As if

1. ORIGINAL ARTICLE
‘As if’ Reasoning in Vaihinger and Pasch
Stephen Pollard
Received: 6 April 2009 / Accepted: 16 December 2009 / Published online: 22 January 2010
Ó Springer Science+Business Media B.V. 2010
Abstract Hans Vaihinger tried to explain how mathematical theories can be
useful without being true or even coherent, arguing that mathematicians employ a
special kind of fictional or ‘‘as if’’ reasoning that reliably extracts truths from
absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence
of core mathematical theories, but right about the utility of fictional discourse in
mathematics. This essay explores this area of agreement between Pasch and
Vaihinger. Pasch’s position raises questions about structuralist interpretations of
mathematics.
1 Introduction
According to one sober authority, Moritz Pasch’s Lectures on modern geometry
(Pasch 1882) inaugurated ‘‘a new epoch in humanity’s knowledge of its own
activity.’’ ‘‘People had only vague ideas about the structure of that most human of
all creations, geometry, until Pasch revealed a firm and well articulated founda-
tion.’’1
Here was a thinker to be reckoned with: the founder of a new epoch no less!
When retirement from the University of Giessen freed Pasch from all administrative
and teaching responsibilities, his zeal for foundational studies must have been
undiminished: he produced a remarkable body of work in which he provided axioms
for arithmetic, attempted to supply arithmetical and geometric axioms with content,
and explored the features essential to mathematical proofs.2
This essay will discuss
S. Pollard (&)
Department of Philosophy & Religion, Truman State University, Kirksville, MO 63501, USA
e-mail: spollard@truman.edu
1
My translation of Dehn (1934, pp. 130, 136). Max Dehn is not to be lightly dismissed: he had the
distinction of solving Hilbert’s third problem the very year Hilbert proposed it.
2
Engel (1934) is a good source of information about Pasch’s later years.
123
Erkenn (2010) 73:83–95
DOI 10.1007/s10670-009-9205-7

2. one theme from this late period: Pasch’s reaction to the mathematical fictionalism of
Hans Vaihinger. We begin with an overview of Vaihinger’s ‘‘as if’’ approach.
2 The ‘‘as if’’ Philosophy
‘‘The true and final purpose of thought,’’ says Vaihinger, ‘‘is action and the
facilitation of action.’’ ‘‘Our conceptual world lies between the sensory and motor
nerves … and serves merely to make the interconnection between them richer and
easier, more delicate and more serviceable.’’3
It is reasonable to employ any idea
that enhances this relationship between sensation and activity. Such an idea is ‘‘an
instrument … by means of which the higher organisms move about’’ (p. 65). Some
ideas perform this service even though we can recognize that nothing answering to
them could be empirically present to us. Such ideas are fictions and ‘‘the duty of a
logical theory of fictions’’ is to discover ‘‘the mechanism by means of which these
constructions perform their service’’ (p. 67).
So, for example, it would not be enough to report that a mathematical idea
surpassing the bounds of sense has played a useful role in the regulation of human
action. The ‘‘as if’’ philosopher is duty-bound to show how it has done so, to reveal
the mechanism by which it has done so. This seems a tall order because, according
to Vaihinger, the fundamental concepts of mathematics do not just violate the
conditions for a possible experience. They are logically incoherent. ‘‘All these
concepts are contradictory fictions, mathematics being based upon an entirely
imaginary foundation, indeed upon contradictions’’ (p. 51). Calculus is particularly
rife with absurdities.
We are very far, even to-day, from having resolved the contradictions
involved in the infinitesimal method. For two hundred years mathematicians,
together with the philosophers, endeavoured to show that there were no such
contradictions. We reverse the position, and insist from the opposite point of
view that these contradictions are not only undeniable but are the very means
by which advances have been made. (p. 61)
If our logic were classical, each inconsistent theory would characterize the same
structures as, say, the sentence ‘0 = 0’: namely none. But: ‘‘It is wrong to imagine
that only what is logically non-contradictory is logically fruitful’’ (p. 65).
