*Journal of the History of Ideas* 57.2 (1996) 233-253
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Symbolic Mathematics and the Intellect Militant: On Modern Philosophy's
Revolutionary Spirit

Carl Page
------------------------------

What makes *modern* philosophy different? My question presupposes the
legitimacy of calling part of philosophy "modern." That presupposition is in
turn open to question as regards its meaning, its warrant, and the
conditions of its applicability.
1<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT1>Importance
notwithstanding, such further inquiries all start out from the
phenomenon upon which everyone agrees: philosophy running through Plato and
Aristotle looks significantly different from philosophy running from
Descartes to Kant.

My concern in this essay is with the phenomenon of the difference itself,
rather than with the second-order questions associated with how properly to
assign it historical meaning. I take the difference between ancient and
modern philosophy to be as significant as differences in philosophy's
history can be: modern philosophy rests on a new interpretation of the
nature and fulfillment of human reason, and disputes about the nature of
human reason are the ultimate battles of philosophy. But the general thesis
is not my main point.
2<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT2>The
focus of this essay falls on what may be called the integrity of the
phenomenon, on the specific interpretation of human reason that lends modern
philosophy its peculiar face. *[End Page 233]*

Modern philosophy's strikingly revolutionary spirit is my point of
departure. When Descartes writes in the first of his *Meditations* that "it
was necessary, once in the course of my life, to demolish everything
completely and start again right from the foundations if I wanted to
establish anything at all in the sciences that was stable and likely to
last,"
3<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT3>he
reveals the same enthusiasm for total reform later found in Kant:
"This
attempt to alter the procedure which has hitherto prevailed in metaphysics,
by completely revolutionizing it in accordance with the example set by the
geometers and physicists, forms indeed the main purpose of this critique of
pure speculative reason."
4<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT4>How
exactly did philosophy become so convinced that its central
tradition--a
sprawling, disorganized, ugly city, as Descartes has it in the
*Discourse*(I:116)--needed razing to the ground in the interest of
some rational
town-planning? Moreover, the calls for revolution have not abated, despite
contemporary disillusionment with both Cartesian rationalism and
Enlightenment philosophy in general; they have grown more shrill. The
confidence with which rationalism, foundationalism, universalism,
logocentrism, Platonism, and so on are currently set at naught for the sake
of contingency, particularity, and difference reveals the same revolutionary
and totalizing spirit that marks the earlier phase of philosophy's
modernity. Such enthusiasm is reason's freedom taken to an extreme.
5<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT5>What
inspires this march of the intellect militant? What, if anything,
justifies its hubristic self-assertions in the domain of philosophy? These
are the questions I address.

Descartes is commonly identified as the father of modern philosophy. While
the full story of modern philosophy's parentage is more complicated than
this, it is fair to say that in Descartes self-consciousness of a new mode
of doing philosophy emerges with a focus and revolutionary sense of purpose
that caught philosophical imagination in his own time and continues to do so
in ours.
6<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT6>Motifs
of modern philosophy may be found in many places--Machiavelli,
Hobbes, Francis Bacon, Nicholas of Cusa, Giordano Bruno, Jakob
Boehme--nonetheless, Descartes's intensely single-minded, even jealous
advocacy commends itself to all but the most stubborn antiquarian mentality
as modernity's almost perfect philosophical representative. *[End Page 234]*

That Descartes stands on a remarkable philosophical cusp is apparent in the
contrast between the title and the subject matter of his most influential
philosophical work: *Meditationes de Prima Philosophiae* (1641). To that
point *prima philosophia* or First Philosophy had been construed as the
metaphysics. It was not concerned with the critical question of how
metaphysical sciences are possible and was not directly related to any
doctrine of the human soul--except perhaps on the one point of the divinity
of *nous*, the human soul's highest part. In Descartes's *Meditations,* on
the other hand, the landscape has altered. Doubt, certainty, knowledge, the
ego *cogitans* and its stream of representations are the new subject matter.
What is first in philosophy is no longer what is first in the order of being
but what is first in the order of knowing. The distinction itself had long
since been recognized, appearing in Aristotle for example as the difference
between what is more knowable to us and what is more knowable in itself. It
is the hierarchical arrangement that is new. The question of justifying
human reason has taken philosophy's center stage.

What does such a shift mean? At one level, it might be thought a reassuring
tale for the legitimation of modern philosophy's own self-assertion, by
seeming to capture a decisive reason for claiming unqualified superiority
over what went before. Descartes is the hero of this tale, an effect
plausibly intended by his intriguing self-presentation as the author of his
own being.
7<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT7>Descartes
thus "discovers" the critical need for establishing a complete
theory of knowledge before any theory of the world.

Apart from the all but insurmountable problem of having to interpret the
previous history of philosophy as blind to the eternally obvious question of
critique, Descartes's own view of the reorientation he promulgates and soon
comes to symbolize indicates a difficulty. That his *Meditations* were
subversive he knew quite well:

I may tell you, between ourselves, that these six *Meditations* contain all
the foundations of my *Physics*. But please do not tell people, for that
might make it harder for supporters of Aristotle to approve them. I hope
that readers will gradually get used to my principles, and recognize their
truth, before they notice that they destroy the principles of Aristotle.
(Letter to Mersenne, 28 January 1641)
8<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT8>

The subversion on behalf of his physics is what most needs to be emphasized
here, especially in light of Descartes's later, undoubtedly playful yet not
entirely misleading recommendation to the Princess Elizabeth of Bohemia *[End
Page 235]* that she spend only a few hours per year on metaphysics--the
topic of the *Meditations*--and a few hours per day on physics and
mathematics (Letter to Elizabeth, 28 June 1643). Such asides suggest that in
the hierarchy of Descartes's own theoretical concerns, science outranks
epistemology. This raises a question about the received view of Descartes's
revolutionary achievement: does the location of epistemology at center stage
count as an independent philosophical insight, argued for on its own
grounds, or is it a philosophical move determined by deeper presuppositions
about the nature and warrant of Cartesian physics?

The rhetorical presentation of the *Meditations* plus nearly four centuries
of modern epistemology make it difficult for contemporary readers to imagine
that placing the critical question absolutely first might not be a matter of
spontaneous philosophical reflection, demanding only an unfettered mind for
the appreciation of its necessity. Yet this is not the only way in which
Descartes has been read. Some commentators have seen in the postulation of
the philosophical primacy of epistemology not an unfettered mind but one
spellbound by a quite particular picture of the proper use of reason.
9<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT9>

No more than sketched, the interpretation runs along the following lines:
Descartes was in the vanguard of the new science--the tradition now
associated with such names as Copernicus, Galileo, Kepler, and Newton. The *
Meditations* are designed to clear a space for the new science, on the one
hand in the face of its special theoretical need for epistemological
foundation and on the other in the face of practical impediments occasioned
by Scholastic authority, both political and academic. In pursuit of these
ends Descartes becomes the first full-fledged modern epistemologist.
Heidegger defends the theoretical arm of this interpretation at some length
in his lectures on Kant's first critique, published in English as *What is a
Thing?* (*WT*, 88-108). He summarizes the main point as follows: *[End Page
236]*

Descartes does not doubt because he is a skeptic; rather, he must become a
doubter because he posits the mathematical as the absolute ground and seeks
for all knowledge a foundation that will be in accord with it. (*WT*, 103)

Heidegger thus forges a crucial link between Descartes's scientific concern
for what is now often called mathematical physics and the philosophical
concerns of Cartesian epistemology. The link depends on the logical
implications of what it means to mathematize physics--not so much that
mathematical order turns out to be its epistemic backbone but that the
homogeneity of mathematical order of itself moves toward a sort of noetic
imperialism. Although I endorse the framework supplied by Heidegger's
interpretation, it will require important qualification that in the next
section takes me beyond his treatment.

