Abstract
In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind’s approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege’s logicism. I also show that from a modern perspective, Frege’s criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege’s and Dedekind’s views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.
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Notes
There is a danger of equivocation in the talk about late 19th century logicism, because Dedekind also described his position as deriving from logic. However, I believe that only Frege’s brand of logicism corresponds to the way the term is usually understood in modern literature.
This is the stand Kitcher (1992), among others, takes and it has been contested by, e.g., Wilson (2010) and Tappenden (1995). As will be seen, I believe that Grundlagen is fundamentally a mathematical work. Nevertheless, there seems to be little doubt that Frege generally comes from a more philosophical background than Dedekind.
It should be noted that Frege (Grundlagen §85) does mention both Cantor’s infinity and ordinal numbers and applauds (§86) his general work on the subject. But, this is the extent of the discussion of those ideas in Grundlagen.
As in all quotations in this paper, the emphasis is in the original text.
Coined by Boolos (1998), although Frege already makes it explicit in Grundlagen that the principle comes from Hume.
For Cantor, this of course led to famous results concerning the cardinalities of infinite sets.
Here, Von Neumann and Zermelo differ, as will be seen. The approach given here is Zermelo’s.
See Burgess (1984) or Boolos (1990).
Not to mention the potentially problematic aspects of employing second-order logic, which does not enjoy the conceptual simplicity (e.g., completeness) first-order logic does. Although in Frege’s time this would not have been considered problematic, in the modern discussion, the difference between first- and second-order logic is a crucial issue.
As is the case with many interpretations of Dedekind, this is not universally accepted. Ferreirós (1999), for example, does not see Dedekind as a proponent of axiomatization. But, again, the research of Sieg and Schlimm (2005) seems to convincingly establish that Dedekind is indeed a key figure in the axiomatic tradition.
For finite numbers, the isomorphism is trivial and the least infinite ordinal ω is identified with the smallest infinite cardinal No. For larger transfinite ordinals, the matter becomes trickier, as between the cardinal of countable infinity No and the cardinal of the smallest uncountable infinity N1, there are (according to the continuum hypothesis) no infinite cardinals, yet there are uncountably many infinite ordinals.
The Dedekind-Peano axioms can be presented in various ways, but in Peano’s (1889) original work, the content of the nine axioms were as follows:
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1.
1 is a number.
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2.
For all numbers a, a = a.
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3.
For all numbers a and b, a = b if and only if b = a.
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4.
For all numbers a, b, and c: if a = b and b = c, then a = c.
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5.
For all numbers a and b: if a = b, then b is a number.
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6.
For all numbers a, a + 1 is a number.
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7.
For all numbers a and b, a = b if and only if a + 1 = b + 1.
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8.
For all numbers a, a + 1 = 1 is false.
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9.
If a set K is such that 0 belongs to K and for every number n: if n belongs to K, then n + 1 belongs to K, then every number belongs to K.
It should be noted that ‘a + 1’ does not mean addition, but the successor of 1, usually notated as S(a). The last axiom is the axiom of induction, and as we notice, in Peano’s original axiomatization it is a second-order sentence. It can also be presented as a first-order axiom schema.
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1.
For Dedekind, one important problem was the existence of infinite systems in general. His “proof” for this in Was sind und was sollen die Zahlen? is infamous in its reliance on the infinity of the number of thoughts. However, in modern philosophy of mathematics, such proofs of actual infinity are often not considered crucial for the position, as seen in, e.g., Hellman (1989). There are other ways of distinguishing the simply infinite system from other types of systems, ranging from the philosophical considerations of potential infinity to the Zermelo (1908) approach of taking the existence of infinite sets (or systems) as an axiom. This latter approach is, however, no doubt the kind of thing Russell (1919, p. 71) criticized as having the advantage of “theft over honest toil”—a criticism which was, incidentally, targeted against Dedekind’s construction of real numbers.
It should be noted that Dedekind’s influence was crucial in this development. Indeed, we will see that Dedekind can also be interpreted as a proponent of the set theoretical approach.
See, e.g., Wang (1957).
Of course, he (§133) also proves the crucial result that any two simply infinite systems are isomorphic.
Here, for the modern reader, it is perhaps easiest to understand Dedekind as stating that we can choose any of the models of arithmetic. Because Dedekind’s arithmetic is second-order, there are no differences between the models. The existence of non-standard models in first-order arithmetic makes this approach problematic in first-order approaches.