Vaihinger is suggesting that the logic of mathematics is, to use today’s parlance,
paraconsistent. Classically, an inconsistent theory implies everything: a result that
would not help us ‘‘move about.’’ Vaihinger suggests, however, that the real logic of
mathematics is not classical. Somehow mathematical absurdities yield, not endless
garbage, but information of a quantity and quality essential to the improvement of
humanity’s behavior. We proceed as if some absurdity were true and, somehow, end
up with theorems that help us build better bridges and toasters. Fictionalists are
duty-bound to explain how this happens.
3
Vaihinger (1935, p. 66). I have relied entirely on C. K. Ogden’s admirable translation. Subsequent page
references in this section will be to this volume.
84 S. Pollard
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3. Vaihinger suggests that mathematicians employ a form of counterfactual, indeed
counterpossible, reasoning. A counterpossible conditional is a subjunctive condi-
tional whose antecedent contemplates something impossible.4
If, as one is naturally
inclined to believe, there are no situations in which an impossibility is true, then
there are no situations in which a counterpossible conditional has a true antecedent
and a false consequent. This would mean that every counterpossible conditional is
true. Yet one’s confidence in this conclusion may waver in the face of some
examples. Consider the following two conditionals.
1. If Fermat’s last theorem were false, then no argument for it would be sound.
2. If Fermat’s last theorem were false, then Andrew Wiles would deserve eternal
fame.
At first glance, conditional 1 seems correct while conditional 2 seems ridiculous.
If we decide that 2 is false, we may also decide that 1 is non-trivially true, that is,
true for some reason other than the impossibility of its antecedent. Vaihinger offers
the following examples (pp. 261–262).
3. If the circle were a polygon, then it would be subject to the laws of rectilinear
figures.
4. If the diamond were a metal, it would be soluble.
He insists that both conditionals are non-trivially true: true because of the
‘‘connection’’ between antecedent and consequent, not because of the impossibility
of the antecedent.
In a hypothetical connection, not only real and possible but also unreal and
impossible things can be introduced, because it is merely the connection
between the two presuppositions and not their actual reality that is being
expressed. (p. 261)
It is, presumably, the ability of mathematicians to recognize such connections that
allows them to pass from contradictory premises to conclusions that follow non-
trivially. They do not pass blindly from absurdity to absurdity.
Still, the non-triviality of conditional 3, does not make its consequent any less
absurd. If our mathematical reasoning is to help us ‘‘move about,’’ it must, at some
point, yield something coherent and, even better, true. According to Vaihinger, the
mechanism that fashions sense from nonsense is the method of ‘‘double’’ (p. 241) or
‘‘antithetic’’ (p. 109) error. ‘‘Thought proceeds to correct the error which it makes.
This simple statement contains the whole principle of fictions’’ (p. 61).
If, in fictions, thought contradicts reality, or if it even contradicts itself, and if
in spite of this questionable procedure it nevertheless succeeds in correspond-
ing to reality, then … this deviation must have been corrected and the
contradiction must have been made good. (p. 109)
‘‘The whole progress of thought rests entirely upon such antithetic operations or
errors’’ (p. 120). When we reason as if some absurdity were true, we first advance
4
For a taste of the literature on this topic, see Mares (1997).
‘As if’ reasoning in Vaihinger and Pasch 85
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4. the absurdity in the form of a ‘‘fictive judgment’’ (p. 261); we then pass selectively
from absurdity to absurdity through counterpossible reasoning; finally, we ‘‘cancel’’
the original absurdity by advancing a new, antithetic absurdity. Since ‘‘all methods
of thought find their purest expression in mathematics’’ (p. 109), Vaihinger offers
a mathematical example: Fermat’s technique for identifying maxima and minima
(pp. 113–115). To put it mildly, Vaihinger’s analysis is not successful. Since it
would be a grim exercise to enumerate Vaihinger’s missteps and since our main
interest here is Pasch’s response to Vaihinger’s fictionalism, we turn now to Pasch
and let him supply a critique.