As Descartes reviews his education in the *Discourse on the Method of
rightly conducting one's reason and seeking the truth in the
sciences*(1637), he tells us that "above all I delighted in
mathematics, because of
the certainty and self-evidence of its reasonings. But I did not yet notice
its real use; and ... I was surprised that nothing more exalted had been
built upon such firm and solid foundations" (*Discourse*, I:114). He is
reporting the intense rational satisfaction that mathematics naturally
provides the curious mind. Its procedures marvelously preserve truth and
bring forth further truth, apparently without recourse to anything obscure
to the intellect. This is the abiding charm of the mathematical, a rational
seductiveness that, as Aristotle warned--using a poetic word that connotes
the familiarity observable in dogs --"fawns [*sainei*] on the soul" (*
Metaphysics*, 14.1090b1). Descartes commends the "certainty and
self-evidence" of such reasonings, thereby restricting his encomium to the
discursive operations of the intellect underway in an already constituted
domain of mathematical objectivity. Nonetheless, the naive experience of
rational satisfaction in ordinary mathematics may be granted. It belongs
only to a relatively sophisticated philosophical consciousness to wonder
about the mysterious origins of how naturally and how, as it were,
thoughtlessly we construe the world in terms of number and figure, the
content of which interpretation transcends the sensate. What is the
experience of discursive certainty in ordinary mathematics?

Certainty in all forms entails a meta-reflection, an assessment that a given
judgment has been made properly and correctly. It is a retrospective
certification that the evidence is in order and that the train of thought
leading up to the judgment has followed adequate procedures. Declarations of
certainty in actual cases are thus relative to the standards of evidence and
ratiocination presupposed for different types of judgment. In the case of
mathematical reasonings--proofs, calculations, solving problems--the aspect
of evidence is a matter of definitions, axioms, postulates, and
constructions, all of which are usually taken for granted as regards their
noetic propriety. In setting the stage for ordinary mathematical
ratiocination, it is *[End Page 237]* considered sufficient to specify the
objects one proposes discursively to manipulate under the constraint that
the set of specifications be mutually consistent. That is, the set-up must
be well-defined.

Why this is rationally satisfying and why it is a noetically proper
beginning is not of direct concern to the one who wants to get on with the
calculation or proof, but it amounts to this: because reason has determined
the specifications in accord with its own design, what reason thus begins
with is to that extent *clear*. Clear means, in this context, that the
identification of the object is exact; it is no more and no less than reason
has determined to be present and presentable for attention. Reason cannot
"mistake" the object its own decision identifies (although its
identifications can yet "mistake" reality). In general mathematical clarity
is a virtue because it means exact identification, and exact identification
means there can be no mistake. That there can be no mistake, however, does
not mean there can be no discovery. Discovery occurs within the horizon set
up by exact identification; it is a matter of articulating or explicating
what has already been identified, a matter of making things *distinct*. The
foregoing is a gloss on the basis for Descartes's well-known criteria of
clarity and distinctness as necessary conditions for rationally certifiable
knowledge.

The customary challenge in ordinary mathematics is to maintain the tacit
clarity of well-defined origins as the mind explicitly manipulates the
objects it has specified. Certainty in the discursive activity of
mathematics is the awareness that nothing has been introduced by the actual
operation of the intellect that it cannot fully identify, that nothing is
being added or subtracted, equated or transformed without the mind's active
warrant that no mistake either has occurred or is possible. In order to
ensure the maintenance of exact identifiability, mathematical vigilance of
this sort admits only a very small set of valid transformations. Discursive
certainty is a matter of seeing exactly how things are staying the same and
exactly how they are being changed. This is what it means to maintain
mathematical precision: that no further play of formal determinations occurs
without the express allowance of reason. There can therefore be no mistakes,
no unhappy surprises--unless, of course, there were mistakes at the outset.

If this is how mathematical reason is certain, it should therefore always
doubt when things are not clear and distinct, when what is present or what
has been transformed cannot be completely identified and articulated. Here
is the generic source of Descartes's methodical skepticism, a skepticism in
service of restructuring opinion in accord with the paradigm of mathematical
clarity. Descartes does not just happen to have a few independently
compelling arguments that show how what is taken for knowledge is not so
firm as usually imagined. He already supposes that knowledge is a matter of
having everything as clearly and distinctly before reason as it is in the
case of mathematical knowledge. His doubt is systematic, directed at
overhauling reason's entire field of epistemic operation. All possible
knowledge is to be *[End Page 238]* measured against the standard of
indubitability implied by the moment of reason's ability to be perfectly
precise in the case of mathematical knowledge. Philosophical extrapolation
of mathematical precision as a noetic standard is part of the "real use"
Descartes did not as a youth notice for arithmetic and geometry.

In the fourth of the *Rules for the Direction of the Mind* (probably
composed in the form we have it by 1628, though never completed or published
in his lifetime), Descartes postulates a *mathesis universalis*, a
"universal learning" that teaches far more than any particular mathematical
science. Despite its single appearance in the Cartesian oeuvre under this
description, the *mathesis universalis* was for Descartes, and has been for
much commentary and philosophy since, the great emblem of the
mathematization of theory.
10<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT10>It
is the discipline that accounts for "why it is that, in addition to
arithmetic and geometry, sciences such as astronomy, music, optics,
mechanics, among others, are called branches of mathematics" (*Regulae*,
I:19). They are all mathematics, according to Descartes, because they each
involve the principles of *order* and *measure*. Descartes takes these
principles to exhaust the systematicity of the sciences mentioned, thus
effacing all aitiological structure save formal causality. It follows from
such a conception that the science dealing with order and measure in their
maximum generality is the architectonic science. Perhaps, too, the art of
its use is the architectonic art of reason's exercise in all domains.

