# Continuity, Topos and Infinitesimals

Clarence Protin

###### Abstract

In this essay we discuss the concepts of _topos_ (place), _sunekhes_
(continuity) and _infinitesimal_ as found in classical philosophy (primarily
in Aristotle’s _Physics_) and their relationship to modern topology. The
connection between the ancient and the modern was important in the later
thought of René Thom, in particular in his _Esquisse d’une Sémiophysique_. We
show how the modern (Leibnizian and Newtonian) concept of infinitesimal has
its roots in classical thought and that the concepts of topos, continuity and
infinitesimal are related. We show furthermore how Aristotle’s notions of
topos and continuity can be given precise definitions in terms of modern
topology and category theory and how they are relevant to modern problems in
the foundations of mathematics and philosophy of science. (Accepted peer-
reviewed article).

##  Aristotle’s Theory of the Continuum

A major preoccupation in the later thought of René Thom was bringing to light
the ”primordial topological intuitions” present in Aristotle (see for instance
[6, 7, 18, 19, 20, 21, 22, 23, 8]) and showing how the Peripatetic system
hinges on certain fundamental intuitions and concepts which only with advent
of modern topology have found a full formal development. According to Thom the
key concept is that of _sunekhes_ , a noun which is usually translated as
”continuum” or ”continuity”. The _Physics_ can be considered as being
concerned with the fundamental conditions of _phusis_ 111For convenience we do
not mark vowel-length in our romanisation of Greek terms. or nature, which
include change, extension or magnitude,(_megethos_), place and time as well as
form (_morphe_ , _eidos_) and matter _hule_. Aristotle lays great emphasis on
proving that these are all continuous quantities. Thom proposed furthermore
that for Aristotle continuity plays a central role in our cognitive
organisation, namely via the ”semantic spaces” of genera and species
[6][Ch.8]: Aristotelic physics is interpreted as a _semiophysics_ , a
semiotics not unlike that of Greimas or Brandt. Aristotle layed the foundation
for much of early modern Western scientific and philosophical thought:
continuity plays a major role in Newton, Leibniz, Kant, Hegel, Peirce and
Husserl, albeit with unique developments proper to each philosopher. What is
peculiar to Aristotle’s concept of continuity which is of contemporary
philosophical relevance222At least for those who hold that there should always
be an interplay between philosophy, mathematics and science. is its affinity
to recent developments in mathematics, mathematical logic and theoretical
physics[33]. We will show how this connection (specially with regards to
Aristotle’s concept of _topos_) sheds light on the present and historical role
of infinitesimals in mathematics and physics even though the modern concept of
infinitesimal is not in fact explicitly present in Aristotle.

To investigate the continuum in Aristotle we must consider the concepts of
point (_monas_), infinity (_apeiron_), limit (_peras_) ( and the allied
concepts of envelop, _periekhomenon_), division (_diareisis_) and contact
(_haphe_) and explore how we should interpret the basic distinction between
the substrate (_hupokeimenon_) and its accidents/attributes (_sumbebekhos_).

The most striking characteristic of the Aristotelic _sunekhes_ which sets it
apart from the modern mathematical notion is that it is not conceived as an
aggregate of points, as formed by the joining together of points
(231a21-b10)333All references are to the _Physics_ and we follow the text of
[12]. Aristotle takes the later Euclidean definition of a point as that which
has no parts. The definition given of a _sunekhes_ involves contiguity: the
extremities of its parts are united (227a6-16 ). The term ”connectedness” is
perhaps a more faithful translation. Since we cannot distinguish between the
extremities of a point and the point itself, the coming together of two points
will result in complete coalescence and thus a continuum could never be
constituted from points. The extremity or limit (_peras_) of something is a
kind of interface, that which allows the entity to relate to other entities or
to the surrounding environment or be capable of receiving diverse predicates.
These considerations have roots in Plato’s _Parmenides_. In the first
hypothesis of the final ”dialectical” section, the One is considered in its
absolute simplicity (exemplifying an idea). It could not manifest (or exist)
for then it would have to be in a certain place. But being in a place it would
be _enveloped_ by that place, that is, its extremities would touch, be in
contact, with the surrounding environment. Plato then continues to argue (in
_Parm_. 138A-B) that since the extremities are a part of the entity and the
One is by definition an entity without parts, the One could not be in a place
and hence could not ”exist”. According to Scolnikov [17][p95-100] this problem
is resolved in the second hypothesis wherein the Existing One is considered as
a complex unity, a composite (_sunolo_ which allows participation (or
predication) whilst preserving its underlying unity. As we shall see, _sheaf
semantics_ offers a striking illustration of such a relativized principle of
non-contradiction. Scolnikov [17][p100] makes the interesting observation that
this complex is in concrete instances always finitely determined or divided
(_diareton_), although it can potentially be divided infinitely - _diareisis_
presupposes the possibility of making distinctions within a form— any form
—and requires, in principle, that division can go on indefinitely. But
potentially endless divisibility is a chief property of the Aristotelic
continuum: In (200b19-20) we read: _to eis apeiron diaireton sunekhes on_ \-
the continuum is what is infinitely divisible.

We are not proposing that in Aristotle substance _upokeimenon_ is to be
conceived as a continuum: in stricr Aristotelism the continuum belongs to the
category of quantity. This a main point of disagreement between Aristotle and
Thom. Thom recognised his interpretation of the substrate as a continuum as
being ”fundamentally unfaithful” [6][p.245].

However Thom is not postulating the Aristotelic continuum to be substance but
rather modern Cartesian space which is not to be confused with place
(_topos_). Aristotle in (208b8-22) seems to be perfectly aware of the modern
concept of orientation (cartesian coordinates) and their _relativity_ though
he also tended to hold the view of the existence of an absolute cosmic
orientation or frame of reference. In (209b11-17) he ascribes a similar view
to Plato : the identification of matter and space, space being what is capable
of receiving forms.

In conclusion we have seen that Aristotle gives us two fundamental
characterisations of _sunekhes_ : that whose parts have united extremities
(and hence cannot consists of points) and that which is infinitely divisible.
The modern topological definition of a continuum involves the assumption that
the continuum consists of points. Yet in this framework the concepts of
connectivity and local connectivity capture the first property and, for the
real line, the second property is captured by that of dense linear order. But
note that there are connected topological spaces which do not have anything
that could correspond to the divisibility property (consider the coarse
topology).