3 The Fermat Example
In 1919, Vaihinger and Raymund Schmidt founded Annalen der Philosophie
(subtitle: mit besonderer Ru¨cksicht auf die Probleme der Als-Ob-Betrachtung; that
is, ‘‘with particular reference to the problems of the ‘As if’ approach’’).5
Since
Vaihinger believed that ‘‘philosophy can only advance in the closest co-operation
with the individual sciences,’’ he encouraged submissions from ‘‘eminent
representatives of the most important branches of science.’’ Contributors included
‘‘the theologian Heim, the lawyer Kru¨ckmann, the doctor Abderhalden, the
mathematician Pasch,’’ and other luminaries (Vaihinger 1935, p. xlviii). Indeed,
Pasch published eight papers in Vaihinger’s Annals (1921b, 1922a, b, c, 1924b,
1925a, b, 1926b), the fifth of which addressed Vaihinger’s analysis of Fermat and,
more generally, Vaihinger’s treatment of infinitesimal methods. Pasch described
this paper as ‘‘a response to Vaihinger’s idea that mathematicians often make
significant progress by using fictions that are logically untenable but, for that very
reason, useful’’ (1924b, p. 183).6
This idea, according to Pasch, is quite wrong.
Pasch takes a gentle approach, blaming Vaihinger’s misconceptions on the poor
standards of exposition prevalent in mathematics.
The ‘‘as if’’ philosophers have, naturally, gathered their impressions of
mathematics from mathematical papers and textbooks. Unfortunately, profes-
sional papers are aimed at mathematicians who can fill various gaps in the
exposition, while textbooks forsake complete rigor to accommodate less
proficient readers. We can only reach a reliable judgment about the logical
soundness of the delicate concepts involved if we have a rigorous analysis of
them that is as complete as possible. (1924b, p. 161)
Interpreters of Fermat face a special challenge. ‘‘Fermat offers a method of
calculation without explaining the underlying theory, though he does favor us with
some hints. Since he expresses himself in the imperfect terminology of his day, he
can easily be misunderstood.’’ (1924b, p. 187)
5
This journal was the precursor of Erkenntnis. For some of the history, see Hegselmann and Siegwart
(1991).
6
All translations of Pasch are by me.
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5. Faced with such a challenge, Vaihinger stumbles badly. ‘‘Vaihinger considers the
expression f(x) = x2
(a - x) where a is a positive constant and x is a variable whose
values range from 0 to a. The task is to determine a value of x where f(x) reaches a
maximum.’’ (1924b, p. 183) As Pasch noted, Fermat supplies a method, but does so
in a way that invites misunderstanding. ‘‘That Vaihinger has, indeed, misunderstood
… is evident from the following fact: in Vaihinger’s reconstruction of the argument,
the key assumption that f(x) reaches a maximum plays no role at all.’’ (1924b,
p. 187) Vaihinger’s flawed approach is not Fermat’s.
Pasch does not undertake a full analysis of Fermat’s ‘‘subtle and far-reaching’’
argument. ‘‘One thing is quite clear however: Fermat does not employ Vaihinger’s
method of advancing a palpable falsehood only to cancel it out with a second
falsehood.’’ (1924b, p. 187) Though Vaihinger offers Fermat’s rule for maxima and
minima as a prime example of the method of antithetic error, no application of that
method is to be found there. This is not the only case where fictionalist analyses
miscarry. After a patient overview of the foundations of differential calculus
(‘‘a favorite topic of the ‘as if’ school’’), Pasch concludes that, here too,
mathematicians have no need to cancel absurdities with absurdities: they manage
to steer clear of absurdities entirely.
4 Pliable and Rigid Mathematics
So far, this episode is not of monumental interest. Though a profound Kant scholar,
Vaihinger was no mathematician. It is not particularly surprising or illuminating to
learn that he had to be corrected by a professional. Of greater interest, and our real
focus, is Pasch’s respect for Vaihinger’s general outlook. Immediately after
demolishing Vaihinger’s analysis of Fermat, Pasch acknowledges our indebtedness
to Vaihinger. Now this might be an empty display of nice manners: a polite bow to a
well-regarded figure who helped supply an outlet for Pasch’s philosophical papers.