This discipline should contain the primary rudiments of human reason and
extend to the discovery of truths in any field whatever ... it is a more
powerful instrument of knowledge than any other with which human beings are
endowed, as it is the source of all the rest. (*Regulae*, I:17)

In the universality of this *mathesis universalis* lie also the seeds for
converting the satisfactions of mathematics into a philosophical paradigm of
reason, a subtle alchemy that rightly marks Descartes as the father (though
not the creator *ex nihilo*) of modern philosophy. But where did this
enthusiasm come from? Why does it appear now? Is it entirely explicable in
terms of the natural (therefore universal) seductiveness of the
mathematical? In a final formulation: how exactly does the *mathesis
universalis* suggest to Descartes "the Plan of a Universal Science to raise
our Nature to its Highest Degree of Perfection," as he once told his friend
Mersenne he was thinking of calling what eventually became the *Discourse*?
(Letter to Mersenne, March 1636). *[End Page 239]* Heidegger approaches this
question through the type of mathematical understanding that he takes to be
at work in modern mathematical physics and the ground of its impressive
cognitive success. In speaking of "mathematical physics" here, I mean the
discipline in its ideal form, the form entailed by the logic of physical
science, a form only partially recognized and only partially realized in
Descartes. Heidegger calls the epistemological centre of the idea of modern
physical science its "mathematical project." The mathematical project is
what, in principle, renders modern physical science systematic and exact.
From this also follow its positive, experimental, predictive, and
quantitative characters (*WT*, 88-95). It is a "project" (*Entwurf*) because
it is an a priori model that opens up a well-defined domain of objectivity,
an explanation framework that determines in advance how things are to appear
and become explicable within it. It is rightly called mathematical because
the structure of the systematic framework and the occasions of its empirical
confirmation contain nothing outside of parameters set in advance by reason,
thus in accordance with the principle of mathematical precision. Heidegger
attributes a natural expansiveness to this mathematical project, an
eagerness for the rational satisfaction it so readily engenders:

There is not only a liberation in the mathematical project, but also a new
experience and formation of freedom itself, i.e., a binding with obligations
which are self-imposed. In the mathematical project develops an obligation
to principles demanded by the mathematical itself. According to this inner
drive, a liberation to a new freedom, the mathematical strives out of itself
to establish its own essence as the ground of itself and thus of all
knowledge. (*WT*, 97)

Heidegger's personification here is exaggerated, but the experience of
rational satisfaction in mathematical precision remains striking. Its
guarantee for avoiding error altogether is a natural and compelling
commendation to the curious, theoretically inclined human mind. But there is
a difficulty in taking this experience to be entirely new.

Aristotle recognized the seductions of mathematics as it "fawns on the
soul," while Descartes himself draws attention to the story of Plato's
academy guarded by a sign announcing that no one who had not studied
geometry could enter. One might also recall the unified hierarchy of the *
Republic*'s Divided Line (6.509d-511e), where the realm of the intelligible
consists of mathematical ideas first and then pure forms, an ordering
through which the soul naturally progresses. The Platonic testimony in
particular, reveals an ancient recognition that the rational satisfaction
obtained in mathematics bears an essential relation to the perfection of
theoretical reason. Yet neither Plato nor Aristotle projected mathematical
precision as an architectonic standard for theorizing. This was not a matter
of oversight. As Aristotle wryly remarks in the *Metaphysics* (translating
colloquially) "philosophy has *[End Page 240]* become mathematics for those
on the cutting edge [*tois nun*]" (A9.992b1), clearly meaning to distinguish
himself from his modish contemporaries.

Descartes's philosophical faith in a *mathesis universalis* as the
instrument of theoretical perfection cannot come from the experience of
mathematical precision alone, for Plato and Aristotle had the same
experience yet resisted its blandishments. This is not to say that they were
unconditionally right and Descartes unconditionally wrong in the
interpretation of such experience; it is to note that more must be involved
in the Cartesian response. In general, the mathematizability of physics is
not a uniquely modern phenomenon. Not only had medieval thinkers such as
Robert Grosseteste (1168-1253) and Roger Bacon (1220-92) stated--long before
Galileo--that mathematics was the key to natural science, Archimedes had
amply demonstrated how practically satisfying a mathematicized science of
nature could be.

This poses the question of why, if the experience of mathematicized physics
is so compelling, earlier thinkers did not project the rational satisfaction
of mathematics as a paradigm for all theory. At least the direction of
Heidegger's analysis of modern philosophy's uniqueness is on this score
sound. He supposes that in Descartes's case the omnivorous enthusiasm of the
extrapolation comes from a philosophical interpretation of the rational
satisfaction made available in mathematical understanding as such and not
merely in its applicability to the physical world. What Heidegger's account
fails to appreciate, on the other hand, is exactly how, in Descartes's
notion of a *mathesis* *universalis*, the fundamental experience of
mathematical understanding has itself been reinterpreted. The applicability
of mathematics to physics has, in the new science, revealed a more powerful
sense of specifically *mathematical* order.

Aristotle did not fall to mathematizing philosophy, let alone the knowable
as such, because the ancients had a different conception of mathematical
discipline and therewith a different conception of its plausibility as a
cognitive paradigm. The decisive difference between ancient and modern
philosophy is not occasioned by qualitative non-mathematical physics versus
mathematical quantitative physics but by a different interpretation of what
is cognitively achieved through the mathematical dimension in any science at
all.

The chronicle of modern European mathematics records the remarkable
innovation of analytic geometry, a new branch of mathematical discipline
established by the work of François Viète, Simon Stevin, Pierre Fermat, and
Descartes, based in part on an appropriation of algebra from Arabic sources.
Innovations do not have to be revolutions, and in this particular case the
innovators themselves thought of their achievement as a renewal and
extension of certain parts of ancient learning, most especially the
teachings of Diophantus, Apollonius, and Pappus. Yet the accuracy of this
self-description has been challenged: the ontological and epistemological
presuppositions *[End Page 241]* at work in the discipline of analytic
geometry and the algebra that makes it possible constitute something akin to
a paradigm shift in the mathematical sciences.
11<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT11>One
immediate symptom of this transfiguration is the installation of
algebra
at the center of mathematical science, where earlier conceptions of
mathematics had kept algebraic techniques at the service of mathematical
disciplines with more determinate, less abstract subject matter. A further
effect, visible over the subsequent history of mathematics, is the
remarkable dilation undergone by the notion of number from what are now
called natural numbers (a designation that itself hints of what is at stake
in the mutation) through rational numbers, fractions, irrationals, negative
numbers, transcendental numbers, complex numbers, and on into transfinite
cardinals.