The infinite (_apeiron_) is very different from our modern conception of an
infinite cardinality. Aristotle distinguishes between an infinity which ”goes
beyond any finite magnitude” (206b20) and a bounded infinity exemplified by
the set obtained by successively adding to each other segments of half the
previous length (expressing the convergence of the geometric series). In
illuminating passages (206b3-12, 207a7-25, 209b8) we see that it is that which
has a limit, not in itself but in something external to itself. It is
something incomplete or incompleted, _aoristos_(209b9), imperfect or not yet
perfect or whole (207a7-15). This is also reflected in the difference between
the aorist and perfect tenses in classical Greek. It can be an expression both
of a completable or incompletable process (involving division or addition). It
can express potentiality (_dunamis_) and generated being (_oude menei…alla
gignetai_ , it does not rest…but is in the process of becoming) and is likened
to time (207b15).

The above passages show that Thom was quite justified in his insight that
_apeiron_ corresponds to the modern concept of _open set_ in topology444For a
good introduction to modern general topology see for instance [10]. Some basic
concepts are also defined in the appendix of this essay. as well in his
insight that there is an intimate relationship between matter _hule_ and the
infinite and between form and limit (_peras_) or the envelop
(_periekhomenos_). It is interesting to note that distinction in (209b8)
between _diasteme_ (extension) and _megethos_ (magnitude). In modern terms the
_diasteme_ would be the _interior_ of the megethos considered as a closed
interval (cf. [4][p.207 Note 5] ). Another notable example of an open set in
Aristotle is found in the discussion related to the inexistence of a first
time wherein a motion started(236b 32 - 237b22). The concept of _peras_ can
correspond to the modern topological concepts boundary or limit of a sequence.
A point functions as a limit and as a medium either for division or junction
(Aristotle distinguishes in (226b18-24) between mere contact and the total
fusion of extremities in continua). In modern topology open sets function like
local homogeneous patches that generate the whole space (cf. the concept of
basis of a topological space). They express the concept of the local spatial
quality around a given point in which the particular size of this
neighbourhood is irrelevant (cf. Hegel’s concept of indifferent quantity. See
the section of [11] on the category of quantity. Hegel’s discussion of the
relationship between quantity and quality in fact anticipates Thom in certain
aspects as well as harkening back to Aristotle.).

In one of Thom’s topological analogies in [6] an open set correponds to matter
(_hule_) and potentiality (_dunamis_) whilst the boundary (which is a closed
set) corresponds to form and to actuality (_energeia_). The genesis of an
entitiy can be seen as a separation from a matter by means of a cut along its
boundary defining and giving form to the entity. But on the other hand the
open set defines its own limit or boundary and the _energeia_ of change takes
place during an open interval of time.

It appears furthermore that Aristotle’s concept of the finite/whole/complete
corresponds to the modern concept of compactness (which in the topology of
ordinary space is equivalent to being closed and bounded).

As we mentioned, _sunekhes_ in Aristotle is close to the modern concept of the
standard (connected) topology over the line (or general euclidean space
$R^{n}$) but with the difference that such a topology is always defined in
terms of a sets of points. We propose that _apeiron/hule_ should be taken as
the fundamental generating element of the Aristotelic continuum (together with
the endless potentiality of division or joining together) and that points and
closed sets be seen as the result of taking limits of indefine processes of
division or joining. Aristotle’s _sunekhes_ is thus characterised in terms of
being built from open sets, not points, by being indefinitely divisible (which
has in fact a certain relationship with standard topological ”separation”
axioms) and by being connected.

We will relate this to recent advances in modern geometry and topology. In
this framework it is possible to interpret rigorously not only the Aristotelic
concept of _topos_ (place) but also that of infinitesimal. In Thom’s
interpretation of _hupokeimenon_ as a continuum or closed bounded subset of
$R^{3}$ (as in [6][p.153]) this continuum though not an absolute _materia
prima_ is an individuated matter which shares with it the characteristic of
not yet being able to exist by itelf. Natural beings always consist of both
matter and form, and for Thom form is a continuum (or space) stratified
according to different locally homogenous qualities separated by ”catastrophe
sets” determined by a fiber bundle (or sheaf of local qualities). This is at
the heart of Aristotelic biological division between _homeomeres_ and
_anomeomeres_. More generally, the passage (228a26-30) shows how Aristotle
clearly recognizes that globally change is not continuous only locally
continuous, a concatenation of _contiguous_ continuous changes and is thus in
this way stratified.

In the _Physics_ we have also the germ of the idea of a continous function, a
quality of how a certain quantity varies in function of another quantity, such
as place or some other quality, in terms of time. One property of continuous
functions is that the image of a compact set is a compact set. This property
relative to compact sets seems to be the intuition present in Aristotle’s
proof that there can be no infinite motion in a finite time
(237b23-238a11-20). Another property is that the image of a connected set is
connected. A particular case of this property is Bolzano’s intermediate value
theorem (taught in elementary calculus classes) which expresses _natura non
datur saltus_ , the fact that a continuous function has no jumps or holes,
cannot skip intermediate values (more intuitively we can see that the graph is
a connected curve in space). It is interesting to note that the continuity of
a function can in fact be defined more intuitively in terms of the arcwise
connectedness of its graph.

The basis of Thom’s Catastrophe Theory is that qualities should always be
_locally_ continuous with respect to space and time. It is quite natural to
replace Thom’s use of fiber bundles with that of sheaves or their equivalent
étale-bundle presentation[28][p.88] which capture exactly this property.

The sheaf-theoretic formulation lends itself more naturally to a formulation
based only on the category of open sets, without any direct appeal to points,
or appeal to the idea of space being made up of points. Thom in facts mentions
sheaves in [6][p.209] to argue for the topological primacy of the substrate
space over its fiber space. We will now show how the above considerations cast
light on Aristotle’s definition of topos and on the concept of infinitesimal.