But Pasch makes clear elsewhere that he means for his responses to the ‘‘as if’’
school to be at least as constructive as they are polemical. He characterizes his very
first Annalen paper (1921b) as a contribution to Vaihinger’s program, not an attack
on it. Furthermore, on more than two dozen occasions in the papers of his late
period, Pasch attributes an ‘‘as if’’ method to mathematicians or employs such a
method himself. Pasch was aware of Vaihinger’s shortcomings as a mathematician
and logician, but he nonetheless credited Vaihinger with insights into mathematical
methodology. We want to identify those insights.
First, a note on Pasch’s broad conception of mathematics. Pasch was unusually
fond of dichotomies. Mathematics can be controversial or durable, delicate or
sturdy, settled or unsettled, pliable or rigid. Here are some characteristic passages.
Mathematics is a system with two parts that must be clearly distinguished.
The first, the properly mathematical part, is devoted entirely to deduction. The
second makes deduction possible by introducing and elucidating a series of
insights that are to serve as material for deduction. (1918, p. 228)
‘As if’ reasoning in Vaihinger and Pasch 87
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6. In the latter, more libertine, part of … mathematics … all are free to do as they
will. In the former part, every move is constrained by the iron laws of
deductive logic. I call these the pliable and the rigid parts of mathematics.
(1921b, p. 152)
The rigid part … proceeds by mathematical proof and mathematical definition.
In the proofs, new propositions, theorems, are derived from the core
propositions …. In the definitions, new concepts are introduced, derived in
each case from those already introduced. Such derived concepts are to be
distinguished from the core concepts … that are not mathematically definable.
(1918, pp. 228–229)
A Kernbegriff, a core concept, is indeed a Begriff: a thought content we can grasp,
not a flatus vocis. Mathematical indefinability does not imply meaninglessness.
Mathematicians doing work of a more ‘‘pliable’’ sort offer ‘‘material for deduction,’’
in the form of axioms, and do their best to supply that material with content. This is
just the sort of work for which Pasch was best suited: he was an axiomatizer of the
first rank, always pursuing a deeper understanding of the subject matter even while
insisting on the purely formal character of mathematical proof.
I mention this dichotomy, pliable vs. rigid, because Pasch identifies and, indeed,
practices one sort of ‘‘as if’’ reasoning in the ‘‘pliable’’ sector of mathematics and
another in the ‘‘rigid.’’ In his search for axioms that provide a ‘‘natural foundation’’
for geometry and arithmetic (1924c), Pasch deals openly in fictions: just so stories
meant to relate the mathematical ideas to sensory experiences. On the rigid side,
Pasch employs a kind of definition that, he insists, provides a ‘‘link to the doctrine of
as-if,’’ though it is somewhat obscure what this link might be. In this essay, I will
only discuss the just so stories. Pasch offered such tales as a prelude to both
geometry and number theory. I will focus on the geometric fables. First, however,
we briefly consider Pasch’s justifications for the distinctive methodology he labeled
‘‘empiricism.’’
5 Pasch’s Empiricism
Pasch endorsed an empiricist approach to both geometry and number theory (1917,
pp. 184–187; 1922a, pp. 188–192; b, p. 155; c, pp. 362–363; 1924a; 1926b, pp. 249–
250).7
He viewed geometry as ‘‘a branch of natural science’’ and ‘‘an empirical
science’’ (1922a, pp. 189–190). He believed in ‘‘the essential unity of geometry and
physics,’’ applauding Einstein’s pursuit of the ‘‘physical meaning of geometrical
propositions’’ as well as Einstein’s insistence that geometric concepts be associated
with procedures that would determine where those concepts apply (1922a, pp. 189–
190). Pasch was not a professional philosopher or epistemologist. He was, however,
a scientist and perhaps it was his faith in the fundamental unity of science, at least as
much as his allegiance to a philosophical theory, that inclined him in an empiricist
direction. In any case, the texts of Pasch’s later period offer little in the way of
7
For a discussion of the effect of Pasch’s empiricism on his geometric work, see Gandon (2005).
88 S. Pollard
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7. a priori justifications for his empiricism. Arguments of a more pragmatic stripe are
easier to identify, with Pasch reporting four respects in which his empiricism did or
could prove beneficial.
