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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So a natural number is any referent of an abstraction nx.Fx<br />
that can be ‘reached’ from 0H by follow<strong>in</strong>g <strong>the</strong> relation<br />
PreH. As did Frege, Neo-Fregeans go on to def<strong>in</strong>e<br />
<strong>in</strong>dividual number terms:<br />
110<br />
0H means nx.(x ≠ x)<br />
1H means nx.(x = 0H)<br />
2H means nx.(x = 0H ∨ x = 1H) …<br />
From <strong>the</strong>se def<strong>in</strong>itions we can derive all of Second order<br />
Peano Arithmetic, which completely characterises a<br />
natural number structure.<br />
3 An alternative abstraction pr<strong>in</strong>ciple<br />
3.1 Benacerraf’s pr<strong>in</strong>ciple<br />
Now I turn to <strong>the</strong> argument that <strong>the</strong> Neo-Fregean ontology<br />
conta<strong>in</strong>s too many abstract objects. I do this by present<strong>in</strong>g<br />
an abstraction pr<strong>in</strong>ciple that is similar to Hume’s pr<strong>in</strong>ciple.<br />
This alternative abstraction pr<strong>in</strong>ciple can do <strong>the</strong> same work<br />
as Hume’s pr<strong>in</strong>ciple <strong>and</strong> <strong>in</strong> a similar way. But, as with <strong>the</strong><br />
(1) <strong>and</strong> (2) above, <strong>the</strong> two abstraction pr<strong>in</strong>ciples yield two<br />
dist<strong>in</strong>ct sequences of abstract objects. As was argued <strong>in</strong><br />
<strong>the</strong> case of sets, I shall argue that <strong>the</strong>re is noth<strong>in</strong>g to<br />
decide which abstraction pr<strong>in</strong>ciple yields <strong>the</strong> ‘true’ natural<br />
numbers.<br />
The new pr<strong>in</strong>ciple is simpler than Hume’s pr<strong>in</strong>ciple,<br />
call it Benacerraf’s Pr<strong>in</strong>ciple:<br />
<strong>the</strong> unitar<strong>in</strong>ess of F = <strong>the</strong> unitar<strong>in</strong>ess of G iff<br />
nei<strong>the</strong>r F nor G are s<strong>in</strong>gletons, or<br />
F <strong>and</strong> G have <strong>the</strong> same extension.<br />
Say that F is unitary if it has exactly one element <strong>in</strong> its<br />
extension. Then Benacerraf’s pr<strong>in</strong>ciple identifies, for each<br />
predicate F, an object that is <strong>the</strong> unitar<strong>in</strong>ess of F. We can<br />
write ‘<strong>the</strong> unitar<strong>in</strong>ess of F’ as ux.Fx, <strong>and</strong> <strong>the</strong>n Benacerraf’s<br />
pr<strong>in</strong>ciple is:<br />
Benacerraf <strong>and</strong> Bad Company (An Attack on Neo-Fregeanism) — Michael Gabbay<br />
∀F∀G [ ux.Fx = ux.Gx ↔<br />
((¬∃!xFx ∧ ¬∃!xGx) ∨ ∀x(Fx ↔ Gx)) ] (10)<br />
Call a property, or concept, unitary when only one th<strong>in</strong>g is<br />
<strong>in</strong> its extension. Then Benacerraf’s pr<strong>in</strong>ciple is to be taken<br />
as a def<strong>in</strong>ition, <strong>in</strong> terms of be<strong>in</strong>g unitary, of <strong>the</strong> b<strong>in</strong>d<strong>in</strong>g<br />
operator ux.(…). The <strong>in</strong>tuition for unitarieness is that one<br />
can abstract out of a unitary property its ‘unitar<strong>in</strong>ess’, or<br />
<strong>the</strong> way <strong>in</strong> which it is unitary. Any non-unitary properties<br />
are unitary <strong>in</strong> <strong>the</strong> same way: <strong>the</strong>y are not. The way unitary<br />
properties are differentiated, <strong>in</strong> <strong>the</strong> spirit of Frege’s Basic<br />
Law V (see (14)) is through <strong>the</strong>ir extension.<br />
Unitar<strong>in</strong>ess is at least as fundamental to our concept<br />
of number as one-one correspondences. After all, a oneone<br />
correspondence is a correspondence between unit<br />
objects; when we count, we count unit <strong>in</strong>dividuals;<br />
variables of first order quantifiers range over unit entities;<br />
<strong>the</strong> symbols of <strong>the</strong> language necessary to express even<br />
basic propositions are discrete, discernable units. Without<br />
<strong>the</strong> concept of a unit, a discrete th<strong>in</strong>g, a s<strong>in</strong>gle entity, we<br />
cannot even beg<strong>in</strong> a logical enquiry let alone ground<br />
arithmetic <strong>in</strong> one-one correspondences. Frege himself<br />
discusses <strong>and</strong> rejects <strong>the</strong> possibility of develop<strong>in</strong>g a <strong>the</strong>ory<br />
of arithmetic based on units. But his compell<strong>in</strong>g refutations<br />
are aimed at <strong>the</strong>ories of numbers as agglomerations or<br />