Jacob Klein identifies two fundamental signs of the transition to
specifically modern mathematics: (1) its symbolic formalism and (2) its
greater emphasis on calculative techniques.
12<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT12>These
features are coordinate with a difference in the objects of
mathematical knowledge on the one hand and, correlatively, the cognitive
stance of the knowing subject on the other. In broadest terms mathematics
moves from being primarily a contemplative science (*epistêmê*) for
demonstrating theorems to primarily an inventive art (*technê*) for solving
problems. At the root of this complex set of differences is a reconstitution
of the mathematical domain as *symbolic*. Symbolic in this context has a
particular meaning over and above the simple fact that modern mathematics
employs certain graphological techniques. Ancient mathematics, too, on
occasion allowed letters to name mathematical objects. The question is: what
do the symbols of modern mathematics signify? In what sense is the signified
itself symbolic? In the originative cases of algebra and analytic geometry,
the objects of those disciplines were arrived at by a process of
generalization, "a new kind of generalization which may be termed
'symbol-generating abstraction' ..." and held to be in themselves general
objects.
13<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT13>This
procedure was in turn the basis of a generalization to the conception
of a comprehensive *ars analytice* in Viète, with "general magnitude" as its
object, corresponding to Descartes's *mathesis universalis* and its subject
matter of pure order and measure. *[End Page 242]*

Symbol-generating abstraction is more than a simple increase in degree of
abstraction. The move from this or that number, or this kind or that kind of
number (e.g., Odd, Even, Triangular, Prime, Perfect), to the algebraic
manipulation of possible magnitudes is also a move from determinacy to
indeterminacy. This latter move presupposes a special operation of the
intellect--symbolization in the sense being here discussed--whose relation
to the objectivity of mathematical science is open to interpretation.

Both ancients and moderns allow that appreciation of the mathematical domain
requires some active operation of the mind. Platonic and Pythagorean strains
within the tradition notwithstanding, for most of subsequent Greek
mathematics Aristotle's doctrine of "abstraction" (*aphairesis*) formulates
the ancient sense in which human reason actively participates in
mathematical appearances.
14<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT14>According
to this view, the objects of mathematics become accessible in
virtue of what the mind contrives to leave out of account in the attention
it pays to what is given as experience. Abstraction in this sense leaves the
immediate objects of mathematical attention determinate and the subject of a
first-order intentionality. It follows from such an account that the primary
representations of mathematical objects are *eikonic*, i.e., images that
imitate to some degree of isomorphism what are supposed to be externally
subsistent and individual realities. Eikonic representation works by
imitating proportions.

Generalization of the sort embodied in algebraic formulations and Cartesian
geometry, on the other hand, involves a reflective, second-order
intentionality. It has to detour through what is common in the notions that
render determinate, first-order objects accessible, where the commonality is
generic and not therefore directly legible off those determinate
appearances. When letters appear in simple algebraic equations (and thus in
the equations that define the figures of analytic geometry), they represent
relations between sets of possible numbers. But a set of numbers requires an
act of conception that is second-order in relation to the conception that
originally generated the numbers afterwards gathered into that set. One of
the most striking illustrations of this second-order style of
conceptualization at its most fundamental is Frege's definition of natural
number.
15<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT15>

Yet this sort of second-order generalization is not all there is to
symbol-generating abstraction. Once having lifted out what is common in the
relevant first-order notions, those common features are themselves
reconceived in a first-order mode, turned into general objects that enter
the same field of mathematical operations inhabited by the original objects.
The latter step *[End Page 243]* takes generalization to the level of
specifically symbolic abstraction. Only by such a route can one arrive, for
example, at the notion of complex numbers, for they emerge as the imaginary
roots of equations whose algebraic generality is presumed to remain
functionally intelligible while yet lacking solutions in the domain of
non-complex numbers. It follows from this account that the representations
of mathematical objects are no longer eikonic but *schematic*, they are
stand-ins for an indefinite array of possible specifications that have yet
to be performed. Mathematical representations are no longer imitations but
functions.

Generalization as an intellectual operation was not lost on the Greeks. It
enabled Aristotle to pose and answer in the negative the question of whether
being (*to* *on*) is substance (*ousia*). More concretely, the problem
presented by symbol-generating abstraction is at the heart of the
Pythagorean discovery of incommensurable magnitudes. The realization that
continuous magnitude did not submit to perfect arithmelogical measurement
originally provoked a conceptual crisis whose significance is easily hidden
from view by our glib assumption that the notion of number may, without
further ado, legitimately be generalized to include fractions and
irrationals. The incommensurability problem revealed a genuine conceptual
aporia. In light of the contemporary resolution and its apparent
self-evidence it must be asked: why did generalization not become a norm for
ancient mathematics? Why was it not obvious to admit irrationals into the
family of numbers?

Klein's account emphasizes one plausible part of the answer to this
question, namely the constraint on mathematical practice imposed by
ontological convictions. Centrally: "*arithmos *never means anything other
than 'a definite number of definite objects.' "
16<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT16>In
ordinary counting the unit is supplied by the kind (
*eidos*) of things counted, while the science of arithmetic deals directly
with pure monads. Hence the Euclidean definition of number as "the multitude
composed from monads" (vii, def. 2). Within the horizon of such assumptions,
the generalization presupposed by algebra becomes difficult, though not
impossible, to countenance. (It cannot be impossible because ontological
commitments are not only revisable, they determine only the boundary
conditions of the more specific sciences.)

David Lachterman corroborates Klein's point with a detailed review of how
Euclid's text handles the issue of incommensurable magnitudes. It emerges
that for the author or authors of the *Elements*, it was a serious
ontological question whether discrete and continuous magnitude were not so
different in kind as to preclude the legitimacy of performing arithmetical
operations on ratios or even whether it was permissible to think of numbers
as ratios at all.
17<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT17>But
Lachterman adds a further dimension. He begins by noting that in
Euclid
the clash between ontological presuppositions and mathematical practice is
by no means resolved with a perfectly clear conscience. *[End Page 244]* The
focal point on this score is the treatment of compound ratios. The technique
of compounding as it is employed in the text visibly outstrips the
ontological conviction, written into the system of Euclid's presentation,
that ratios are not really numbers but relations between numbers and do not
therefore permit manipulation by arithmetic operations. So, even in the
magisterial Euclid ontological convictions only uneasily govern the ethos of
geometry. Sometimes the mathematical suggestiveness of technique gets the
upper hand.

This raises the question of what other factors might have constrained
Euclid's presentation. Lachterman proposes a certain sort of prudence about
theoretical matters, a "mathematical *phronesis*" which is "the fitting of
appropriate means to ends worthy of choosing, rather than the determination
of ends by the accessibility of means."
18<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT18>The
exercise of such mathematical
*phronesis* is chiefly in the service of psychagogy. The ethos of geometry,
the way it is done as a worthwhile human activity, includes situating any
proof "in a dialogue, not in the solitary monologue of the *ego* *cogitans.*"
19<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT19>The
ancient reason for didactic circumspection is that responsible
introduction to mathematical knowing needs very carefully to reveal how such
knowing, amidst the well-acknowledged need for construction and therewith
the need to mobilize the mind's own artful resources, is nonetheless also
and essentially an acquisition or a discovery. Euclid wished to ensure that
his pupils not get carried away with their own cleverness. He intended that
their understanding keep pace with their facility.

