Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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imperatives, or comm<strong>and</strong>s <strong>in</strong> <strong>the</strong> form of admonish<strong>in</strong>g<br />
adherence to rules that we assent to follow. The <strong>in</strong>tuitionist<br />
Dummett, whose position Wittgenste<strong>in</strong>’s resembles, refers<br />
to ma<strong>the</strong>matical statements as quasi-assertions:<br />
Quasi-assertions are declarative sentences which<br />
are not associated with determ<strong>in</strong>ate conditions of<br />
truth <strong>and</strong> falsity but share with assertions properly<br />
so-called <strong>the</strong> feature that <strong>the</strong>re is such a th<strong>in</strong>g as<br />
assent<strong>in</strong>g to <strong>the</strong>m; where such assent is<br />
communally understood as a commitment to some<br />
def<strong>in</strong>ite type of l<strong>in</strong>guistic or non-l<strong>in</strong>guistic conduct,<br />
<strong>and</strong> receives explicit expression precisely by <strong>the</strong><br />
mak<strong>in</strong>g of <strong>the</strong> quasi-assertion. 13<br />
The subtle aspect of underst<strong>and</strong><strong>in</strong>g <strong>the</strong> dist<strong>in</strong>ction between<br />
ma<strong>the</strong>matical statements as <strong>in</strong> pr<strong>in</strong>ciple verifiable aga<strong>in</strong>st<br />
an objective reality, versus hav<strong>in</strong>g <strong>the</strong> character of be<strong>in</strong>g<br />
ratified by voluntarily acceptance, is that although we seek<br />
to preserve some sense of non-arbitrary structure, we<br />
must locate its apparent “necessity” <strong>in</strong> our discretionary<br />
compliance ra<strong>the</strong>r than <strong>in</strong> some facet of extra-mental<br />
reality. This necessity has <strong>the</strong> form of will<strong>in</strong>gly b<strong>in</strong>d<strong>in</strong>g<br />
ourselves to a normative correctness that we enact <strong>in</strong> our<br />
practice. Hence we have <strong>the</strong> sufficient leverage to not only<br />
ask “[o]f someone who is tra<strong>in</strong>ed [<strong>in</strong> a specific type of rulefollow<strong>in</strong>g]<br />
‘How will he <strong>in</strong>terpret <strong>the</strong> rule <strong>in</strong> this case?’”, but<br />
fur<strong>the</strong>r to raise <strong>the</strong> question, “How ought he to <strong>in</strong>terpret <strong>the</strong><br />
rule for this case”? 14<br />
This view of ma<strong>the</strong>matics as hav<strong>in</strong>g a humanly<br />
devised comm<strong>and</strong> structure <strong>in</strong>stead of a structure <strong>in</strong>sured<br />
by objective reality alters our picture of <strong>the</strong> type of<br />
normative guidance underly<strong>in</strong>g ma<strong>the</strong>matical judgment.<br />
Instead of be<strong>in</strong>g guided <strong>in</strong> mak<strong>in</strong>g ma<strong>the</strong>matical<br />
statements by facts, we consider that “all ma<strong>the</strong>matical<br />
propositions [are] expressed <strong>in</strong> <strong>the</strong> imperative, e.g., ‘Let 10<br />
x 10 be 100.’” 15 The significance is that this depiction of<br />
ma<strong>the</strong>matics makes <strong>the</strong> consistency of its structure<br />
dependent on our voluntary commitment to uphold<br />
conceptual relations <strong>in</strong> specific ways:<br />
Such an account is exactly what we should<br />
<strong>in</strong>tuitively propose for sentences express<strong>in</strong>g <strong>the</strong><br />
mak<strong>in</strong>g of a promise. No one would ord<strong>in</strong>arily<br />
suppose that <strong>the</strong> use of sentences of <strong>the</strong> form, ‘I<br />
promise to …’ is best understood as <strong>the</strong> mak<strong>in</strong>g of a<br />
statement, true or false; though <strong>the</strong>ir be<strong>in</strong>g prefixed<br />
by ‘it is true that …’ is grammatical sense. 16<br />