Consider a space $E$ representing the environment on which is defined a
certain sheaf of qualities. Consider an individual entity given by a closed
bounded subset $B$ of $E$. What is the topos of $B$ in $E$ in the Aristotelic
sense ? At first sight we are faced with a problem. We wish to find ”the limit
of what is enveloping” (212a5). If we consider the complement of $B$, $c(B)$,
then we get an open set. This open set will have its own boundary which will
(for sufficiently tame kinds of set) coincide with the boundary of $B$ so that
this boundary cannot be seen as the Aristotelic topos which is essentially
”separable”, being like the limit of the container (the inner surface of a
bottle containing water). As Leon Robin wrote[13][p.142] the topos is ”always
qualified”. If we consider a sheaf of qualities then the topos will certainly
not be determined merely by the behaviour of the sheaf on the induced topology
of $B$, for this ignores completely the _situation_ of $B$ relative to $E$ (to
use an analogy from elementary calculus, it is as if we were to try to
calculate the lateral limit of a function by only considering the value of the
function at the given point). The Aristotelic concept of topos points directly
to the modern concept of limit and to a particular concept of infinitesimal.
The topos of $B$ in $E$ is captured by a limit (in the modern category-
theoretic sense) of a sequence of nested open neighbourhoods of $B$ in $E$, or
rather, by the behaviour of the sections of a sheaf of qualities on such a
sequence. The topos of $B$ in $E$ is determined by its infinitesimal
neighbourhood. If $B$ is taken as a point $p$ then its topos is the _stalk_ at
$p$. The exact concept of topos if found remarkably in modern algebraic
geometry in the concept of a formal completion of a subvariety $Y$ in a
variety $X$. In [2][p. 190] Hartshorne writes:

_The formal completion of $Y$ in $X$ (…) is an object which carries
information about all the infinitesimal neighbourhoods $Y_{n}$ of $Y$ at once.
Thus it is thicker than any $Y_{n}$, but is contained inside any actual open
neighborhood of $Y$ in $X$._

Aristotle was at least not always aware of this construction as in (209a7-13)
where the point is stated to be identifical to its topos. Waterfield
[5][p.253] also comments on the connection to the first hypothesis of the
Parmenides.

An infinitesimal is something that is not a point yet is smaller than any open
neighbourhood of this point. A sequence of nested neighbourhoods can be
conceived as ”flowing” (cf. Newton’s _fluxiones_) and any finite number of
elements of this sequence can be removed without affecting the defined
infinitesimal. This can be also interpreted as an essential local homogeneity
or self-similarity of the standard topology of ordinary euclidean space. All
closed intervals are qualitatively the same. As we shall see in the next
section, in a category-theoretic, point-free, presentation of topology, closed
intervals or infinitesimals should be seen a central objects, ”generators” in
some sense, of all major topological properties. In the following sections we
will try to give more detailed technical illustrations of the ideas discussed
in this section in what pertains to recent advances in geometry and topology.

##  Topos, Infinitesimals and Sheaves

Leaving aside for a moment the problem of a ”point-free” definition of the
continuum we present some considerations on how both infinitesimals and the
concept of topos can be captured using modern Sheaf Theory. An excellent
introduction to Sheaf Theory can be found in [28, 15]. How can we define an
infinitesimal on the line at point $a$ ? One possible answer is saying that it
is simply the set of all open neighborhoods of $a$ \- for containing the
infinitesimal neighbourhood of $a$ is what they all have in common - and not
simply the point $a$.

We can also consider the nested sequence $U_{1}\supset U_{2}\supset...\supset
U_{i}\supset..$ of open subsets containing $a$ as a definition of the
infinitesimal at $a$, provided that they form a countable basis of the
topology. It is interesting that the infinitesimal is not any of the $U_{i}$’s
but the very sequence of them, that is, the flow of such $U_{i}$’s tending to
but never identifying with the point $a$. If we remove finitely many $U_{i}$’s
we still are defining the same infinitesimal. It is all of them and none of
them in particular. The classical paradox of infinite cardinalities
(illustrated by the paradox of Hilbert’s hotel expressing the invariance of
infinite cardinalities under finite subtraction) creates this paradox of the
definition of the infinitesimal. The infinitesimal can only be defined using
an infinite number of terms, any finite number of which are dispensable. The
infinitesimal is simply a local (sub)basis for a topology. Aristotle’s
sequence of halved lines in the _Physics_ furnishes an exampleof such a basis
for the real line. The homogeneous nature (any local open part has the same
quality as the whole) of the continuum is related to the fact that it is
generated by the same infinitesimal. The infinitesimal is an ideal element
which can only enter into relationship to the continuum and the point $a$
through reference to an infinite sequence of subsets containing $a$, but being
at the same time neither the point $a$ itself nor any infinite sequence of
open subsets or indeed any single one. No finite collection of open sets or
points can define an infinitesimal, nor can the point itself, for the same
”point” can be ”in” a space of any dimension.

As we have seen, the nested sequence of open sets $U_{1}\supset
U_{2}\supset...\supset U_{n}\supset...$ in $R^{n}$ all containing a certain
point $x$ defines (let us just assume we have uniqueness) the _infinitesimal_
$dx$ which we take to be the ”limit”, the infinitesimal ”ball” with center $x$
or the flowing vanishing neighbourhood of $x$, not equal to $x$ yet contained
in all open sets containing $x$. Now $x$ can be replaced with any closed
compact subset $K$ of $R^{n}$ and we get the ”infinitesimal” $dK$ defined by a
sequence of nested open sets all containing $K$. If we view $K$ as a ”body”
within the ”body” $R^{n}$ then $dK$ is precisely what Aristotle means by place
topos : _to peras tou periekhontos somatos kath’ho sunaptei to
periekhomeno_(212a5-6), _the limit of the enveloping body according to where
it touches what it envelopes_. Indeed $dK$ is something outside the body $K$
which is yet a limit of what is outside and that touches _sunaptei_ (without
identifying with) $K$. In (211b10) we have _en tauto gar ta eskhata tou
periekhontos kai tou periekhomenou_ , _the limits of the container and
contained coincide_ (211b12-14), but without fusing and being identical.