As well as Dedekind’s structuralism, there is some debate as to what kind of structuralist we should understand Dedekind to be, as seen in Reck (2003). In Reck’s analysis, we should take Dedekind’s notion of “free creation” seriously and thus advocate an interpretation that Dedekind is a logical structuralist rather than an ante rem one. While Reck’s arguments are in many parts persuasive, I still believe that Dedekind is best understood as not taking a stand on the metaphysical issue. Thus, I do not read Dedekind here as an ante rem structuralist who believes that there exist a universal platonic structure of numbers, but neither do I want to strictly deny that possibility.
This part of Dummett’s interpretation of Frege is rather controversial. Angelelli (1994), for example, criticizes Dummett’s extreme view of Frege as an enemy of abstraction. Although Angelelli’s criticism is justified, there is little doubt that Frege had a different—and more reluctant—attitude toward abstraction than Dedekind.
Instead of a series, which refers to a sum in modern mathematics, we would now talk of a sequence. But, in this paper, I will use the term “series” in its meaning for Frege and Dedekind.
This is meant to be as an analogy: of course, we are not putting the same pebble in the row many times, so there are other distinguishing characteristics in the pebbles.
Frege (Grundlagen §9) makes a similar point.
For more on the Mill-Whewell debate, see Gillies (1982, pp. 20–26).
At the stage of proto-arithmetical cognition, it is better to speak of “numerosities,” rather than natural numbers, in order to distinguish its primitive nature from developed arithmetical thinking.
This is a standard method in the study of infants. To put it simply, infants get bored when they see something that they expect. When they see something surprising, they look at it longer.
See Dehaene (2011), pp. 41–44. Although Wynn’s result seems valid, we should be careful about postulating needlessly developed cognitive capacities for the subjects. Her paper, for example, was called “Addition and subtraction by human infants.” But it seems quite problematic to assume that the infants are doing additions or subtractions in the process. It seems more likely that they are keeping one numerosity in their working memories. The infant and animal ability only applies to small quantities—usually no larger than four—and gets increasingly inaccurate as quantities become larger.
See Pantsar (2014) for a more detailed account of the empirical data and its relevance to the epistemology of arithmetic.
See Dehaene (2011), pp. 10–11.
As seen in the work of Lakoff and Núñez (2000). What they suggest is no doubt a possible story, in fact quite plausible, but we can hardly say with much confidence that this is how mathematics actually developed.
Obviously, I do not want to claim that all laws of thought are about quantities. Indeed, Frege’s laws of logic can retain their status in everything else, but in the Dedekindian approach suggested here, there is simply no need to derive numbers in terms of them.
See Pantsar (2014) for more on such contextual a priori characterization of arithmetical knowledge.
References
Agrillo, C., Dadda, M., Serena, G., & Bisazza, A. (2009). Use of number by fish. PloS One, 4(3), e4786.
Angelelli, I. (1994). Abstraction and number in Michael Dummett’s Frege: philosophy of mathematics. Modern Logic, 4, 308–318.
Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70, 661–679.
Boolos, G. (1990). The Standard of Equality of Numbers. In G. Boolos (ed.): Meaning and Method: Essays in Honor of Hilary Putnam , Cambridge: Cambridge University Press, pp. 261–277
Boolos, G. (1998). Logic, logic and logic. Cambridge: Harvard University Press, pp. 202–219
Cantor, G. (1878). Ein Beitrag zur Mannigfaltigkeitslehre. Journal für reine und angewandte Mathematik, 84, 242–258.
Corry, L. (1996). Modern algebra and the rise of mathematical structures. Basel: Birkhäuser.
Dedekind, R. (1887). Was sind und was sollen die Zahlen?, translated by W.W. Berman. In R. Dedekind (Ed.), Essays in the Theory of Numbers. New York: Dover.
Dedekind, R. (1890). Letter to Keferstein. In J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1937. New York: Harvard University Press, pp. 98–103.
Dehaene, S. (2011). The number sense—revised & expanded edition. New York: Oxford University Press.
Dehaene, S., & Brannon, E. (Eds.). (2011). Space, time and number in the brain. London: Academic Press.
Dummett, M. (1991). Frege: philosophy of mathematics. Cambridge: Harvard University Press.