None of this is to say that either Euclid or the ancients managed a perfect
balance between the seductive power of symbolic technique, adequate
ontology, and the circumspection of psychagogical concern. Euclid in
particular, and ancient mathematics in general, had in some important
respects, though not all, an unnecessarily restrictive view of their
discipline's ontology: Cantor's realm of the transfinite is indeed a
paradise from which no mathematician can justifiably be expelled. The
difference between ancient and modern philosophy hinges not on a "discovery"
of symbol-generating abstraction but on the question of what difference its
technical possibilities should be allowed to make, first to the discipline
of mathematical science, and second to all responsible reflection. The
mathematical power of techniques rooted in symbolic generalization convinced
early moderns that it ought to be allowed to make all the difference in the
world to everything, a conviction that permitted Descartes to express the
suspicion that Pappus and Diophantus had concealed "with a kind of
pernicious cunning" their awareness of a *mathesis universalis* as if it
were obvious that they must have known that the real center of mathematics
was the comprehensive analytic art (*Regulae*, I.19). *[End Page 245]*

Yet, whatever the considerations by which the ancients resisted the charms
of symbolic techniques, that the moderns saw no reason to be modest on this
score creates an essential difference: it sets free the entire mathematical
domain to reconstruction along symbolic lines. The difference is not simply
that a few new sub-disciplines with more general subject matter are added to
the already constituted domain of mathematics. The whole of mathematical
reality gets reconceived. Thus, when Descartes defines all conic sections
through the equation Ax
2<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT2>+
Bxy + Cy
2<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#FOOT2>+
Dx + Ey + F = 0, he means to have more clearly revealed their ultimate
mathematical significance and reality than could have been apparent in the
mathematics that depended on images of circles, ellipses, and so on as
ultimate species of mathematical objects.
20<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT20>By
the time we arrive at the formalization of arithmetic in the Peano
axioms, the eikonic representation of natural number has been eclipsed in
favor of a schematic web of functional relations. The impulse to reconstruct
mathematics in symbolic terms reaches a limit in the unsuccessful program of
logicism (the attempt to found the whole of mathematics on formal logic),
but just short of that the foundation of mathematics on the axioms of
set-theory is a monument to the mathematical power of symbolic technique. On
this one point the moderns win the quarrel with the ancients hands down.

That monument was not erected without difficulty, as the famous "crisis of
foundations" makes clear.
21<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT21>The
crisis brought recognition that enthusiasm for the symbolic paradigm
needed qualification, though it has not--despite the profound (and
different) misgivings of Poincaré or Brouwer for example--disturbed the
paradigm itself. One sees here a partial resurgence of the abiding and
incompletely resolved tensions between ontology, mathematics, and pedagogy
(in Lachterman's broad sense) left behind by the initial enthusiasm. This is
just what should be expected if that enthusiasm depended on
over-interpreting the meaning of certain cognitive experiences. There was
nonetheless a sound basis to the extreme interpretation, and dialectical
reexamination of those experiences has not moved that foundation. *[End Page
246]* No party to the recent dispute doubts that the core reality of the
mathematical is symbolic rather than eikonic; the disagreement is about the
reality of the symbolic.

Regardless of the subtler, philosophical misgivings that should be felt
about the disconnection of symbolic technique from the wisdom that would
assign it a proper place within human life as a whole, liberation of the
symbolic imagination is an outstanding gain in the field of mathematics and
the positive natural sciences. Viète, Descartes, and later Leibniz, Frege,
and Russell, are all visibly excited by the possibility, the last three
going so far as to seek a conversion of logic into symbolic form. And so
they should be excited, for admission into the symbolic domain opens up
whole new realms for exploration in which human reason may be assured of the
deepest cognitive satisfaction and, concomitantly as it happens, the most
flattering practical results. Without underestimating the charms of such
flattery--as Descartes did not in his shameless evangelizing to the effect
that the new science would make human beings "the lords and masters of
nature" (*Discourse*, I:142-43)--it is the theoretical enthusiasm,
enthusiasm for the new way of knowing, rather than enthusiasm for the new
way of controlling, engendered by symbolic mathematics that is of most
interest here. It has the most direct bearing on the topic of reason's
self-interpretation because control is only an outward sign, a symptom of
wisdom or the good employment of reason.

Descartes doubtless exaggerated his originality, but the innovation he
championed was world-shattering anyway. The hubris cannot, therefore, have
been entirely vain. Setting aside the rhetoric of self-promotion, is there
anything intrinsic to the character of the newly liberated way of knowing
that encourages Descartes and his modern philosophical followers to suppose
that they are set to eclipse the ancients not only in point of knowledge,
which may be granted, but in point of intellectual virtue and wisdom as
well? How is it that mathematics which used to guard the vestibule to
philosophy has now, in the guise of a *mathesis universalis*, usurped its
inner sanctum? Less figuratively put: what philosophical ideas about reason
does the symbolic conception of the mathematical induce? Mathematical
precision is seductive, in addition to satisfying, because it gives the
impression that reason has to do with nothing besides itself while yet
maintaining objectivity or genuine cognitive achievement. Symbolic
mathematics magnifies this impression on two main counts.

(1) While both symbolic and eikonic mathematics admit that the intellect is
active in the constitution of mathematical appearances, i.e., that
mathematical science must be representational, in the case of
symbol-generating abstraction, so much more is made to appear through the
direct ministrations of the human mind and therefore so much more seems
immediately accessible to certification by the standard of mathematical
precision. The difference on this score lies not with the precision of
discursive ratiocination once *[End Page 247]* reason is underway in an
already specified domain of mathematical objectivity, for that is the same.
Rather, the difference is felt in the original constitution of mathematical
objectivity itself.

All mathematical procedure establishes clarity at the outset through
exactness in its definitions and so on. The determinations specified by
eikonic abstraction are not themselves mind-induced, which is what it means
to call them imitative, but the determinations specified by
symbol-generating abstraction are indeed mind-induced, they are in their
objectivity *constructions*. The qualification "in their objectivity" is
needed because eikonic mathematics makes use of constructions too, but not
for the constitution of its primary objects, i.e., the things it may be said
to know. There is therefore in eikonic mathematics a sort of slavishness to
the forms its abstract representations seek to imitate, whereas symbolic
mathematics is more masterful in setting up a web of functional relations
that are not, at least in the eikonic manner, so directly about anything.
Ultimate mathematical form emerges holographically, as it were, from out of
the mind's own motions as it traces the structural implications of its
original, self-contrived set-up.

Construction, it should be added, is not exactly the same as creation, for
construction works within an already given context. The crux of the
comparison is not that the soul is seduced by the chance to create truth *ex
nihilo*, to be for a moment an *intellectus originarius*. Rather, the
constructive component remains cognitively satisfying because it allows the
maintenance of precision in the sense of exact identification. There is, in
comparison, something still quite obscure in the eikons and definitions with
which traditional mathematics begins--as simple to identify as they
naturally seem--whereas the pure structure of contemporary axiom systems
appears to have almost its entire being from stipulative, and therefore in
one sense rationally transparent, origins.

