily.” \(\Diamond\) means “possibly.”

If necessarily \(P\), then necessarily necessarily \(P\):

\[ \Box P \rightarrow \Box \Box P \]

De Morgan duality:

\[ \Diamond P = \lnot \Box \lnot P \]

\[ \Box P = \lnot \Diamond \lnot P \]

Alternative logics

Criticism:

Proof theory

See also:

Model theory

Introduction

Model theory is the theory of interpretations of formal languages (a model of a formula of a language is an interpretation of the language for which the formula comes out true).30

Incompleteness

Fom the SEP:

The first incompleteness theorem states that in any consistent formal system \(F\) within which a certain amount of arithmetic can be carried out, there are statements of the language of \(F\) which can neither be proved nor disproved in \(F\). According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent).34

From Wikipedia:

Related:

Chaitin:

[M]y information-theoretic approach to incompleteness makes incompleteness appear pervasive and natural. This is because algorithmic information theory sometimes enables one to measure the information content of a set of axioms and of a theorem and to deduce that the theorem cannot be obtained from the axioms because it contains too much information.

This suggests to me that sometimes to prove more one must assume more, in other words, that sometimes one must put more in to get more out.35

More:

Relationship to mind:

See also:

Complexity theory

Category theory

Introduction

See also:

History

Grothendieck:

If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither “number” nor “size,” but always form. And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things.40

Homotopy type theory

Cobordism hypothesis

Topos theory

Unification programs

Introduction

Yanofsky, N.S. (2016). Why mathematics works so well:

When you have two distinct fields of mathematics and they are shown to be intimately related in a way that results of one field can be used to get results of another field, you have a type of unification. An example of such unification is monstrous moonshine. This subject describes the shocking connection between the monster group and modular functions. Another example is the Langlands program which connects Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups. The Erlangen program can also be seen as a way of unifying different types of geometries using group theory. In all these examples of unifications, there is a function (or an isomorphism) from the domain of discourse of one field to the domain of discourse of another field. The symmetries of one field (the true mathematical statements) will than map to symmetries of the other field. Category theory is an entire branch of mathematics that was created to describe such unifications. The founders of category theory invented a language that was based on algebraic topology, which is a branch of mathematics that unifies algebra and topology. Category theory is now used in many areas to show that seemingly different parts of mathematics (and physics and theoretical computer science) are closely related.43

Erlangen program

Langlands program

Univalent foundations

See also:

Platonism

Introduction

What has been is what will be,
and what has been done is what will be done;
there is nothing new under the sun.
Is there a thing of which it is said,
“See, this is new?”
It has already been,
in the ages before us.44

Pythagoreanism

Figure 2: Bronnikov, F. (1869). Pythagoreans celebrate sunrise. (Wikimedia, 2007).

Plato

Figure 3: Detail of School of Athens by Raphael (1511), showing Plato and Aristotle (Wikimedia, 2013).

I think you know that the students of geometry, calculation, and the like hypothesize the odd and the even, the various figures, the three kinds of angles, and other things akin to these in each of their investigations, as if they knew them. They make their hypotheses and don’t think it necessary to give any account of them, either to themselves or to others, as if they were clear to everyone. And going from these first principles through the remaining steps, they arrive in full agreement.

I certainly know as much.

Then you also know that, although they use visible figures and make claims about them, their thought isn’t directed to them but to those other things that they are like. They make claims for the sake of the square itself and the diagonal itself, not the diagonal they draw, and similarly with the others. These figures that they make and draw, of which shadows and reflections in water are images, they now in turn use as images, in seeking to see those others themselves that one cannot see except by means of thought.45

Contemporary platonism

Whatever we say with the help of names can be said in a language which shuns names altogether. To be assumed as an entity is, purely and simply, to be reckoned as the value of a variable. In terms of the categories of traditional grammar, this amounts roughly to saying that to be is to be in the range of reference of a pronoun. Pronouns are the basic media of reference; nouns might better have been named propronouns. The variables of quantification, ‘something,’ ‘nothing,’ ‘everything,’ range over our whole ontology, whatever it may be; and we are convicted of a particular ontological presupposition if, and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of our affirmations true.47

a theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true.48

Visual proofs

Composite numbers can be arranged into rectangles but prime numbers cannot (source: Wikimedia).
A visual proof that \sum_{k=1}^{n} k = (n^2+n)/2.
A visual proof that \sum_{k=1}^{\infty} \frac{1}{4^k} = \frac{1}{3}.