The promissory quality, <strong>the</strong>n, of ma<strong>the</strong>matical normativity<br />
is that ma<strong>the</strong>matical rules suggest what we “ought to<br />
conclude,” <strong>and</strong> <strong>in</strong> participat<strong>in</strong>g <strong>in</strong> <strong>the</strong>se rule-follow<strong>in</strong>g<br />
exercises we accede to draw <strong>the</strong> conclusion implied by <strong>the</strong><br />
rule. It is not that some feature of an objective world of<br />
numbers <strong>in</strong>tercedes to form <strong>the</strong> basis of our judgment <strong>in</strong> a<br />
necessary fashion. Ra<strong>the</strong>r, <strong>in</strong> ma<strong>the</strong>matical rule-follow<strong>in</strong>g,<br />
we agree to abide by <strong>the</strong> rules as prefigur<strong>in</strong>g or<br />
comm<strong>and</strong><strong>in</strong>g our judgment. If we consider <strong>the</strong> role proofs<br />
play <strong>in</strong> ma<strong>the</strong>matics, “it marks not a discovery of certa<strong>in</strong><br />
objective liaisons between concepts, but someth<strong>in</strong>g more<br />
like a resolution on our part so to <strong>in</strong>volve <strong>the</strong>m <strong>in</strong> <strong>the</strong><br />
future.” 17<br />
13 Wright, Wittgenste<strong>in</strong>, 155.<br />
14 Ludwig Wittgenste<strong>in</strong>, Remarks on <strong>the</strong> Foundations of Ma<strong>the</strong>matics, ed. by<br />
G.H. von Wright, R. Rhees, <strong>and</strong> G.E.M. Anscombe, trans. By G.E.M. Anscombe<br />
(Cambridge, MA: MIT Press, 1996) (RFM), V-9, p. 267.<br />
15 Wittgenste<strong>in</strong>, RFM, 155.<br />
15 Ludwig Wittgenste<strong>in</strong>, RFM, V-17, p. 276.<br />
16 Wright, Wittgenste<strong>in</strong>, 157.<br />
17 Wright, Wittgenste<strong>in</strong>, 135.<br />
Wittgenste<strong>in</strong> on Count<strong>in</strong>g <strong>in</strong> Political Economy — Sonja M. Amadae<br />
If our underst<strong>and</strong><strong>in</strong>g of <strong>the</strong> normativity structur<strong>in</strong>g<br />
apparently necessary truths <strong>in</strong> ma<strong>the</strong>matics rests on our<br />
commitment to follow <strong>the</strong> rules of ma<strong>the</strong>matics, <strong>the</strong>n it is<br />
possible to see that <strong>the</strong> rule-follow<strong>in</strong>g nature of math is<br />
little different from o<strong>the</strong>r rule-follow<strong>in</strong>g <strong>in</strong>stitutions<br />
throughout our society. This opens <strong>the</strong> possibility of<br />
consider<strong>in</strong>g that social-norms that st<strong>and</strong> as a system of<br />
rules have as much sanctity as do <strong>the</strong> rules of<br />
ma<strong>the</strong>matics. Typically, social norms are regarded as<br />
subject to preference; ei<strong>the</strong>r an <strong>in</strong>dividual prefers to follow<br />
a social norm or not; if she chooses to follow a social<br />
norm, this is because she prefers to do so. However, <strong>in</strong> <strong>the</strong><br />
case of ma<strong>the</strong>matical judgment, preference is seldom<br />
<strong>in</strong>voked as a source of decision over <strong>the</strong> result of a<br />
calculation or proof.<br />
This recast<strong>in</strong>g of <strong>the</strong> foundation, as it were, of<br />
ma<strong>the</strong>matics from fact <strong>and</strong> objective truth to socially<br />
constructed <strong>and</strong> ratified laws suggests <strong>the</strong> possibility for<br />
draw<strong>in</strong>g a parallel between legal systems of rule-follow<strong>in</strong>g<br />
<strong>and</strong> ma<strong>the</strong>matical systems. In his essay, “The Groundless<br />
Normativity of Instrumental Rationality,” Donald Hub<strong>in</strong><br />
argues that neo-Humean <strong>in</strong>strumentalists “must engage <strong>in</strong><br />
<strong>the</strong> same ‘lower<strong>in</strong>g of expectations’ [of <strong>the</strong> source of<br />
normativity of <strong>in</strong>strumental rationality to <strong>the</strong> same level]<br />