This concept of topos make sense whenever we have an embedding of a manifold,
analytic space or variety (the formal neighbourhood discussed in the previous
section ) within another one. Take for instance instance the ”tubular
neighbourhood” or normal bundle used in differential geometry and
topology555Elementary algebraic topology offers us many striking embodiments
of the Aristotelic topos, situations in which we are interested in the
behaviour in the neighbourhood of a boundary. This is the main intuition
behind the concept of relative homology of a pair $(X,A)$ and the excision
theorem. According to [34][p.44] : ”In particular a chain in X is a cycle
modulo A if its boundary is contained in A. This reflects the structure of
$X-A$ and the way that it is attached to A. In a sense, changes in the
interior of A, away from its boundary with $X-A$, should not alter these
homology groups”. A very useful property of CW-complexes is that they are
locally strong deformation retracts around the ”place” in which the cells are
attached. . But the above concept of topos has not yet any physical sense for
we must consider space as a substratum of qualities, that is, we need to
consider a sheaf defined over it. As we mentioned, a fundamental feature of
the thought of Thom (which is later expanded by Jean Petitot in for instance
[14] is that Aristotelic categories are reinterpreted in a Cartesian form, in
a way characteristic of modern mathematical physics, wherein modern ”space”
corresponds to the Aristotelic ”substrate” whilst qualities (in modern terms,
fields) are captured by functions over that space, or more precisely by
sections of certain bundles or, equivalently, sheaves. Sheaves allow us to
glue together local information and define in a rigorous way the quality of a
certain substrate at a given point. This is called taking the germ of a
section at a point and is a generalisation of the original concept of
derivative of a function at a point developed by Leibniz and Newton.

The concept of a _constructible sheaf_[16][p.320] involves a stratification of
the underlyng space into locally homogenous qualities in a way the captures
admirably the Aristotelic concept of _anomeomere_ and general scheme of
Catastrophe Theory.

Consider the sheaf $C$ of continuous functions over $R^{n}$, which means we
consider all the continuous functions defined on open sets $U$, which we
denote by $C(U)$. Then if we consider a nested sequence defining the
infinitesimal $dx$ there is a corresponding sequence

$C(U_{1})\rightarrow C(U_{2})\rightarrow...\rightarrow C(U_{n})\rightarrow...$

which defines the ”stalk” $C_{x}$ which we denote by $\frac{dC}{dx}$. What
about for the _situation_ of a closed compact $K$ ? If we denote by
$j:K\rightarrow R^{n}$ the inclusion, then there is a well-defined sheaf (the
_inverse image sheaf_) that satisfies precisely the Aristotelic property:

$j^{-1}C(K)=\lim_{U\supset K}C(U)=\frac{dC}{dK}$

The meaning of this expression is that the qualities expressed by the sheaf
$j^{-1}C$ all depend on the qualities of $K$ considered as enveloped by some
neighbourhood $U$, in fact they depend of $K$ and its infinitesimal
neighbourhood in $R^{n}$, its _situation_ and immersion in the larger space.
It is thus a limit of the qualities on enveloping neighbourhoods of $K$. In
conlusion, the concept of topos corresponds to a rigorously defined
mathematical concept which is used in a wide range of areas of applied
mathematics.

Aristotle considers also the situation in which both the contained and the
container can change (as a boat in a river). The final definition of topos is
_to tou periekhontos peras akinêton proton, tout’estin ho topos_ , _the first
immobile limit of the enveloping entity, that is the topos_(212a20). This
suggests that alternatively we consider the diffeomorphic action
$\phi_{t}:T\times R^{n}\rightarrow R^{n}$ of a dynamical system and define the
topos to be the intersection of all invariant open sets ($\phi_{t}(U)=U$) such
that $K\subset U$ and hence $\phi_{t}(K)\subset U$.

In continuum mechanics _Cauchy’s stress principle_ is related to the ”act” of
infinitesimal homeomeric volumes of the substance on its immediate environment
\- its topos - through the defining region of the act, the surface. This is
the concept of _surface force_. The infinitesimal volume will always be
dynamic and have a surface; Cauchy’s principle states that we can take the
limit of the force/area ratio of the force acting on a given small area of the
surface determined by a normal direction because the moments will vanish at
the limit. Considering each direction we thus can define a (symmetric) tensor,
the stress tensor, at each infinitesimal point.

Cauchy’s principle is an example of the dynamical physical nature of the
infinitesimal. Our direct physical experience of the world around us is very
much like the topos of a small volume in continuum mechanics. We feel our body
directly immersed and affected by the qualities of the surrounding environment
(or, in J. v. Uexküll’s terms, the _Umwelt_) through the medium of
mechanoreceptors and sense organs.

We mentioned above the concept of stratification associated to a constructible
sheaf. One of Thom’s most important papers in topology[9] concerns stratified
sets and morphisms. The theory involved - which deals with a generalisation of
smooth manifolds and algebraic and semi-algebraic spaces - dates back to the
work of Hassler Whitney (1907-1989) on singularity theory . It is perhaps the
most elaborate development of the original topological intuition present in
the Aristotelic concept of topos. The topos is found in the way in which in a
stratification a given stratum $X$ is related to the adjacent strata $Y$ which
are contained in its closure $\bar{X}$. This induces a decomposition of the
boundary of $X$ into such adjacent strata which can be seen as forming the
_topos_ of $X$. These strata $Y$ in general have a very complex topological
relationship to $X$. Whitney’s conditions (A) and (B) are imposed in order to
maintain a tame, intuitive, non-pathological situation, ruling out what can
happen with spirals or in the crossings of the stratification of the Whitney
umbrella. These Whitney stratifications came closer to the intuitive notions
of topos in the smooth way in which the adjacent strata envelop and contain
$X$. Many more mathematical examples that could be adduced regarding the
importance of the behaviour on the ”boundary” of object such as for example
the Dirichlet problem in partial differential equations in which a function on
a given region is completely determined by its values on the boudary of this
region.

In all the previous discussion we placed ourselves in the framework of
continuous, differentiable or smooth (infinitely differentiable) functions or
maps. The smooth category is clearly at the heart of Thom’s mathematical work
and indeed of modern differential topology and differentiable dynamical
systems. In a future essay we will argue for the fundamental importance of the
analytic category (and some extensions thereof) both for mathematics and
natural science and explain how it is distinct yet closely related to the
smooth category. The analytic category and the algebraic category (of which we
have already mentioined the formal completion of a subvariety) have the
interesting feature that the infinitesimal and topos concepts discussed above
can be given purely algebraic definitions. There is no need to appeal to
intuitions of ”flowing quantities” or ”infinitesimally small quantities”. It
was O. Zariski and other who gave rigorous foundations to the geometric
intuitions of the early Italian school of Algebraic Geometry. Thom recognised
that his concept of ”transversality” stems from their notion of ”general
position”.