(2) Once eikonic mathematics is reconstructed in symbolic terms, the
mind-constructed generality of its primary objects makes of all possible
mathematical knowledge a homogeneous totality to whose exploration the mind
is peculiarly fitted. This adds to the common rational satisfaction of
precision hope for the also rational satisfaction, not available to eikonic
mathematics, of certifying *completeness* in accord with the standards of
mathematical exactness. The satisfaction of completeness in this
mathematical sense is not available to eikonic mathematics because the
latter subordinates itself to differences in ontological kind, such as the
difference between continuous and discrete magnitude or the one between
cardinal numbers and ratios, that it takes itself to be not constructing but
contemplating. The *mathesis universalis* makes no demands on ingenuity save
artful, self-controlled ones aimed at maintaining exactness. There is a
general system the mind can trace entirely by its own resources, without
losing control over more specific, external determinations.

Hence one finds throughout modern thought the boast, not merely of having
found a superior instrument of knowing, but of being in the position *[End
Page 248]* of being able to solve all the problems there are. Viète closes
his *In Artem Analyticem Isagoge* (1591)--the foundational text for modern
algebra--with the claim, "the analytical art ... appropriates to itself by
right the proud problem of problems, which is: *to leave no problem unsolved
*."
22<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT22>With
this may be compared both Kant's original announcement concerning his
critique that "in this inquiry I have made completeness my chief aim, and I
venture to assert that there is not a single metaphysical problem which has
not been solved, or for the solution of which the key at least has not been
supplied" (*CPR*, Axiii, cf. Bxxiii) and Wittgenstein's claim in the preface
to his *Tractatus Logico-Philosophicus* that "I therefore believe myself to
have found, on all essential points, the final solution of the
problems."
23<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT23>In
comparison with the assumption that ultimate philosophical
respectability
consists in being able to certify completeness, Aristotle's casual and
inexact enumeration of his categories appears scandalous. To feel sensitive
to that scandal is to be already partial to a mathematical image of
theoretical reason, as opposed to Aristotle's contemplative alternative. As
it happens, mathematical and contemplative interpretations are not the only
options; pragmatic and dialectical images are amongst the other
possibilities.

In sum, symbolic mathematics has both greater mastery over its beginnings
and greater mastery over its ends. This mastery is represented by the two
notions of construction and completeness, a combination that lends to all
modern interpretations of reason their militant, revolutionary character.
Directly correlated with construction and completeness are two further
features that stand out as typical of the new way of knowing: *invention*and
*method*.

The reconception of mathematical reality as symbolic, general, and
functional, brings with it a reassessment of the intellectual virtues proper
to the discipline of mathematical science. The skill of discovery, the *ars
inveniendi*, becomes valued over the patience of understanding, the *ars
demonstrandi*; proving theorems gets overshadowed by the solving of
problems. The *novum organum* is indeed more an instrument, an organon, than
a body of knowledge. Thus Descartes explains to Mersenne that the *Discourse
* is simply a discourse, a preface to, and not a treatise on, the topic,
because "it is a practice rather than a theory" (Letter to Mersenne,
February 1637). The perfection of reason in the *mathesis universalis* is
the masterly establishment of appropriate order and measure, rather than the
slavish hunt for alien form. Technique and invention are thus the highest
intellectual virtues.

From this point on, method takes center stage with respect to scientific
discipline in general. Descartes's notion of method is an elaboration of
Viète's earlier conviction that the ultimate aim of the general analytic art
was "the art of finding, or the finding of finding."
24<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT24>This
exaltation of the mind's
*[End Page 249]* disciplined ingenuity for getting at its now general
objects is at the bottom of the contemporary obsession with the rationality
of inquiry. While fed by other soils as well, the pressing need for method
is centrally rooted in the symbolic conception of the mathematical. In
effect it names the virtue required for disciplining symbol-generating
abstraction, the operation of intellect newly integrated into the
establishment of mathematical and physical knowledge. That new operation is
in need of a different discipline because it is not immediately guided by
the appreciation of external, determinate kinds. The human mind, while it
certainly makes more than is specifically given, cannot be allowed merely to
make things up. Yet not only does method stand for discipline, it also
stands for the assurance of success. To proceed methodically is to be
guaranteed of uncovering all that is relevant, without being beholden to
external forms and their haphazard discovery for one's noetic progress.

With the anticipation of a wholly constructed, completely surveyable domain
of systematic knowledge under the sovereignty of invention and method, the
transformation of reason's picture of its theoretical self has already been
achieved: mathematical order, which is taken to ground the systematicity of
all epistemic cognition, belongs in the first place to the self-directed,
self-originating dianoetic motions of the human intellect.

Objectively, this is expressed as a relocation of the
*mathemata*themselves. Known by Aristotelian abstraction (
*aphairesis*), their primary home is the real world in which they reside *in
potentia*. Known in virtue of symbolization, it is a mind-dependent space
that becomes their primary metaphysical locus. In consequence, a new species
of intellectual act is demanded for negotiating the passage back to the
world or to the reality that grounds mathematical objectivity, one capable
of establishing identity between symbolic structure and the structure of
what it represents. In model-theory this emerges as the satisfaction
relation. More generally, it determines the modern meaning of intuition (*
Anschauung*), which must therefore be distinguished from ancient doctrines
of noetic insight.

Subjectively, the exercise of theoretical or logos-seeking reason becomes
self-involved to the extent that its primary noetic achievement now consists
in its inventive constructions, in the order it might be able to produce
from out of itself. Its ultimate virtue is no longer illumination but the
figuring out of possibilities which may later be compared with actualities
or deployed to control them. To be the better theoretician is to be clever
at manipulating one's thoughts rather than patient in conducing them to the
appreciation of what lies beyond the immanent circle of ideational content.

Modern philosophy's most striking emblem of rational self-involution is the
reification of the *ego cogitans*. Yet while that classic image of Cartesian
dualism has been a constant target for philosophers seeking to assert their
independence from the father of modernity, the abuse of its alleged
metaphysical naiveté has made not one jot of difference to the deeper
presupposition that theoretical reason reaches its perfection in
self-originating inventiveness *[End Page 250]* and imaginative, symbolic
construction. Philosophical modernity is thus the era of concepts and
theories, of rational templates designed to anticipate and dominate the
real. Post-modernity is simply the degeneration of that orientation.
Underlying all of this is not so much an image of the mind as a thing, a *
res*, or a substance, as the image that the mind's primary mode of
theoretical operation is self-contained, that intelligibility is forged
entirely in the workshop of reason's own, internal ideas. Call this the
hypostasis of theorizing reason.

There are two important consequences of this picture. First, the theorizing
mind is oriented inward. Its attention falls first and foremost on the
ideational content of its representations rather than on that to which such
content may be supposed to be pointing and rendering intelligible. It is the
*cogito*, viewing by an inner natural light the stream of its *cogitationes*.
Hence the primary space of rational activity and critical self-consciousness
lies in the self-reflexive relation of consciousness to itself rather than
in the relationship of the intellect to reality. Contemporary hermeneutics
is only a small step away.

