Abstract
In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the third one: why arithmetical knowledge appears to be necessary. A Kripkean analysis of necessity is used as an example to show that a proper analysis of the relevant possible worlds can explain arithmetical necessity in a sufficiently strong form.
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Notes
- 1.
It is not possible here to enter the discussion of what—if anything—constitutes a priori knowledge. In this paper, I will accept the possibility of a priori knowledge and follow the classical understanding that it is knowledge that can be gained in an essentially non-empirical manner.
- 2.
For Kant, there was of course also the important distinction between synthetic and analytic knowledge. In this paper that distinction, namely the possible differences between arithmetical knowledge being analytic a priori or synthetic a priori, will not be considered.
- 3.
The empirical details and references can be found in Pantsar (2014). For an introduction to the empirical studies, see Dehaene (2011) and Dehaene and Brannon (eds) (2011). In the latter, the paper by Nieder (2011) is particularly recommended as a good window to the state of the art in the empirical research of proto-arithmetical cognition.
- 4.
Hence at the level of proto-arithmetic, it is better to talk of numerosities rather than numbers. This distinction is too rarely made in the empirical literature. Unfortunately, it is also common in the empirical literature to postulate needlessly strong abilities to the test subjects. For example, in one of the most famous papers, Wynn (1992), it is shown that infants react to unnatural numerosities in experimental settings. When they see one and one dolls put behind a screen but only one doll when the screen is removed, they are surprised. This was an important result and further experiments have shown the proto-arithmetical ability involved in the process to be a particularly important one. The unnatural numerosity actually surprises infants more than changes in the position, sizes, or even the character of the doll. Nevertheless, when Wynn called her paper “Addition and subtraction by human infants”, she seems to postulate needlessly strong ability to her test subject. The infants could simply have the ability to hold one numerosity in their working memory and be surprised when the observations don’t match that.
- 5.
My epistemological theory is related to, but different from, other accounts emphasizing the empirical foundations of arithmetic, e.g. Kitcher (1983), Lakoff and Núñez (2000) and Jenkins (2008). In all the accounts perhaps the key question is how we can combine the position that arithmetical concepts are empirically grounded with the image of arithmetic as an a priori pursuit. The solutions range from rejecting the apriority (Kitcher) to accepting that arithmetical knowledge is essentially both empirical and a priori (Jenkins). One key feature of my account is to soften the notion of a priori applicable to arithmetic. However, it should be noted that the account of contextual a priori is fundamentally different from the ones by Kuhn (1993) and Putnam (1976). For me, arithmetic is not simply a paradigm that is a priori in one context but could be overthrown when that context changes. Rather, arithmetical knowledge is based on biological primitives and as such it is based on an effectively inevitable way of experiencing the world.
- 6.
Kripke (1980) p. 35 states that in contemporary (meaning around the year 1970) discussion very few philosophers even made the Kantian distinction between a priori and necessary.
- 7.
Here I am ignoring what I see as trivial empirical elements, like the fact that we need to see or hear (or feel) an explanation of the words involved in order to see the truth of sentences like “all bachelors are unmarried”.
- 8.
By “sufficiently developed”, I mean sufficiently developed along the lines that evolution took in our actual world. I do not wish to engage here in speculation about highly developed biological organisms with completely different cognitive systems.
- 9.
The neo-Fregeanism of Wright (1983) comes perhaps closest to fulfilling the logicist idea but in addition to using second-order logic, which is sometimes seen as going against the logicist ideal, he uses a non-logical axiom—the so-called Hume’s principle—in deriving the axioms of arithmetic. Yablo’s own solution is to interpret universal quantifiers in arithmetic as infinite chains of conjunctions.
- 10.
It should be noted that Kripke also introduces (1980, p. 28) the term “strong rigid designation” for terms referring to necessary existents, presumably including numbers. It is unclear to me whether there are conceivable cases of strong rigid designation, but it should be clear by now that natural numbers should not be considered to be such.
- 11.
This research was funded by the Academy of Finland, whose support is acknowledged with great gratitude. A big thank you is in place to the participants of the FilMat conference Philosophy of mathematics: objectivity, cognition, and proof in Milan. The discussions there helped formulate the arguments of this paper. The final version of this paper was written during a visit to the University of California, Irvine. I am grateful for the discussions with the staff there, especially with Sean Walsh, Penelope Maddy and Kai Wehmeier. Finally, with gratitude I note that the paper as it appears here benefited greatly from the thorough referee reports of two anonymous referees.
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Pantsar, M. (2016). The Modal Status of Contextually A Priori Arithmetical Truths. In: Boccuni, F., Sereni, A. (eds) Objectivity, Realism, and Proof . Boston Studies in the Philosophy and History of Science, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-31644-4_5
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