In algebraic and analytic geometry we can define many infinitesimal concepts
such as infinitesimal neighbourhood (formal completion of a subvariety or
formal scheme), tangent space, smooth (non-singular) variety and smooth map in
purely algebraic terms involving quotients , localisations and completions of
rings and modules and algebras of formal differential operators. A central
role is played by _nilpotent elements_ of a ring, that is, elements $a$ for
which some power $a^{n}$ is zero. This algebraic approach is also at the heart
of modern singularity theory and the proofs of Thom’s theory of Elementary
Catastrophes. We shall say something more about this in a further section were
we discuss ”space as a quotient”.

Nilpotents and zero-divisors can be seen as giving an interpretation to the
procedure of ”ignoring” infinitesimals of a higher order to obtain equalities,
often excused as ”approximations”. To see how this relates to niloptency,
consider Leibniz’s rule for differentiation
$(f(x)g(x))^{\prime}=f(x)g^{\prime}(x)+f^{\prime}(x)g(x)$. But for
differentials this is expressed as $d(yz)=zdy+ydz$.

By definition $d(yz)=(y+dy)(z+dz)-yz$. But this is $yz+ydz+zdy+dydz-
yz=ydz+zdy$ because $dydz$ is an infinitesimal of higher order than the other
terms and so can be ”neglected” or equated to zero $dydz=0$. Geometrically,
the rectangle $(y+dy)(z+dz)$ is divided into a constant rectangle $yz$, two
rectangles with one infinitesimal and one constant side, $ydz$ and $zdy$, and
a rectangle with both sides infinitesimal $dydz$, which is thus ”infinitely
smaller” than the other components and may be ”neglected”.

##  Topology without Points

We have seen how the Aristotelic _sunekhes_ corresponds roughly to a
characterisation of the standard topology on the real line (or euclidean
space) in terms of (local) connectivity and divisibility. In standard terms,
this _divisibility property_ could be described (for the line $R$) as follows:
given any point $p$ there are two disjoint open sets whose union is dense in
$R$ and for which $p$ belongs to both their closures. If we consider the
standard algebraic structure on the real line then we have a linear order and
divisibility can be expressed by the standard _density property_. For higher-
dimensional spaces instead of a point we could consider hyperplanes for
instance.

Following Thom we wish to capture the primordial intuition of space before it
receives any further order-theoretic or algebraic determinations. We wish to
find a formulation of these basic properties which does not involve an appeal
to points (as in the standard definition of topological space).

In modern mathematics there are two (closely interrelated) ways in which this
might be achieved. The order-theoretic one (in terms of locales) and the
category-theoretic one, in terms of Topos Theory (in particular Grothendieck
Toposes and Synthetic Differential Geometry). There is furthermore also an
algebraic approach in which points correspond to prime (or in particular
maximal ideals), used in algebraic geometry and functional analysis.

The first two approaches are essentially relational. We start from open sets
or objects in a category and consider the relationships between them. In
Category Theory itself we find in a sophisticated way some of the primordial
Aristotelic concepts: that of limit (and colimit) and of adjoint functors
where the left adjoint can be seen as expressing a vast generalisation of the
idea of an open interior whilst the right adjoint is seen as a kind of closure
as seen in the case of adjoint functors between posets. All the constructions
we discussed to capture the notion of topos and infinitesimal in the last
section involve limits.

Inspired by the characterisation above we propose the following definition of
an _Aristotelic Locale_ L. A locale is a complete distributive lattice (which
hence has a top element $1$ and bottom element $0$) and is way of defining
topology without points (see chapter II of [3] for an introduction). Since
there are no points there is no obvious way to define density or closure. We
might attempt to capture a point-free description of the Aristotelic
properties of the standard topology of Euclidean space by the following axioms
(the quantifiers are to be understood as having scope $L$):

$\sim\exists u,v.1=u\vee v\&u\wedge v=0$ $1=\bigvee\\{z:\sim\exists
u,v.z=u\vee v\&u\wedge v=0\\}$ $\forall u\exists w,v:w\wedge v=0\&\forall
z.w\vee v\leq z\rightarrow z=u$

Local connectivity means that we can cover any open set with connected open
sets.

Sheaves are defined in terms of the (opposite) category of open sets of a
given topology. The infinitesimals we considered in the last section do not
naturally belong to this category although the associated stalks belong to the
target category.

Grothendieck’s philosophy[26] is that the category of sheaves on a space gives
a more fundamental description of a space than its topology. A Grothendieck
topology is precisely a way of defining a ”topology” on an arbitrary category
without any need of an underlying set of points. The underlying concept is
that of ”cover”, another intuition related to continuum which can be traced to
many places of Aristotle’s _Physics_ , as the idea of measuring a given
interval by a smaller one clearly corresponds to a cover. We can define
sheaves on categories with a Grothendieck topology (called _sites_) and we
obtain again infinitesimal concepts which allow us to define the germ of a
sheaf. The category of sheaves over a site is called a _Grothendieck Topos_
and satisfies a remarkably small and elegant set of axioms. The more general
concept of Topos was discovered independently by Lawvere where it is seen as
defining an intuitionistic set theory without points. This also allows us to a
have temporal or spatially localised concept of truth and to give meaning to
such sentences as ”The cat is black and white” and furnish a rigorous basis
for the relativized form of the principle of non-contradiction. The suboject
classifier object of a topos has the structure of a Heyting algebra which can
be interpreted as giving the possible logical values of propositions. In fact,
the poset of subobjects $Sub(A)$ of an object $A$ in a category can have
diverse properties depending on the type of category. In the cases of a topos
these posets are Heyting algebras. We can thus think of $Sub(A)$ as a kind of
generalised space. In[29] there is the concept of a molecular (also called
locally connected) topos. This means that every object E of E can be written
as categorical sum $E=\Sigma M_{i}$ where $M_{i}$ is a molecule (or connected
object) meaning that $M_{i}$ cannot be written as a categorical sum of two
proper subobjects. This is the analogue of our Local Connectivity Axiom. There
is also an analogue of the Global Connectivity Axiom. Interestingly[31] we can
characterise connected objects in a topos $T$ in terms of their associated
hom-functor $hom_{T}(A,-)$ preserving finite coproducts (this appears to have
been first discovered by Grothendieck in [26]). And it turns out that the
topos of sheaves over a topological space $X$ is locally connected if and only
if $X$ is locally connected in the classical sense.

In order to obtain a complete formulation of the Aristotelic continuum it
would be interesting to extend likewise to topoi the Divisibility Axiom above.