Second, and most important, the mind's representations as objects of such an
inner gaze are construed as independently intelligible signs that may or may
not manage to signify other realities. Ideas become the mind's counters,
produced and pushed around by an internal reckoning, building-blocks piled
into edifices of the mind's own design. Theorizing is thus converted to
modeling. For all that later moderns disdain substantialization of the
thinking self, the image of the mind as constructive lurches unavoidably
toward picturing thinking as the will at work on noetic clay. Husserl will
later characterize opinion as sedimentation. To summarize with a distinction
from Plato's *Sophist* (219c): the establishment of epistemic order becomes
a matter of production (*poiêsis*) rather than acquisition (*ktêsis*).

The symbolic paradigm for reason's successful operation bequeaths to modern
philosophy an unstable dialectic of noetic freedom and epistemic
responsibility. Noetic freedom is not the license to imagine whatever one
likes and have it count as knowledge. Freedom for Descartes is the freedom
to know all there is to know, and to that end he directs his jealous
husbandry of reason's self-mastery and autonomy. The liberation implied by
the symbolic imagination is a liberation from slavishness to external
determination in the matter of rendering knowledge exact and complete. Yet,
order that is to count as knowledge has to be more than wishful thinking.

Although not a slave to external order, symbolic construction seeks mastery
of appearances that announce an independent reality not of the mind's own
making, nature is put to the test--to recall Bacon's iconoclastic,
Promethean phrase--not annihilated. In other words, models must still be
true. This raises the general question of what it means that the products of
symbolic imagination and the real order of things should exhibit the hidden
harmony they do, a harmony suggested most obviously by the power of *[End
Page 251]* symbolic mathematics over the natural world.
25<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#FOOT25>In
fact it forces reflection on whether one can forgo adversion to the
real
order of things as a merely honorific, metaphysical, mathematically
unnecessary middle term for the more salient fact that symbolic technique
reliably produces control.

Descartes recognized this possible interpretation of the radically
hypothetical character of symbolic order, yet he still held to a traditional
notion of theoretical satisfaction: "I would think I knew nothing in Physics
if I could only say how things could be, without proving that they could not
be otherwise" (Letter to Mersenne, 11 March 1640). As a metaphysician,
Descartes sought to negotiate the difference between symbolic and real order
with the deductive train of his *Meditations*; as a mathematician he relied
on the power of intuition (*Regulae* 1.14). Both resolutions have failed to
remain as certain to later moderns as they were to Descartes, but the
assumptions that theoretical order is in the first place mind-generated and
that the mind is a self-contained workshop of ideas, has by and large not
been questioned. Consequently, modern philosophy is marked by a congenital *
immanentism* and modern philosophers have been left with the problem of
stretching invention--without intuition or deductive proof of reason's
adequacy to first principles--as far as possible toward the ideal of
universality. Little wonder they should soon forsake the straight jacket of
governing invention by standards of certainty and algorithmic method. The
problem of interim stability, the need to show how contingent, historical
inventions count as rationally acceptable approximations to universal
understanding, is the latest manifestation of that evil against which
Euclid's mathematical phronesis guarded: the attempt to make facility do for
understanding.

My final point concerning the cognitive discipline of symbolic construction
retreats a little from the murkier philosophical depths of what it
ultimately means to call models true, to the relative daylight of
methodological questions about reason's procedures for the certification of
systematic knowledge. Where in the first point the topic was truth, here it
is rationality. Both are matters of justification. Even if the general
adequacy of the symbolic mathematical domain to the real be presupposed, the
dianoetic machinations that go on within its horizons are still in need of
discipline, for imagination issues in fantasy and delusion as readily as it
does in order and measure. Left to its natural inclinations within the
freedom of the symbolic domain, reason either gropes around at random,
hobbled by lack of method, or goes mad with possibility--hence Descartes's
project of establishing *regulae ad directionem ingenii*, "rules for the
direction of ingenuity." Crucial to observe, however, is that Descartes's
rules are self-legislated; they are the edicts of an already self-contained
and now autonomous rationality. This leaves to modern philosophy the
question of how such self-determined *[End Page 252]* rules justifiably
govern an activity supposed to issue in universal knowledge or the best
approximation to it available to human beings. Tradition and the historical
forms of inquiry are the latest candidates for grounding such mortal
self-governance in the matter of knowledge. But this recent historicism is
already showing signs of intolerable strain. It is time to think again about
the fourth part of Plato's Divided Line.

* St. John's College *

Notes

I should like to thank the editor and referees of this journal for their
helpful comments on an earlier draft.

1<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF1>.
Some of these questions are taken up in Richard Rorty, J. B. Schneewind, and
Quentin Skinner (eds.), *Philosophy in History* (Cambridge, 1984); and Jorge
J. E. Gracia, *Philosophy and Its History: Issues in Philosophical
Historiography* (Albany, 1992). With respect to modernity in general see
Reinhart Koselleck, *Futures Past: On the Semantics of Historical Time*, tr.
Keith Tribe (Cambridge, Mass., 1985); Hans Blumenberg, *The Legitimacy of
the Modern Age*, tr. Robert M. Wallace (Cambridge, Mass., 1983), Robert
Pippin, "Blumenberg and the Modernity Problem," *Review of Metaphysics*, 40
(1987), 535-57, and *Modernism as a Philosophical Problem: On the
Dissatisfactions of European High Culture* (Cambridge, Mass., 1991).

2<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF2>.
For the larger context see Carl Page, *Philosophical Historicism and the
Betrayal of First Philosophy* (University Park, Penn., 1995).

3<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF3>.
René Descartes, *The Philosophical Writings of Descartes*, tr. John
Cottingham, Robert Stoothoff, and Dugald Murdoch (2 vols.; Cambridge, 1985),
II, 12 (citations will be given in the text by volume and page).

4<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF4>.
Immanuel Kant, *Critique of Pure Reason*, tr. Norman Kemp Smith (New York,
1965), 25 (Bxx*ii) (citations in the text by standard edition and B-edition
pagination).

5<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF5>.
See Carl Page, "From Rationalism to Historicism: the Devolution of Cartesian
Subjectivity," *St. John's Review*, 42 (1994), 95-111.

6<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF6>.
See Alexandre Koyré, *Descartes und die Scholastik* (Bonn, 1923); Etiènne
Gilson, *Etudes sur le rôle de la pensée médiévale dans la formation du
système Cartésien* (Paris, 1951); Hiram Caton, *The Origin of Subjectivity:
An Essay on Descartes* (New Haven, 1973); and Dalia Judovitz, *Subjectivity
and Representation in Descartes: The Origins of Modernity* (Cambridge,
1988).