Nominalism

Introduction

History

Antisthenes:

A horse I can see, but horsehood I cannot see.

Bayes:

It is not the business of a mathematician to show that a straight line or circle can be drawn, but he tells you what he means by these; and if you understand him, you may proceed further with him; and it would not be to the purpose to object that there is no such thing in nature as a true straight line or perfect circle, for this is none of his concern: he is not inquiring how things are in matter of fact, but supposing things to be in a certain way, what are the consequences to be deduced from them.51

Johwn Stuart Mill:

Nominalists, who, repudiating Universal Substances, held that there is nothing general except names.52

See also:

Science Without Numbers

Field: Abstract platonic entities, were they to exist, could not interact causally with the world.55

significant in context but naming nothing.59

Logicism

Introduction

Principia Mathematica

ZFC

Neologicism

Linsky & Zalta:

Our knowledge of mathematics is to be explained in terms of the faculty we use to understand language, since that is the only faculty we need to understand object abstraction.65

Formalism

Introduction

Hilbert:

We must know. We shall know.

Hilbert:

Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.66

Hilbert’s program

Hilbert wanted math to be

See also:

Intuitionism

Introduction

Criticism

Hilbert:

Taking the Principle of the Excluded Middle from the mathematician … is the same as … prohibiting the boxer the use of his fists.68

Fictionalism

Structuralism

Naturalism

Unreasonable effectiveness

Philosophy is written in that great book which ever lies before our eyes—I mean the Universe—but we cannot understand it if we do not learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures without whose help it is impossible to comprehend a single word of it, without which one wanders in vain through a dark labyrinth.

– Galileo Galilei. (1623). Il Saggiatore (The Assayer).70

Quine-Putnam indispensability thesis

Putnam:

[Q]uantification over mathematical entities is indispensable for science…; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.76

Mathematical naturalism

My thoughts

Annotated bibliography

Wigner, E.P. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

  • Wigner (1960)

My thoughts

  • TODO.

Field, H. (1980). Science Without Numbers.

  • Field (2016)

My thoughts

  • TODO.

Snapper, E. (1979). The Three Crises in Mathematics: Logicism, Intuitionism, Formalism.

  • Snapper (1979)

My thoughts

  • TODO.

  • TODO.