We noted how infinitesimals do not in general belong to the base category on
which sheaves are defined. But in a topos theoretic framework we may wish for
infinitesimals to actually belong to the topos. This is precisely the road
taken by Lawvere’s Synthetic Differential Geometry[32]. Lawvere, in his quest
for the adequate categorical foundations for continuum mechanics, was
influenced by Grothendieck’s project of a ”tame topology” which seeks to avoid
the unintuitive paradoxes of standard topology (such as space-filling curves
and the Banach-Tarski paradox)666There is also a model-theoretic approach[25]
. In [35] Lawvere gives an extraordinarily interesting category-theoretic
philosophical analysis of space and allied concepts employing an
interpretation of the concepts of Hegel’s Logic.

There is an interesting and promising development of category theory in
relationship to philosophy and logic which has the most wide-ranging
implications for understanding the Aristotelic concept of topos (and to
theoretical physics in general) : _homotopy type theory_ and _higher category
theory_ , in particular _higher topos theory_. Homotopy type theory, an
extension of the constructive type theory of Martin-Löf, was first proposed by
Voevodsky as an alternative foundations for mathematics suitable for computer-
assisted formalisation and verification. For each type there is a whole
hierarchy different levels of ”equality” analogue to the series of homotopy
groups $\pi_{i}(X)$ of a topological space777 The linearly orderered real
interval $I$ plays a central role in homotopy theory. The concept of
connectivity associated to it is pathwise connectivity (which can also be
extended to Topos Theory[30]). This seems particularly relevant to Thom’s
Semiophysics, the attempt to described the topological architecture of
semantic spaces based the Aristotelic theory of genera and species wherein two
species belong to the same genus if they can be continuously deformed into
each other[6][p.199]. The divisibility property of the continuum is what
allows the division of the genus into species, the boundary of such a division
corresponding to the _difference_ which allows a _definition_. . The adequate
model for homotopy type theory involves higher category theory in which we
have not only morphisms, but morphisms between morphisms and then morphisms
between these and so forth. For equality types all such morphisms are
invertible and we have what is called an $\infty$-groupoid. Models of homotopy
type theory are a higher-order analogue of toposes called homotopy toposes or
$(\infty,1)$-toposes. In [36] Urs Schreiber uses higher toposes to formalise
quantum field theories. It would be very interesting to analyse the crucial
property of being ”cohesive” in terms of the Aristotelic continuity concepts
analysed in this paper.

##  Topos as a Quotient

In Aristotle’s _Categories_ a division is made between discrete and continuous
quantity. If we visualize numbers - discrete quantity - in a spatial way, then
spatiality is related to divisibility, to the formation of quotients involving
equivalence relations. If we take $0$ and $1$ then there is a space between
them and we would like to name the ”points” in this space and are thus forced
to ”invent” $1/2$ with the additional caution that $1/2\cong 2/4\cong 4/8...$.
This abstract construction is often a stumbling block for children learning
mathematics. Modern algebraic geometry springs from Descartes’ theory of
studying geometrical figures by means of algebraic equations defining them.
These form an ideal in a commutative ring and we can consider the behavior of
such functions in a small vicinity of a point - the topos of the point - and
this is done by entirely analogous quotient constructions such as localisation
and completion. There is also a similar concept of localization of a category.
A full subcategory $i:D\rightarrow C$ is a localization if there is a left
adjoint $r$ to $i$ preserving limits. If furthermore $r$ is left exact then
giving a localization is the same as giving a _calculus of fractions_ , a
construction analogous to the construction above where the morphisms of the
category are replaced by equivalence classes of certain pairs of morphisms
$(f,g)$. We discussed in the last section how the subobjects $Sub(A)$ of an
object $A$ can be seen as a generalized space. It turns out that giving a
localization on a category is the same as giving a _universal closure
operator_ on $Sub(A)$ for every object $A$. This closure operator is a
generalisation of the closure operator of ordinary topology. For a detailed
discussion of the topics of this section see [3][Ch.7]. We note that the
inverse image functor we considered in the previous section has a powerful
expression in the context of derived categories of Abelian sheaves and these
are given by a localization of the homotopy category of complexes [16][ch.1].
In fact modern homotopy theory is centered around the interplay between
different kinds of simplicial structures and model categories888The notions of
fibration and co-fibration can be seen as generalisation of what it means for
a space to be respectively ”inside” and ”over” another space.. The latter
”model” a homotopy theory which can be defined as a category with a class of
morphisms - weak equivalences - we wish to take as isomorphisms. We form the
homotopy category by localising such categories at weak equivalences. Certain
classes of models form themselves a category and have a natural notion of weak
equialence. Thus they can be seen as a ”homotopy theory of homotopy theories”
[1] [p.82] which can be seen as the most complex evolution of the ancient
concepts of topos and space to date.

The category $Set^{C^{op}}$ of presheaves over a category $C$, which is simply
the category of functors from $C^{op}$ to $Set$ which are not required to
satisfy any ”gluing conditions”, can be seen as a sort of _materia prima_.
Given a category $C$, endowing $C$ with a Grothendieck topology is the same as
defining a universal closure operator in $Set^{C^{op}}$ and the associated
localization is precisely the Grothendieck topos, the category of sheaves on
$C$.

Topos theory and sheaf theory are rich enough to generalise and unite most of
our classical spatial intuitions. Besides quotients, associated to the
rational naumbers, there is also the classical concept of defining the real
line by a process of ”completion”. The construction of the reals via Dedekind
cuts on the rational numbers can in fact be generalised to the topos of
sheaves over a topological space $X$. We can also view a topos as a model for
higher-order intuitionistic logic and in this topos we formulate all the
standard concepts of the differental calculus. The real numbers in this topos
correspond to the sheaf of continuous functions over $X$ and this topos is a
model for intuitionistic analysis in which all functions are
continuous[28][p.324], thus furnishing a model which comes close to the
Aristotelic theory of change wherein continuity is omnipresent. In view of
what we discussed about local continuity in Aristotelic Physics and
Catastrophe Theory it would be interesting to extend these constructions to
the case of merely locally continuous functions.

##  Conclusion

We have seen how Sheaf Theory (or more generally, Topos Theory) captures the
kindred concepts of topos and infinitesimal as well as furnishing a framework
for defining the topology of the continuum in a point-free Aristotelian way

This illustrates how ancient philosophy holds insights which are only recently
being clarified in the light of modern knowledge.

It invites us to reconsider the essence of philosophy and its relationship to
mathematics and natural science.