7<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF7>.
David Lachterman, "Descartes and the Philosophy of History," *Independent
Journal of Philosophy*, 4 (1983), 31-46; Jean-Luc Nancy, "Larvatus Pro Deo,"
*Glyph*, 2 (1977), 14-37.

8<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF8>.
Translations of the letters are from *Descartes: Philosophical Letters*, tr.
Anthony Kenny (Oxford, 1970).

9<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas/\
v057/57.2page.html#REF9>.
Martin Heidegger, *What is a Thing?* tr. W. B. Barton Jr. and Vera Deutsch
(Lanham, 1985), henceforth *WT*; Edmund Husserl, *The Crisis of European
Sciences and Transcendental Phenomenology: An Introduction to
Phenomenological Philosophy*, tr. David Carr (Evanston, 1970); John Herman
Randall, Jr., *The Career of Philosophy from the Middle Ages to the
Enlightenment* (New York, 1962); Ulrich Hommes, "Sicherheit als Maß der
Freiheit? Descartes Idee der Mathesis Universalis und die praktische
Philosophie der Neuzeit," C. Fabro (ed.) *Gegenwart und Tradition:
Festschrift für Bernhard Lakebrink* (Freiburg, 1969), 105-24; Caton, *The
Origin of Subjectivity*; Jean-Luc Marion, *L'ontologie grise de
Descartes*(Paris, 1975); Leo Strauss and Karl Löwith, "Correspondence
concerning
Modernity," *Independent Journal of Philosophy*, 4 (1983), 105-19; Bernard
Flynn, "Descartes and the Ontology of Subjectivity," *Man and World*, 16
(1983), 3-23; Walter Soffer, *From Science to Subjectivity: An
Interpretation of Descartes' Meditations* (New York, 1987); Tom Sorrell, *
Descartes* (Oxford, 1987); Judovitz, *Subjectivity and Representation in
Descartes*; R. Thomas Harris, "Mathematics, Descartes, and the Rise of
Modernity," *Philosophia Mathematical*, 2nd Series, 3 (1988), 1-20.

10<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF10>.
Lewis J. Beck, *The Method of Descartes: A Study of the Regulae* (Oxford,
1952); Frederick Van De Pitte, "Descartes' *Mathesis Universalis*," *Archiv
für Geschichte der Philosophic*, 61 (1979), 154-74; Pamela Kraus, "From
Universal Mathematics to Universal Method: Descartes' 'Turn' in Rule IV of
the *Regulae*," *Journal of the History of Philosophy*, 21 (1983), 159-74.

11<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF11>.
Jacob Klein, *Greek Mathematical Thought and the Origin of Algebra*, tr. Eva
Brann (Cambridge, Mass., 1968), David R. Lachterman, *The Ethics of
Geometry: A Genealogy of Modernity* (New York, 1989). See also A. G.
Holland, "Shifting the Foundations: Descartes' Transformation of Ancient
Geometry," *Historia Mathematica*, 3 (1976), 21-49; and *Jacob Klein:
Lectures and Essays*, ed. Robert B. Williamson and Elliot Zuckerman
(Annapolis, 1985). The question of how ontological reassessments count as
paradigm shifts in mathematical science is addressed in Donald Gillies
(ed.), *Revolutions in Mathematics* (Oxford, 1992). While this volume
principally deals with how to assign historical meaning to the history of
mathematics (including an excellent bibliography), the essays by Caroline
Dunmore, "Metalevel Revolutions in Mathematics" and Jeremy Gray, "The
Nineteenth-century Revolution in Mathematical Ontology" are the most
congenial to the eidetic orientation of Klein and Lachterman.

12<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF12>.
*Greek Mathematical Thought and the Origin of Algebra*, 122.

13<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF13>.
*Ibid*., 125.

14<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF14>.
See John J. Cleary, "On the Terminology of 'Abstraction' in Aristotle," *
Phronesis*, 30 (1985), 13-45.

15<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF15>.
Gottlob Frege, *The Foundations of Arithmetic: A Logico-Mathematical Enquiry
into the Concept of Number*, 2nd Revised Edition, tr. J. L. Austin
(Evanston, 1980), 79-80: "The Number which belongs to the concept F is the
extension of the concept 'equal to the concept F'...."

16<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF16>.
*Ibid*., 7.

17<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF17>.
*Ethics of Geometry*, 29 ff.

18<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF18>.
*Ibid*., 32.

19<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF19>.
*Ibid*., 115.

20<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF20>.
Emily Grosholz, *Cartesian Method and the Problem of Reduction* (Oxford,
1991), argues for an important qualification here regarding Descartes, that
he is so seduced by symbolic generalization he fails to appreciate
ontological heterogeneity in both the purely mathematical and the physical
domains, mistakenly reducing to relations of discursive order what
nonetheless absolutely requires an intuitive component for its accurate
comprehension. This point does not, however, effect the distinction between
eikonic and symbolic abstraction.

21<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF21>.
Herbert Mehrtens, *Moderne-Sprache-Mathematik: Eine Geschichte des Streits
um die Grundlagen der Disziplin und des Subjekts formaler
Systeme*(Frankfurt am Main, 1990); Jean Van Heijenoort,
*From Frege to Gödel: A Source Book in Mathematical Logic* (Cambridge,
Mass., 1967), and Stephan Körner, *The Philosophy of Mathematics: An
Introductory Essay* (London, 1960). See also Janet Folina, *Poincaré and the
Philosophy of Mathematics* (London, 1992); and Mary Tiles, *Mathematics and
the Image of Reason* (New York, 1991).

22<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF22>.
Klein, *op. cit*., 353.

23<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF23>.
Ludwig Wittgenstein, *Tractatus Logico-Philosophicus*, tr. D. F. Pears and
Brian F. McGuinness (London, 1961), 4.

24<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF24>.
Klein, *op. cit*., 169.

25<http://muse.jhu.edu.ezp1.harvard.edu/journals/journal_of_the_history_of_ideas\
/v057/57.2page.html#REF25>.
Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural
Sciences," *Communications on Pure and Applied Mathematics*, 13 (1960),
1-14.


On 8/29/06, Lancelot R. Fletcher <lrfletcher@...> wrote:
>
> I am trying to find the following article from the Journal of the
> History of Ideas which, based on an abstract, seems to discuss Klein's
> ideas about mathematics.
>
> Page, Carl 1957- "Symbolic Mathematics and the Intellect Militant: On
> Modern Philosophy's Revolutionary Spirit"
> Journal of the History of Ideas - Volume 57, Number 2, April 1996, pp.
> 233-253
> University of Pennsylvania Press
> <http://muse.jhu.edu/about/publishers/upenn>
>
> If anybody here has a hard copy of the article and could fax it to me
> at: 1-202-478-0278 I would be very grateful. Or if you have access to
> the electronic version via Project Muse (which some academic libraries
> have), please send it to me and I will post it to the group file library.
>
> Lance Fletcher
>
>
> This is one of the lists sponsored by The Free Lance Academy, home of
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>
>
>
>
>
>
>


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