that <strong>the</strong> legal positivist must.” 18 For Hub<strong>in</strong>, practical<br />
rationality, of which <strong>in</strong>strumentality is part, is not an<br />
objective matter. In mak<strong>in</strong>g his po<strong>in</strong>t, he draws on legal<br />
positivism’s retreat from natural law <strong>the</strong>ory, <strong>and</strong> draws on<br />
H.L.A. Hart to exp<strong>and</strong> on this view. 19 Hub<strong>in</strong> is mak<strong>in</strong>g <strong>the</strong><br />
po<strong>in</strong>t that even though a legal system provides a<br />
normative basis for action, it cannot ground its ultimate<br />
pr<strong>in</strong>ciples. I am rework<strong>in</strong>g Hub<strong>in</strong>’s parallel between<br />
positive law <strong>and</strong> <strong>in</strong>strumental reason to contrast a realist<br />
account of math with an alternative declarative<br />
underst<strong>and</strong><strong>in</strong>g. In an anti-realist ma<strong>the</strong>matics, <strong>the</strong> b<strong>in</strong>d<strong>in</strong>g<br />
quality of rules only holds <strong>in</strong>sofar as we assent to <strong>the</strong>m.<br />
It has traditionally been <strong>the</strong> case <strong>the</strong> social <strong>and</strong><br />
political normativity has been viewed as of a lesser<br />
pedigree than <strong>in</strong>strumental <strong>and</strong> ma<strong>the</strong>matical normativity<br />
<strong>in</strong>sofar as <strong>the</strong> former is conditional, <strong>and</strong> <strong>the</strong> latter is nonnegotiable.<br />
For example, Phillip Pettit provides an<br />
explanation for how social norms may be derived from<br />
<strong>in</strong>strumental agency as <strong>the</strong> former is conditional on<br />
<strong>in</strong>dividual rational self <strong>in</strong>terest. 20 In his Theory of Justice,<br />
John Rawls was widely criticized from with<strong>in</strong> rational<br />
choice <strong>the</strong>ory for plac<strong>in</strong>g action accord<strong>in</strong>g <strong>the</strong> “<strong>the</strong><br />
reasonable,” which <strong>in</strong>cluded <strong>the</strong> political <strong>the</strong>oretic concept<br />
of fair play, on par with agency conform<strong>in</strong>g to <strong>the</strong> dictates<br />
of expected utility <strong>the</strong>ory. 21 It was not automatically<br />
obvious from with<strong>in</strong> rational choice <strong>the</strong>ory that agents had<br />
a duty to uphold <strong>the</strong> rules of government if <strong>the</strong>y did not<br />
fur<strong>the</strong>r an agent’s ends <strong>in</strong> each <strong>and</strong> every circumstance of<br />
action. 22 Therefore, without some sanction<strong>in</strong>g device that<br />
alters payoffs, <strong>the</strong> rule of law does not <strong>in</strong> <strong>and</strong> of itself<br />
provide a reason for action that trumps agents’<br />
preferences over end states. Rawls concludes of his<br />
contrast<strong>in</strong>g approach to justice as fairness, “There is no<br />
thought of try<strong>in</strong>g to derive <strong>the</strong> content of justice with<strong>in</strong> a<br />
18 Donald Hub<strong>in</strong>, "The Groundless Normativity of Instrumental Rationality",<br />
The Journal of <strong>Philosophy</strong> 98:9(2001), 445-468, 466.<br />
19 Hub<strong>in</strong>, “Groundless Normativity,” 463.<br />
20 Philip Pettit, “Virtus normativa: Rational Choice Perspectives,” <strong>in</strong> his Rules,<br />
Reasons, <strong>and</strong> Norms (Oxford University Press, 2002), 308-343.<br />
21 John Rawls, A Theory of Justice (Harvard University Press, 1971); John<br />
Rawls, “Justice as Fairness: Political not Metaphysical,” <strong>Philosophy</strong> <strong>and</strong> Public<br />
Affairs, 14:3 (summer, 1985), 223-51.<br />
22 This is <strong>the</strong> problem David Gauthier faces <strong>in</strong> Morals by Agreement (Oxford<br />
University Press, 1985).<br />
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