First, it supplied him with a kind of indispensability argument for the consistency
of arithmetic. Pasch’s pursuit of an empirical basis for number theory led him to a
system K of combinatorial principles (presented in 1919/1921a) whose consistency,
he was convinced, was unprovable in any system weaker than K (1924c, pp. 235–
236). Pasch recognized that K yields ‘‘arithmetic as a whole’’ (1924c, p. 239; 1919,
p. 20) and, indeed, we can show that K is at least as strong as Peano Arithmetic
(PA). Since Pasch was convinced that a consistency proof for arithmetic would
require all of K (1924c, p. 236), he would insist that the consistency of PA is
unprovable in any system weaker than PA. Though Pasch claims nonetheless to
have established the consistency of arithmetic (1926a, p. 167), he is careful to point
out that his argument does not take the form of a mathematical proof (1924c,
p. 238). Instead, he directs us to his empiricist analysis of combinatorial ideas, an
analysis in which he associates each idea with the possible experiences of a human-
like creature. This analysis, he says, will lead a careful and impartial judge ‘‘to
concede that none of these ideas can be eliminated from scientific or even everyday
thought’’ (1921a, p. 155). Furthermore, his analysis reveals that the core
propositions in which these ideas figure, the axioms of K, ‘‘express insights
essential not only for the construction of arithmetic and, indeed, mathematics
generally: they prove indispensable well beyond the bounds of the mathematical
sciences’’ (1919, p. 20). It does not follow that K is consistent. It does follow,
according to Pasch, that we have no choice but to accept the axioms of K and that
‘‘we cannot seriously entertain the question of whether K is consistent’’ (1924c,
p. 235). The idea seems to be that to do otherwise would entangle us in a paralyzing
skepticism about reason itself. Pasch emphasizes that it was his empiricist
reconstruction of combinatorics that allowed him to reach this important conclusion.
So his empiricism revealed a significant characteristic of arithmetic.
Second, an empiricist analysis ‘‘prepares the way for investigations into
mathematics meant to contribute to a general science of thought.’’
To this study belongs the question of the relation between mathematics and
experience. This relation is much more evident in geometry than in number
theory. In the latter field, you are likely to overlook it altogether unless you
undertake the most detailed analysis—an analysis that, however, inevitably
makes the relation manifest. (1921a, pp. 155–156)
Pasch does not claim to be an epistemologist or psychologist, but he is confident that
his empiricist reconstructions of mathematical theories will contribute to these
fields.
Third, such reconstructions help to improve mathematical exposition and
teaching (1918, p. 231). ‘‘A decisive consideration in my choice of core concepts
was my belief that an explanation of a mathematical concept should mirror, as far as
possible, the way the concept actually arose or could have arisen.’’ (1917, p. 190) It
is important to note here that Pasch is quite willing to settle for just so stories about
‘As if’ reasoning in Vaihinger and Pasch 89
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8. how a concept could have arisen. Max Dehn’s assessment seems correct. Discussing
Pasch’s analysis of the geometric notion of segment, he observes:
Pasch does not try to investigate the actual experiences that might have led to
the concept of a segment. That would be a kind anthropological inquiry.
Instead he uses a series of possible experiments to lead the reader to this
concept.8
When Einstein spins a yarn about someone using a pole to measure the height of a
cloud (Einstein 1920, p. 6), no one interprets this as a contribution to empirical
psychology or cultural anthropology. It is evident, however, that even the wildest of
Einstein’s fables help us better understand the physical meaning of the notions he is
discussing. This, too, was the purpose of Pasch’s thought experiments: to help
human creatures, so dependent on their sensory receptors, better understand
fundamental mathematical ideas.
Fourth, Pasch claimed that his empiricist inquiries also help us better understand
the applicability of mathematics in empirical research. Indeed, he claimed there is
no other way to develop an adequate account of applied mathematics (Pasch 1917,
pp. 185–186; 1922a, p. 188).
Methods are to be judged by their fruits. If Pasch’s empiricism yielded good
results, we should not be too troubled by the absence of a philosophical grounding.
6 Pasch’s Fables
We now consider some of the ‘‘as if’’ thinking Pasch employed in his ‘‘pliable’’ pre-
axiomatic work: the work where he attempted to link mathematical concepts to
sensory experiences. We focus on geometry: the science of bodily shape according
to Pasch (1922a, p. 192; b, p. 156).