Until the days of Husserl it was normal for a philosopher to have a good
knowledge of mathematics andl physics and to not only reflect upon the
foundational questions of mathematics and physics but to see these questions
as fundamental to epistemology, ontology and the philosophy of mind as they
are for the philosophy of nature.

The subsequent divorce of science and mathematics from philosophy has had the
tragic consequence that the independent technical development of the latter
has often led to it missing out on a valuable enriching interplay with the
philosophical thought of the past.

Topological and geometric concepts have shown themselves to be potent factors
in the progress of physics. It is true that Thom held formal logic in small
regard and once wrote that the main goal of Catastrophe Theory was to
”geometrize thought and linguistic activity” and in his _Esquisse_ he wrote
that the logical was derivative of the geometric. The geometric and the
topological, with their extension to spatial-temporal processes brought about
by the revolutions of Poincaré and Einstein, allows us to encompass many
fundamental philosophical categories such as causality and possibility (phase-
space), and furnish a powerful foundation for our rationality which is
mirrored equally in the structure of the mind and in natural science.

It is difficult to think of a domain in which we are given as much power at
once to construct models for the natural world and to shed light on the
working of our own mind _and which at the same time yields so readily to
formal axiomatic treatment_ 999We could also mention the axiomatic treatments
mereology.. This can be seen in the recent developments of homotopy type
theory inspired by ideas of Lawvere. Modal homotopy type theory has been
proposed by Corfield[37] as a refinement of the (multi)modal logics usually
considered by philosophers and the concept of cohesive higher topos is used by
Urs Schreiber to axiomatise certain aspects of theoretical physics.

Aristotle the topologist and phenomenologist is also Aristotle the formal
logician101010His disciple Theophrastus developed the basis for propositional
logic. and we find therein a perfect harmonisation and mutual enrichnment
between proof theory and semantics. We find a promising domain for unveiling
”a genealogy of the understanding” with applications not only to natural
science but to cognitive science and the formalisation of natural language.

Experimental data in physics has been an occasion for the both the deployment
and refinement of our geometric-dynamic concepts and there are currently many
active areas of investigation which propose that the classical model of space-
time collapses at the Planck scale and a new concept of space is called for.
Topos Theory has been applied to theoretical physics (cf. [33]). Zeno’s
paradoxes of motion addressed by Aristotle in the _Physics_ retain their
relevance today: can space and time by ultimately discrete ?

The movement towards abstraction and axiomatisation described in this essay is
also at once a movement towards logical clarity (we might call it ”logical
phenomenology”) and a movement towards a progressive vision and intuition of
the pure essence of our primordial topological-dynamic intuitions perceived so
keenly by Aristotle and Thom.

## Appendix

We give here some definitions of a few important concepts in modern topology.

Let $X$ be a set. Then a _topology_ over $X$ is a collection $\tau$ of subsets
of $X$ such that $X$ and the empty set belong to $\tau$, the union of any
collection of sets in $\tau$ belongs to $\tau$ and the finite intersection of
sets in $\tau$ is still in $\tau$. The elements of $\tau$ are called _open_
sets. A _closed_ is the complement of an open set. A _basis_ for a topology
$\tau$ is a collection $\mathcal{B}$ of subsets of $X$ such that every open
set can be expressed as a union of sets in $\mathcal{B}$. A basis for the
standard topology of the real line is given by the open intervals $(a,b)$.
Given a point $x\in X$ a _neighbourhood_ of $x$ is an open set containing $x$.
Given a subset $Y$ of $X$ the interior of $Y$ is the set of all points that
have a neighbourhood contained in $Y$. The closure of $Y$ is the intersection
of all open sets containing $Y$. It is always a closed set. The correspondence
which associates to each subset $Y$ its closure is an example of a _closure
operator_. For instance, the closure of the open interval $(a,b)$ is the
closed interval $[a,b]$. The _boundary_ of a set is the set of all points $x$
that satisfy the condition: any neighbourhood of $x$ will contain points both
in $Y$ and not in $Y$. There is a general concept of _compactness_ but in the
case of the ordinary topology on the real line (or Euclidean space $R^{n}$) it
corresponds to being both closed and bounded (all points at a finite distance
from the origin).

A _category_ can be thought of as a collection of _objects_ and for any two
objects (possibly equal) a (possibly empty) collection of _arrows_ (or
morphisms). Each arrow has a source and a target. If the target of an arrow
$f$ is equal to the source of an arrow $g$ then we can compose them to form a
new arrow $g\circ f$ which will have the source of $f$ and the target of $g$.
Given an object $A$ there is a unit arrow $u_{A}$ which goes from $A$ back to
$A$. The formal definition of category requires an associative condition on
composition and that the unit arrows function as units for composition, for
instance $f\circ u_{A}=f$ when $f$ has source $A$. Given two categories $C$
and $D$ a _functor_ is a correspondence between objects of $C$ and objects of
$D$ and arrows of $C$ and arrows of $D$ which preserves composition and unit
arrows. Examples of categories are the category $Set$ of sets whose objects
are sets and morphisms are maps between sets and the category $\mathcal{O}(X)$
of open sets of a topology on $X$ whose objects are the open sets for any two
open sets $U$ and $V$ there is either a single morphism if $U$ is contained
and $V$ or no morphism at all if otherwise. In the category $Set$ we can form
the _cartesian product_ $X\times Y$ of two sets $X$ and $Y$ as well as the
_disjoint union_ $X\sqcup Y$. The properties of these constructions can be
generalised to obtain the concepts of product and coproduct for any category
(or more generally limit and colimit). A _Topos_ can be understood as a
category which possesses generalisations of many fundamental properties of
$Set$.