SEP

IEP

Wikipedia

Others

Videos

References

Aguirre, A., Foster, B., & Merali, Z. (2016). Trick Or Truth?: The mysterious connection between physics and mathematics. Springer.
Atiyah, M. et al. (1994). Responses to "Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics", by A. Jaffe and F. Quinn. Bulletin of the American Mathematical Society, 30, 178–207. https://arxiv.org/abs/math/9404229
Azzouni, J. (2015). Nominalism, the nonexistence of mathematical objects. In E. Davis & P. Davis (Eds.), Mathematics, Substance and Surmise (pp. 133–145). Springer.
Baez, J. C. (2002). The octonions. Bulletin of the American Mathematical Society, 39, 145–205. https://arxiv.org/abs/math/0105155
Baez, J. C. & Dolan, J. (1995). Higher‐dimensional algebra and topological quantum field theory. Journal of Mathematical Physics, 36, 6073–6105. https://arxiv.org/abs/q-alg/9503002
Baez, J. C. & Stay, M. (2009). Physics, topology, logic, and computation: A Rosetta Stone. https://arxiv.org/abs/0903.0340
Bayes, T. (1736). An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst. London. (published anonymously).
Bonevac, D. (2003). Deduction: Introductory to Symbolic Logic (2nd ed.). Blackwell.
Bronstein, M. M., Bruna, J., Cohen, T., & Velickovic, P. (2021). Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. https://arxiv.org/abs/2104.13478
Brouwer, L. E. J. (1908). Unreliability of the logical principles. https://arxiv.org/abs/1511.01113
Bueno, O. (2013). Nominalism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/nominalism-mathematics/
Burgess, J. P. (1983). Why I am not a nominalist. Notre Dame Journal of Formal Logic, 24, 93–105.
Button, T. & Walsh, S. (2018). Philosophy and Model Theory. Oxford University Press.
Carnap, R. (1958). Introduction to Symbolic Logic and its Applications. New York: Dover Publications.
Carter, N. (2009). Visual Group Theory. Mathematical Association of America.
Chen, E. K. (2018). The intrinsic structure of quantum mechanics. http://philsci-archive.pitt.edu/15140/
Cooper, J. M. & Hutchinson, D. S. (1997). Plato: Complete works. Hackett Publishing.
Cubitt, T., Perez-Garcia, D., & Wolf, M. (2015). Undecidability of the spectral gap. Nature, 528, 207–211.
Drake, S. (1957). Discoveries and Opinions of Galileo. New York: Doubleday and Co.
Enderton, H. B. (2009). Second-order and higher-order logic. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/archives/sum2019/entries/logic-higher-order/
Ewald, W. (2018). The emergence of first-order logic. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/logic-firstorder-emergence/
Field, H. (1989). Realism, Mathematics, and Modality. Oxford: Blackwell.
———. (2016). Science Without Numbers (2nd ed.). Oxford University Press. (Originally published in 1980 by Princeton University Press).
Fong, B. & Spivak, D. I. (2018). Seven sketches in compositionality: An invitation to applied category theory. https://arxiv.org/abs/1803.05316
Franzén, T. (2005). Gödel’s Theorem: An incomplete guide to its use and abuse. A K Peters.
Frè, P. G. (2013). Gravity, a Geometrical Course, Volume 1: Development of the Theory and Basic Physical Applications. Springer.
Goldstein, R. (2005). Incompleteness: The Proof and Paradox of Kurt Gödel. Norton.
Goodman, N. & Quine, W. V. O. (1947). Steps toward a constructive nominalism. The Journal of Symbolic Logic, 12, 105–122.
Hamming, R. W. (1980). The unreasonable effectiveness of mathematics. The American Mathematical Monthly, 87, 81–90.
Hilbert, D. (1967). The foundations of mathematics. In J. van Heijenoort (Ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (pp. 464–479). Harvard University Press. (From a lecture given by Hilbert in 1927).
Hunter, G. (1971). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Press.
Kripke, S. A. (1959). A completeness theorem in modal logic. The Journal of Symbolic Logic, 24, 1–14. http://naturalthinker.net/trl/texts/Kripke,Saul/Kripke%20S.%20-%20A%20Completeness%20Theorem%20in%20Modal%20Logic.pdf