When we adopt a geometric perspective, when we manipulate or imagine
ourselves manipulating physical bodies with a view to studying their shapes, we
sometimes find it convenient to make dubious assumptions about the bodies and to
attribute fanciful powers to ourselves. For example, from a geometric standpoint,
‘‘no noticeable change counts as no change at all’’ (1922b, p. 156). So if close
inspection indicates that a certain body retains its shape when handled and moved
about, we proceed as if the shape were unchanged, not worrying about the
molecules its surface may have gained or lost or about the imperceptible
deformations we may have caused.
As we move bodies about, we are able to bring them together and fasten them to
one another forming composite bodies.
If … I am to limit myself to what is actually feasible, then I cannot extend the
process of composition arbitrarily. In geometry, however, we are accustomed
to act as if we could compose arbitrarily many arbitrary objects one after
another. (1922b, p. 157)
8
My translation of Dehn (1934, p. 138).
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9. Our geometric axioms ‘‘determine what constructions are possible’’ (1922c, p. 363).
Those axioms will allow for the composition of any objects at all, since fretting
about practicalities would betray an un-geometric outlook. To take a particular case,
we allow for rods of arbitrary length.
… two rods AB and BC could always be composed in a way that yields another
rod, a lengthening of AB past B … We can repeat this procedure. Indeed, we
act as if there is no limit to our lengthening of a straight segment and we
assume this process will never carry us back to the original segment. (1922b,
pp. 175–176)
If any of Einstein’s imaginary physicists hope to construct a pole to measure the
height of a cloud, we geometers will say nothing to discourage them. Similarly:
We can ‘‘widen’’ a planar surface by adding on one or more planar surfaces.
We act as if there is no limit to our widening of a planar surface and we
assume this process will never carry us back to the original surface. (1922b,
p. 181)
Sometimes bodies stand in the way of our lengthenings and widenings, but ‘‘we act
as if such barriers can always be removed’’ (1922c, p. 363). We let the engineers
worry about the feasibility of our imaginary constructions, much as we let the
nutritionists worry about whether the bodies we construct are good to eat. Mass,
temperature, nutritional value: all are irrelevant to the science of shape.
It is important to emphasize that Pasch considers tall tales about idealized
constructors ‘‘pre-geometric’’ (1926b, p. 265) or even ‘‘pre-mathematical’’ (1926b,
p. 263).9
Once our axioms are in place, we can get down to the ‘‘genuinely
mathematical’’ work of deriving theorems, a task in which the content provided by
our fables plays no logically essential role. Indeed, ‘‘The most reliable way to check
a proof is to formalize it’’ (1926b, p. 265), a process ‘‘in which content words with
definite meanings are replaced by arbitrary symbols’’ (1926b, p. 263). This
guarantees that:
… the meaning of the content words will play no role in the mathematical
proof. The proof will rely only on the structural elements and, so, will have a
purely formal character. This formalist approach … I have called the lifeblood
of mathematics. (1926b, p. 263)
Still, lifeblood or not, we need to strike the right balance here. Pasch’s slogan might
be: formalism for mathematical proofs, empiricism for mathematical concepts.
Pasch considers it essential to supply mathematical theories with content intelligible
to real human creatures. To characterize this work as pre-mathematical is not to
denigrate it. (Quite the opposite, in fact.)
Pasch was clear about the role of his just so stories. He describes his ‘‘pliable’’
contributions to geometry as follows.
9
As we have seen, Pasch was not of one mind about this, elsewhere locating his tall tales within ‘‘pliable
mathematics.’’ Pliable mathematics would be a form of mathematics, not pre-mathematics. The issue is
terminological, however, not substantive.
‘As if’ reasoning in Vaihinger and Pasch 91
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10. I had to preface the genuinely mathematical work with discussions of the
content of geometry in order to connect the pure mathematical construction
with physical applications and to explain my renderings of the fundamental
facts. (1926b, pp. 264–265)
Again, geometric Kernbegriffe (core concepts) are thought contents we must grasp
if we are to understand the geometric Kernsa¨tze (axioms). Creatures such as we,
with sensitive surfaces and agile fingers, most easily manage this mental grasping
when told a tale of beings who experience and manipulate a world of shapes in
human-like ways. Or, at least, Pasch believed so. He believed, too, that such a tale
helps us understand how geometry is to be applied in the world we actually inhabit.