Given a category $C$ we can form the _opposite category_ $C^{op}$ which is
obtained by inverting the direction of all the arrows. A _presheaf_ is simply
a functor $F$ from $C^{op}$ to $Set$. Given an object $A$ of $C$ the elements
of $F(A)$ are called _sections_. If $U\subset V$ then this arrow corresponds
to an arrow $F(V)\rightarrow F(U)$ (note that source and target are switched
because we are considering the opposite category) called the _restriction_
map. Presheaves can be made into a category with a suitable definition of
morphism (natural transformations between functors). If $C$ is the category of
open sets of a topology on a space $X$ then a _sheaf_ over $X$ is a presheaf
on $C$ which satisfies the _gluing_ condition. This says that if we express an
open set $U$ as a union (cover) of open sets $U_{i}$ and we consider sections
$s_{i}$ on each $U_{i}$ such that they agree on all the intersections
$U_{i}\cap U_{j}$ then there is unique section on $U$ such that the sections
$s_{i}$ are given by the restrictions corresponding to the inclusions
$U_{i}\subset U$. An example of a sheaf is the correspondence which associates
to each open set $U$ of a topological space $X$ the set of _continuous
functions_ defined over $X$. An example of a presheaf which is not a sheaf is
given by constant functions on open sets. Considering two disjoint open sets
it is easy to see why the gluing condition fails. The concept of
_Grothendieck_ topology is a generalisation of the concepts involved in this
property to an arbitrary category. The idea of covering $U$ by the open sets
$U_{i}$ becomes the concept of _sieve_.

## References

  * [1] J. Bergner, _The Homotopy Theory of $(\infty,1)$-Categories_, Cambridge, 2015.
  * [2] R. Hartshorne, _Algebraic Geometry_ , Springer, 1977.
  * [3] M. C. Pedicchio, W. Tholen (Eds.), _Categorical Foundations: Special Topics in Order, Topology, Algebra and Sheaf Theory_ , Cambridge University Press, 2004.
  * [4] P. Pellegrin (tr.), _Aristote, Physique_ , GF Flammarion, 2002.
  * [5] R. Waterfield, _Aristotle, Physics_ , Oxford University Press, 1996.
  * [6] R. Thom, _Esquisse d’une Sémiophysique_ , InterÉditions, Paris 1991.
  * [7] R. Thom, _Prédire N’est Pas Expliquer_ , Flammarion, 2009.
  * [8] R. Thom, _Biologie Aristotélicienne et Topologie_. Conférence à l’Université de Coimbra, 14-15 août 1995. Boletin da Sociedade Portuguesa de Matemática, 32, Agosto 1995, pp. 3-14.
  * [9] R. Thom, _Ensembles et morphismes stratifiés_ , Bull. Amer. Math. Soc. 75(2): 240-284 (March 1969).
  * [10] J. Kelley, _General Topology_ , Van Nostrand, 1955.
  * [11] G. W. Hegel, _Enzyklopädie der Philosophischen Wissenschaften_ , Felix Meiner Verlag, Hamburg, 1969.
  * [12] W. D. Ross, _Aristotle’s Physics, A Revised Text with Introduction and Commentary_ , Oxford,1936.
  * [13] L. Robin, _Aristote_ , Presses Universitaires de France, 1944.
  * [14] J. Petitot, _Sheaf mereology and Husserl’s morphological ontology_ , Int. J. Hum. Comput. Stud. 43(5-6): 741-763 (1995).
  * [15] R. Godement, Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris, 1959.
  * [16] M. Kashiwara, P. Shapira, _Sheaves on Manifolds_ , Springer, 1990.
  * [17] Plato, _Parmenides, Translated with an Introduction and Commentary by Samuel Scelnicov_ , University of California Press, 2003.
  * [18] R. Thom, _Quid des stratifications canoniques. Quid of canonical stratifications_. Singularities. London Math. Soc., Lecture Note Ser. 201. Édition du colloque de Lille de 1991, par J.-P. Brasselet. Cambridge Univ. Press, Cambridge, 1994, pp. 375-381.
  * [19] R. Thom, _L’antériorité ontologique du continu sur le discret_. Le labyrinthe du continu, Colloque de Cerisy, 11-20 sept. 1990, J.-M. Salanskis et H. Sinaceur éd., Paris, Springer, 1992, pp. 137-143.
  * [20] R. Thom, _Aristote Topologue_. Conférence tenue au mois de mai 1996, ENS Ulm, séminaire Ch. Alunni. Revue de Synthèse, t. 120, 4ème série, 1, janvier-mars 1999, Paris, Albin Michel, pp. 39-47.
  * [21] R. Thom, _Topoi de l’Utopie_ (Biennale Cascais, Portugal, 11-12 août 1995).
  * [22] R. Thom, _Qualitatif et quantitatif dans les sciences de la nature_. Conférence à la Maison franco-japonaise, Tokyo, 12 oct. 1977. Bulletin de la Société franco-japonaise des Sciences pures et appliquées, 28, Mai 1978, pp. 34-53. (Voyage au Japon, 7-23 octobre).
  * [23] R. Thom, _Aristoteles redivivus_ , Paper presented September the 5th, 1996, for the meeting ”Cartesio et la Scienza”.
  * [24] J. Petitot, _Apories Fondatrices et Dialectique Mathematique: L’Exemple Du Structuralisme Catastrophiste_.
  * [25] L. van der Den Dries, _Tame Topology and O-minimal Strutures_ , London Mathematial Society Lecture Notes, Cambridge University Press, 2003.
  * [26] M. Artin, A. Grothendieck, J-L. Verdier, _Theorie de Topos et Cohomologie Etale des Schemas I, II, III_ , Lecture Notes in Mathematics, Springer, 1971.
  * [27] Marquis of L’Hôpital, _Analyse des Infiniment Petits_ , 1768.
  * [28] S. MacLane, I. Moerdijk, _Sheaves in Geometry and Logic: A First Introduction To Topos Theory_ , Springer, 1992.
  * [29] M. Barr, R. Diaconescu, _On locally simply connected toposes and their fundamental groups_ , Cahiers de topologie et géométrie différentielle catégoriques, tome 22, no.3 (1981), p. 301-314.
  * [30] I. Moerdijk, G.C. Wraith, _Connected Locally Connected Toposes Are Path-Connected_ , AMS Vol.295, No. 2, June 1986.
  * [31] https://ncatlab.org/nlab/show/locally+connected+topos
  * [32] https://ncatlab.org/nlab/show/synthetic+differential+geometry
  * [33] C. J. Isham, _Topos Methods in the Foundations of Physics_ , https://arxiv.org/abs/1004.3564.
  * [34] J. W. Vick, _Homology Theory: An Introduction to Algebraic Topology_ , 2nd Ed. , Springer.
  * [35] F. W. Lawvere, _Some Thoughts on The Future of Category theory_.
  * [36] Urs Schreiber, _Differential cohomology in a cohesive infinity-topos_ , https://arxiv.org/abs/1310.7930v1
  * [37] David Corfield, _Modal Homotopy Theory: The Prospect of a New Logic for Philosophy_ , Oxford University Press, 2020.