Ferreirós, J. (1999). Labyrinth of thought—a history of set theory and its role in modern mathematics. Basel: Birkhäuser.
Frege, G. (1879). Begriffsschift. In van J. Heijenoort, (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1937. New York: Harvard University Press, pp. 5–82.
Frege, G. (1884). The foundations of arithmetic. Oxford: Basil Blackwell.
Frege, G. (1893). Grundgesetze der Arithmetik. Band I. Jena: H. Pohle.
Frege, G. (1903). Grundgesetze der Arithmetik. Band II. Jena: H. Pohle.
Frege, G. (1914). Logic in mathematics. In G. Frege (Ed.), Posthumous Writings, Oxford: Wiley-Blackwell, pp. 203–250.
Gillies, D. A. (1982). Frege, Dedekind, and Peano on the foundations of arithmetic. Assen, Netherlands: Van Gorcum.
Hale, B., & Wright, C. (2001). The reason’s proper study. Oxford: Clarendon Press.
Hellman, G. (1989). Mathematics without numbers. Oxford: Oxford University Press.
Hilbert, D. (1925). On the Infinite. In P. Becacerraf & H. Putnam (Eds.), Philosophy of mathematics (2nd ed.). Cambridge: Cambridge University Press.
Husserl, E. (1891). Philosophie der Arithmetik. Halle: Pfeffer.
Kitcher, P. (1983). The nature of mathematical knowledge. New York: Oxford University Press.
Kitcher, P. (1992). Frege’s epistemology, Philosophical Review 88, pp. 235–262.
Lakoff, G., & Núñez, R. (2000). Where mathematics comes from. New York: Basic Books.
Mill, J. S. (1843). A system of logic. In J. M. Robson (Ed.), Collected works of John Stuart Mill, vols. 7 & 8. Toronto: University of Toronto Press.
Nieder, A. (2011). The neural code for number. In Dehaene & Brannon (Eds.), Space time and number in the brain (pp. 107–122). London: Academic Press.
Pantsar, M. (2014). An empirically feasible approach to the epistemology of arithmetic. Synthese, 191, 4201–4229.
Peano, G. (1889). The principles of arithmetic, presented by a new method. In J. van Heijenoort, (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1937. New York: Harvard University Press, pp. 85–97.
Peano, G. (1891). Sul concetto di numero. In G. Peano (Ed.), Opere Scelte Vol. III. Rome: Cremonese, pp. 80–109.
Reck, E. (2003). Dedekind’s sructuralism: an interpretation and partial defense. Synthese, 137, 369–419.
Resnik, M. (1981). Mathematics as a science of patterns: ontology and reference. Noûs, 15, 529–550.
Russell, B. (1903). Principles of mathematics. New York: Norton.
Russell, B. (1919). Introduction to mathematical philosophy. New York: Allen & Unwin.
Schröder, E. (1873). Lehrbuch der Arithmetik und Algebra für Lehrer und Studirende I. Leipzig: Teubner.
Sieg, W., & Schlimm, D. (2005). Dedekind’s analysis of number: systems and axioms. Synthese, 147, 121–170.
Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. New York: Oxford University Press.
Tait, W. (1997). Frege versus cantor and Dedekind: on the concept of number. In W. Tait (Ed.), Early analytic philosophy: Frege (pp. 213–248). Russell, Wittgenstein, Chicago: Open Court.
Tappenden, J. (1995). Extending knowledge and “fruitful concepts”, Noûs 29, pp. 427–467.
von Neumann, J. (1923). An axiomatization of set theory. In J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1937. New York: Harvard University Press, pp. 346–354.
Wang, H. (1957). The axiomatization of arithmetic. Journal of Symbolic Logic, 22, 145–158.
Wilson, M. (2010). Frege’s mathematical setting. In T. Ricketts & M. Potter (Eds.), The Cambridge companion to Frege. Cambridge: University Press.
Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358, 749–751.
Zermelo, E. (1908). Investigations in the foundations of set theory. In J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1937. New York: Harvard University Press, pp. 199–215.
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Pantsar, M. Frege, Dedekind, and the Modern Epistemology of Arithmetic. Acta Anal 31, 297–318 (2016). https://doi.org/10.1007/s12136-015-0280-x
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DOI: https://doi.org/10.1007/s12136-015-0280-x