So our geometric fictions promise to help us ‘‘move about’’ in that world, mediating
‘‘between the sensory and motor nerves’’ just as Vaihinger said.
7 Pasch and Vaihinger
Vaihinger and Pasch agreed, then, that mathematicians fabricate fictions that help us
navigate and control our environments. This is a substantial instance where Pasch
would have credited Vaihinger with insight into mathematical methodology. On the
other hand, there are major points of disagreement.
Vaihinger was convinced that core mathematical theories are not only false, but
logically false. Pasch dissented. He claimed to have shown that we cannot seriously
question the consistency of arithmetic without succumbing to a debilitating
skepticism. At one time, he thought his argument extended to the whole classical
theory of real numbers (1921a, p. 155) and, hence, to geometry (1919, p. 18). Later
he had doubts (1924c, p. 234; 1925a, p. 364); but these doubts about the reach of his
argument never caused him to doubt the consistency of geometry and analysis.
Vaihinger thought core mathematical theories were themselves works of fiction.
Pasch found it useful to imagine the exploits of fabulous creatures when formulating
his axioms. He found it helpful to recount those exploits when explaining his
axioms to others. He did not, however, insist that his axioms chronicled those
exploits. Pasch introduces his axioms for arithmetic by spinning a yarn about an
idealized combinatorial reasoner blessed with ‘‘eternal life and unlimited memory’’
(1919, p. 21). While he hoped this would help readers understand his axioms, there
is no indication that Pasch wanted his readers to treat arithmetic itself as a fictional
tale about the activities of calculators with powers and abilities far beyond those of
mortal men. Since Pasch’s fables help us understand his axioms without necessarily
asserting just what those axioms themselves assert, the literal falsehood of Pasch’s
fables is no guarantee that the axioms to which they supply the prelude assert
something false. On the contrary, Pasch hints that his axioms assert something true
about the possibility of certain constructions (1921a, p. 149; 1926b, p. 250). He does
not develop this point in detail, however. Since Pasch believed in ‘‘the essential
unity of geometry and physics’’ and considered geometry ‘‘a branch of natural
science,’’ it seems likely he thought geometric axioms had as strong a claim to truth
as physical principles.
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11. 8 Empirical Imagination and Mathematical Understanding
Hans Vaihinger hoped to explain how mathematical theories can be useful without
being true. Contemporary nominalists with this same aspiration can recognize in
him a kindred spirit, with good philosophical intuitions, who, as Pasch helped to
demonstrate, went often astray. While contemporary philosophers have interpreted
mathematics in various ways, Vaihinger thought the point was to leave mathematics
unchanged, acknowledging its absurdity while explaining its utility.
Moritz Pasch was a practicing mathematician, not a professional philosopher.
Among the figures philosophers now consider important he showed awareness only
of Hilbert. In his later papers, there is not one reference to Brouwer, Frege, Russell,
or Weyl. Though he thought it important to understand the general character of
mathematics and its relationship to other sciences, the greater part of his
foundational work aims to improve our mathematical understanding of particular
mathematical theories. His elaborate just so stories were instruments he found useful
in this project.
Just as Einstein’s tale about someone measuring the height of a cloud with a pole
helps people grasp the physical content of statements about length, so Pasch’s
stories about immortal calculators and tireless constructors could help people grasp
the mathematical content of arithmetical and geometric statements. The stories
indicate the content; they do not necessarily express it. To borrow a phrase Pasch
applies to Euclid, his stories are ‘‘illustrations that point the reader’s imagination in
a certain direction’’ (1921b, p. 151). They offer no more than ‘‘preliminary
definitions’’ that are ‘‘not mathematically rigorous’’ and lack ‘‘the precision we
usually expect in mathematics’’ (1925b, p. 110). Pasch provides no technique for
expressing mathematical content through paraphrase, a technique that would allow
us to say, for example: ‘‘there are infinitely many primes’’ means so-and-so. He
does, however, hope to turn our minds in the right direction.