sums of (dist<strong>in</strong>ct) units [Frege 1953, §29-§44]. Frege<br />
objects that such accounts ei<strong>the</strong>r make no sense, or fail to<br />
generate arithmetic. He did not consider <strong>the</strong> possibility that<br />
<strong>the</strong> unit, thought of as a property of properties, <strong>and</strong> derived<br />
(8)<br />
(9)<br />
by a similar abstraction method to Frege’s own, could do<br />
<strong>the</strong> same work as his favoured <strong>the</strong>ory of number.<br />
3.2 An analogue of Frege’s <strong>the</strong>orem<br />
I now sketch how Benacerraf’s pr<strong>in</strong>ciple can be used to<br />
def<strong>in</strong>e <strong>the</strong> numbers along Neo-Fregean l<strong>in</strong>es. To<br />
dist<strong>in</strong>guish <strong>the</strong> def<strong>in</strong>ed terms of this section with those of<br />
Section 2.2 I subscript <strong>the</strong>m with B for ‘Benacerraf’. We<br />
beg<strong>in</strong> with zero:<br />
0B means ux.(x ≠ x)<br />
1B means ux.(x = 0B)<br />
2B means ux.(x = 1B) …<br />
(11)<br />
It is not hard to show that <strong>the</strong> iB are derivably dist<strong>in</strong>ct. For<br />
example suppose that 0B = 1B, <strong>the</strong>n ux.(x ≠ x) = ux.(x = 0B).<br />
So by Benacerraf’s pr<strong>in</strong>ciple ei<strong>the</strong>r ¬∃!x(x ≠ x) ∧<br />
¬∃!x(x ≠ 1B) or ∀x(x = 0B ↔ x ≠ x). Each of <strong>the</strong>se is<br />
derivably false <strong>in</strong> even first order logic.<br />
We can go on to def<strong>in</strong>e <strong>the</strong> predecessor relation as<br />
follows:<br />
PreB(t, t') means ∃F[t = ux.Fx ∧ t' = ux.(x = t)] (12)<br />
A version of Frege’s <strong>the</strong>orem now arises out of adopt<strong>in</strong>g<br />
Benacerraf’s pr<strong>in</strong>ciple ra<strong>the</strong>r than Hume’s pr<strong>in</strong>ciple. We<br />
use (6) to def<strong>in</strong>e <strong>the</strong> natural numbers to be exactly <strong>the</strong><br />
entities <strong>in</strong> <strong>the</strong> transitive closure of PreB . This yields <strong>the</strong><br />
second order Peano axioms.<br />
*<br />
NatB(t) means PreB (0B, t') (13)<br />
I omit <strong>the</strong> rema<strong>in</strong><strong>in</strong>g details here as <strong>the</strong>y are almost<br />
identical to those of <strong>the</strong> proof of Frege’s <strong>the</strong>orem <strong>in</strong> [Wright<br />
1983].<br />
3.3 The attack on Neo-Fregeanism<br />
Let 0H, 1H, 0H… denote <strong>the</strong> entities abstracted <strong>and</strong> def<strong>in</strong>ed<br />
us<strong>in</strong>g Hume’s Pr<strong>in</strong>ciple, call <strong>the</strong>m <strong>the</strong> Hume-numbers. Let<br />
0B, 1B, 0B… be <strong>the</strong> entities abstracted <strong>and</strong> def<strong>in</strong>ed us<strong>in</strong>g<br />
Benacerraf’s pr<strong>in</strong>ciple call <strong>the</strong>m <strong>the</strong> Benacerraf-numbers. It<br />
should be clear that an analogue of Benacerraf’s orig<strong>in</strong>al<br />
challenge arises. Benacerraf’s orig<strong>in</strong>al argument now<br />
applies, both <strong>the</strong> Hume-numbers <strong>and</strong> <strong>the</strong> Benacerrafnumbers<br />
serve as characterisations of <strong>the</strong> natural<br />
numbers. Fur<strong>the</strong>rmore <strong>the</strong>re is no reason for <strong>the</strong> natural<br />
numbers to be identified with <strong>the</strong> Hume-numbers ra<strong>the</strong>r<br />
than <strong>the</strong> Benacerraf-numbers. Therefore Neo-Fregeanism<br />
is to be rejected alongside set <strong>the</strong>oretic reductionism by a<br />
variant of Benacerraf’s orig<strong>in</strong>al argument.<br />
The penultimate claim, that <strong>the</strong>re is no choos<strong>in</strong>g<br />
between <strong>the</strong> Benacerraf <strong>and</strong> <strong>the</strong> Hume numbers, is <strong>in</strong><br />
need of justification. I sketch a justification of it <strong>in</strong> Section 4<br />
by compar<strong>in</strong>g <strong>the</strong> systems obta<strong>in</strong>ed from <strong>the</strong> two<br />
abstraction pr<strong>in</strong>ciples <strong>and</strong> show<strong>in</strong>g that <strong>the</strong>re is little that<br />
can be done with Hume’s pr<strong>in</strong>ciple that cannot also be<br />
done with Benacerraf’s pr<strong>in</strong>ciple.<br />
4 A comparison of two abstraction<br />
pr<strong>in</strong>ciples<br />
4.1 How to avoid bad company<br />
Formally, Hume’s <strong>and</strong> Benacerraf’s pr<strong>in</strong>ciples are<br />
acceptable abstraction pr<strong>in</strong>ciples. I argue for this here by<br />
present<strong>in</strong>g a condition on good abstraction pr<strong>in</strong>ciples (i.e. I