Lewis, C. I. (1917). The issues concerning material implication. The Journal of Philosophy, Psychology and Scientific Methods, 14, 350–356. https://www.jstor.org/stable/2940255
Lewis, D. (1986). On the Plurality of Worlds. Oxford: Blackwell.
Linsky, B. & Zalta, E. N. (2006). What is Neologicism? The Bulletin of Symbolic Logic, 12, 60–99. http://mally.stanford.edu/Papers/neologicism2.pdf
MacFarlane, J. (2021). Philosophical Logic: A contemporary introduction. Routledge.
Martínez-Ordaz, M. R. (2021). The ignorance behind inconsistency toleration. Synthese, 198, 8665–8686. http://philsci-archive.pitt.edu/19672/
Mill, J. S. (1877). An Examination of Sir William Hamilton’s Philosophy, vol II. New York: Henry Holt and Co.
Open Logic Project. (2020). The Open Logic Text. https://openlogicproject.org/
Post, E. L. (1921). Introduction to the general theory of elementary propositions. American Journal of Mathematics, 43, 163–185.
Priest, G. (1998). What is so bad about contradictions? Journal of Philosophy, 95, 410–426.
Putnam, H. (1971). Philosophy of Logic. New York: Harper & Row.
———. (1975a). Mathematics, Matter, and Method. Cambridge University Press.
———. (1975b). What is mathematical truth? Historia Mathematica, 2, 529–543.
Quine, W. V. O. (1948). On what there is. Review of Metaphysics, 2, 21–38.
———. (1986). Philosophy of Logic (2nd ed.). Harvard University Press.
Raatikainen, P. (2020). Gödel’s incompleteness theorems. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/goedel-incompleteness/
Reid, C. (1996). Hilbert. Copernicus.
Roberts, D. A. (2021). Why is AI hard and physics simple? https://arxiv.org/abs/2104.00008
Russell, B. (1905). On Denoting. Mind, 14, 479–493.
———. (1912). The Problems of Philosophy. Oxford University Press. (with Introduction by John Perry 1997).
Russell, G. (2015). The justification of the basic laws of logic. Journal of Philosophical Logic, 44, 793–803.
Shapiro, S. (1991). Foundations without Foundationalism: A Case for Second-Order Logic. Oxford University Press.
———. (2000). Thinking about Mathematics: The Philosophy of Mathematics. Oxford University Press.
Smith, P. (2020). An Introduction to Formal Logic (2nd ed.). Cambridge University Press.
———. (2022). Beginning Mathematical Logic: A Study Guide. https://www.logicmatters.net/resources/pdfs/LogicStudyGuide.pdf
Snapper, E. (1979). The three crises in mathematics: Logicism, intuitionism and formalism. Mathematics Magazine, 52, 207–216. https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1980/0025570x.di021111.02p0048m.pdf
Spivak, D. I. (2013). Category theory for scientists. https://arxiv.org/abs/1302.6946
Stillwell, J. (2004). Emil Post and his anticipation of Gödel and Turing. Mathematics Magazine, 77, 3–14.
Sussman, G. J. & Wisdom, J. (2013). Functional Differential Geometry. MIT Press. https://mitpress.mit.edu/books/functional-differential-geometry
Tait, W. W. (1983). Against intuitionism: Constructive mathematics is part of classical mathematics. Journal of Philosophical Logic, 12, 173–195.
Tao, T. (2007). Differential forms and integration. https://www.math.ucla.edu/~tao/preprints/forms.pdf
Tegmark, M. (2008). The mathematical universe. Foundations of Physics, 38, 101–150. https://arxiv.org/abs/0704.0646
Teller, P. (1989). A Modern Formal Logic Primer. Prentice Hall. https://tellerprimer.ucdavis.edu/
Varadarajan, V. S. (2003). Vector bundles and connections in physics and mathematics: some historical remarks. In A Tribute to CS Seshadri (pp. 502–541). Gurgaon: Hindustan Book Agency.
Viteri, S. & DeDeo, S. (2022). Epistemic phase transitions in mathematical proofs. Cognition, 225, 105120. https://www.sciencedirect.com/science/article/pii/S0010027722001081
von Fintel, K. (2011). Conditionals. https://dspace.mit.edu/handle/1721.1/95781
Weiss, W. & D’Mello, C. (2015). Fundamentals of Model Theory. https://www.math.toronto.edu/weiss/model_theory.pdf
Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13, 1–14. Richard courant lecture in mathematical sciences delivered at New York University, May 11, 1959. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