A point of particular relevance for the philosophy of mathematics today is that
Pasch was no structuralist. He insisted that a thorough understanding of his
geometric and arithmetical theories involves more than a grasp of the structures of
their possible models.10
That is one reason he puts us through a program of pre-
mathematical exercises: to turn us toward extra-structural mathematical content by
stimulating our imaginations. Even a scholar as fair and meticulous as Stewart
Shapiro (1997, p. 149) can miss this point.
[Pasch] was a straightforward, old-fashioned empiricist, holding that the
[geometric] axioms are verified by experience with bodies. Pasch’s empiri-
cism died out; the need for formality and rigor did not. Structuralism is the
result.
10
See Pasch’s remarks (1914, pp. 142–143) about the duality between points and lines in projective
geometry. Although the terms ‘‘point’’ and ‘‘line’’ are interchangeable, it is part of our mathematical
understanding of projective geometry that the corresponding concepts are distinct. If we leave the content
terms uninterpreted, then, Pasch concedes, the projective axioms will still characterize a system of
relations. He denies, though, that they provide a definition of the fundamental concepts.
‘As if’ reasoning in Vaihinger and Pasch 93
123

12. Did Pasch believe that experience with bodies verified his axioms? It is not so clear
that this was his mature view. His acknowledgment of the ‘‘as if’’ element in his pre-
mathematical stories suggests, on the contrary, an awareness of how much
geometric and arithmetical principles diverge from human experience.11
On the
other hand, Pasch’s remarks about ‘‘the essential unity of geometry and physics’’
support Shapiro’s reading, though the verification at issue might be a Quinean
corroboration of the whole system of geometric and physical principles, not a
sentence by sentence comparison of mathematical postulates with empirical
evidence. There is another question, however. Did Pasch’s empiricism consist in
a belief that mathematical axioms are to be verified by experience with physical
objects? That is certainly wrong. Pasch appeals to us as creatures of flesh with
faculties of sense, creatures who manipulate and modify physical bodies, because he
thinks this will help us understand his axioms. This understanding could involve a
grasp of a distinctively mathematical content (and mathematical principles, when so
understood, will not necessarily be subject to direct empirical test).
A view Michael Detlefsen (1993, p. 31) attributes to Henri Poincare´ fits Pasch as
well.
… the inferences of the mathematical reasoner reflect a topic-specific
penetration of the subject being reasoned about that is not reflected in the
topic-neutral inferences of the logical reasoner. Logical mastery of a set of
axioms, on this view, does not in itself bespeak any significant mathematical
insight into the subject thus axiomatized.
My mastery of the logical vocabulary of a categorical theory may provide me with a
grasp of the structure shared by all of that theory’s models; but, according to Pasch,
it will not necessarily provide me with a full mathematical understanding of that
theory. It is not evident that this view has died out among mathematicians. At any
rate, it would be harder to show that than to confirm that mathematicians are
disinterested in the empirical verification of their theories.
Shapiro (2008, p. 289) recently offered a very reasonable statement about what is
or should be the purpose of the philosophy of mathematics.
… the goal of our enterprise is to interpret mathematics, and articulate its
place in our overall intellectual lives … one desideratum of our enterprise is to
provide an interpretation that takes as much as possible of what mathema-
ticians say about their subject as literally true, understood at or near face
value.
From this point of view, a correct understanding of the views of a mathematician of
Pasch’s stature is philosophically important. Eminent mathematicians are sources of
data: data about the science we philosophers are struggling to interpret.
Mathematicians sometimes say foolish things. So some data can reasonably be
ignored; but we have to understand what the mathematician is saying before we
11
Hans Freudenthal (1962, p. 617) is quite wrong to say that, ‘‘Pasch was anxious … to postulate no
more than experience seems to grant.’’ Every time Pasch appeals to ‘‘as if’’ reasoning he is postulating
more than experience grants.
94 S. Pollard
123

13. make that judgment. We should contemplate Pasch’s literary remains with care
before issuing a death certificate.
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