  1. Carter (2009).↩︎

  2. Baez (2002).↩︎

  3. Frè (2013), ch. 2.↩︎

  4. Varadarajan (2003).↩︎

  5. Tao (2007).↩︎

  6. Sussman & Wisdom (2013).↩︎

  7. Bronstein, Bruna, Cohen, & Velickovic (2021), p. 56–60.↩︎

  8. Hunter (1971).↩︎

  9. Teller (1989).↩︎

  10. Bonevac (2003).↩︎

  11. MacFarlane (2021).↩︎

  12. Smith (2020).↩︎

  13. Smith (2022).↩︎

  14. Carnap (1958).↩︎

  15. Open Logic Project (2020).↩︎

  16. Post (1921).↩︎

  17. Stillwell (2004).↩︎

  18. von Fintel (2011).↩︎

  19. Ewald (2018).↩︎

  20. Enderton (2009).↩︎

  21. Shapiro (1991).↩︎

  22. G. Russell (2015).↩︎

  23. C. I. Lewis (1917).↩︎

  24. Kripke (1959).↩︎

  25. D. Lewis (1986).↩︎

  26. Priest (1998).↩︎

  27. Martínez-Ordaz (2021).↩︎

  28. Quine (1986).↩︎

  29. Viteri & DeDeo (2022).↩︎

  30. Hunter (1971), p. 6.↩︎

  31. Button & Walsh (2018).↩︎

  32. Weiss & D’Mello (2015).↩︎

  33. Goldstein (2005).↩︎

  34. Raatikainen (2020).↩︎

  35. Atiyah, M. et al. (1994), p. 182.↩︎

  36. Franzén (2005).↩︎

  37. Cubitt, Perez-Garcia, & Wolf (2015).↩︎

  38. Spivak (2013).↩︎

  39. Fong & Spivak (2018).↩︎

  40. TODO↩︎

  41. Baez & Dolan (1995).↩︎

  42. Baez & Stay (2009).↩︎

  43. Yanofsky (2016) In Aguirre, Foster, & Merali (2016), p. 153.↩︎

  44. Ecclesiastes 1:9–10 (NRSV).↩︎

  45. Plato, Republic VI 510c, Cooper & Hutchinson (1997), p. 1131.↩︎

  46. Quine (1948).↩︎

  47. Quine (1948), p. 7.↩︎

  48. Quine (1948), p. 9.↩︎

  49. Tegmark (2008).↩︎

  50. Chen (2018).↩︎

  51. Bayes (1736), p. 9–10.↩︎

  52. Mill (1877).↩︎

  53. Goodman & Quine (1947).↩︎

  54. Azzouni (2015).↩︎

  55. Field (1989), p. 68. TODO: Get exact quote.↩︎

  56. Field (2016).↩︎

  57. Burgess (1983).↩︎

  58. Bueno (2013).↩︎

  59. Goodman & Quine (1947), p. 105.↩︎

  60. B. Russell (1905).↩︎

  61. B. Russell (1912), p. 91–118.↩︎

  62. Snapper (1979).↩︎

  63. Shapiro (2000).↩︎

  64. Linsky & Zalta (2006).↩︎

  65. Linsky & Zalta (2006), p. 43.↩︎

  66. Hilbert (1967), p. 479. Discussion by Michael J. O’Donnell:

    Well, the tone sounds like hubris again, but this is actually a relatively modest claim, and largely substantiated by events. Mathematics may arguably be understood as the science of forms, which have an objective quality independent of individual beliefs. What is missing here is the claim that a single formal system may embrace all of mathematics at once, and that we may prove that it contains no error. Notice that Hilbert objects to the “actual, contentual assumptions” of Russell and Whitehead. But, recall that Hilbert claims to treat numerical equations contentually, and even seems to regard that as a virtue. It is not the mere contentual quality of Russell’s and Whitehead’s assumptions that Hilbert objects to, but the fact that each particular assumptions is not verifiable by a single computation or finite observation, and furthermore that there is not even a proof that they are consistent with basic numerical equations. Of course, Hilbert’s proposed system never gets its consistency proof either. Oh well.

    ↩︎
  67. Brouwer (1908).↩︎

  68. Reid (1996), p. 149.↩︎

  69. Tait (1983).↩︎

  70. Drake (1957), p.237-8.↩︎

  71. Wigner (1960).↩︎

  72. Putnam (1975b).↩︎

  73. Hamming (1980).↩︎

  74. Roberts (2021).↩︎

  75. Putnam (1975a), p. TODO.↩︎

  76. Putnam (1971), p. 57.↩︎