73
Filozofski vestnik | Volume XLII | Number 1 | 2021 | 73–94 | doi: 10.3986/fv.42.1.04
* Marie Skłodowska-Curie Fellow. Department of Politics, University of York, United Kingdom 
Uroš Kranjc*
Magnitudes in Badiou's Objective Phenomenology 
and Economic Consumer Choice
1
Since the dawn of human civilization in Egypt and Mesopotamia, man has been 
concerned with measuring different quantities as part of his quest for contin-
ued existence and progress. By antiquity, humanity had already created numer-
ous measuring systems, based on various units of standard, for concepts like 
length, surface, volume, mass, and time. Not much changed until the 4
th
 century 
BC when Euclid in his Elements proposed the first systematic contribution to 
measurement by introducing the idea of ratios between different homogenous 
(numerical) magnitudes of attributes –
1
the measure of one magnitude rela-
tive to another. This rekindled interest in measurement theory, since thinkers 
now needed different generalisations of qualities to quantities to cover the vast 
landscape of measurable phenomena, for example Aristotle’s work on temper-
ature. He observed it to be a qualitative phenomenon that can be described in 
quantitative intervals of hotness and coldness. Progress made in the Middle 
Ages demonstrated that certain objects can attain new homogenous additional 
parts. This means they can be lengthened or made heavier – supplied with an 
extensive attribution. It follows that we can then have “extensive magnitudes” –  
quantities of parts observed and ascribed in numerical form and used to per-
form basic operations, such as addition, subtraction, and multiplication. On the 
other hand, the human capacity for sense-perception and intuition also allows 
for interaction with external objects. This in turn gave birth to a set of “inten-
sive magnitudes”, which related various qualitative attributes to determinate 
objects. The concept was introduced as far back as the late 13
th
 century by John 
Duns Scotus and stood the test of time, still being considered valid during the 
scientific revolutions of the 16
th
 and 17
th
 century and influencing the philoso-
phers of the time. We see traces of it in Leibniz’s assertion that “nature never 
makes leaps”, pointing to the natural change occurring in degrees (The Prin-
1
 This project has received funding from the European Union’s Horizon 2020 research and 
innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 893027.

74
uroš kranjc
ciple of Continuity), a fact also later picked up by Kant in his Critique of Pure 
Reason as the Axioms of Intuition and the Anticipations of Perception.
On this point, early modern philosophy has been, in a very general sense, di-
vided between two different lines of argument: (1) Descartes and Locke are as-
sociated with those who emphasize that secondary qualities, such as colour, 
taste or smell, vary according to the state of the experiencing subject, while the 
objects can objectively be determined solely by their primary qualities, such 
as size, shape, and weight. On the opposing side (2) are philosophers, such as 
Berkeley and Hume, who unify the primary and secondary qualities – main-
taining that the primary qualities of objects can be experienced only through 
the mediation of secondary ones – “fusing” them into an entirely subjective act 
of cognition. In his Axioms and Anticipations, Kant attempted to reconcile this 
by attributing both aspects simultaneously to the subject; he ascribes primary 
qualities as extensive magnitudes to the first, denying the exclusive subjective 
perception of the second, rather choosing to define the object’s correspondence 
(in space and time) to the subject’s sensory qualities through intensive magni-
tudes. However, later scientific discoveries, notably in physics, electrical engi-
neering, and chemistry, unambiguously undermined Kant’s proposed unified 
extensive and intensive magnitudes. This spurred the search for new funda-
mentals in the philosophy of measurement, resulting in a branch of mathemat-
ics we now call Measurement Theory. Some fields in the social sciences, how-
ever, chose to retain the distinction. One such example is the merging of the 
economic theory of utility with preference logic
2
 in the second half of the 20
th
 
century. It seems as though the issue has now once again been pushed to the 
forefront, this time in contemporary philosophy.
In his phenomenological project, Logics of Worlds (Being and Event II), Alain 
Badiou proposes an entirely new way of thinking of an object. His separation of 
the onto-logical part of the first Being and Event (1988) from onto- logy, i.e. pure 
logics pertaining to the theory of appearing – the being’s localization in a world 
– is now well established. Whereas the first project of pure being relied on the 
set-theoretic realm of (in)consistent multiplicities, the second project draws on 
2
 For more on the (economic) logic of preference, see the seminal works of Sören Halldén, 
On the Logic of Better, Lund, Library of Theoria, 1957, and Georg H. von Wright, The Logic 
of Preference, Edinburgh, Edinburgh University Press, 1963.

75
magnitudes in badiou's objective phenomenology and economic consumer choice
a broader and richer area of mathematics – category theory (CT) – to explain 
the appearance of a multiple-being in the world. Badiou’s reliance on category 
theory also required he introduce a different, purely relational conception of 
the object. This is where Badiou strays the farthest away from Kant’s concep-
tion of the transcendental object. With Badiou, we instead get a subject-less 
objectivity, an objective phenomenology of being-there. However, what centrip-
etally pulls the two back together is the immanence of “transcendental magni-
tudes” (degrees), attributes pertaining to the regimes of appearance. Although 
the transcendental subject is never present in Badiou’s conception, the void is 
filled by the ubiquitous transcendental indexing of appearing in every particu-
lar situation (i.e. a world). He concedes: “In fact my conception fuses together, 
in the guise of a general algebra of order, ‘extensive magnitude’ and ‘intensive 
magnitude’.”
3
 What is here referred to as the “general algebra of order” is actu-
ally an object of a category that complies with what mathematicians call a com-
plete Heyting algebra, meaning it is a complete lattice, i.e. a partly ordered set 
(or poset), where each of its subsets has a supremum and an infimum. It allows 
Badiou to counter Kant on the grounds that the current advances in mathe-
matics “justify enveloping the two notions in the single one of ʻtranscenden-
tal degreeʼ[…]”.
4
 This implies a mode of existence, which, in the final instance, 
relies precisely on the concept of magnitude (quantum), measuring degrees of 
identities and differences.
When Badiou expressed his philosophical dispositive in terms of mathematical 
discourse, he had in fact opened up countless nexus points to various disci-
plines in the natural and social sciences. This article will focus on the contem-
porary scientificity of economics. We will focus on the transition from classical 
political economy, overcome by the subjective “marginalist revolution” (and the 
advent of preference logic), to the contemporary doctrine of methodological in-
dividualism and rational consumer choice theory.
5
 The inculcation of subjective 
preference plans and its coextensive mathematical and logical apparatus lies at 
the centre of this epistemological break. It can also be understood as a kind of 
‘general algebra’, in this case of an economic equilibrium, resting on the opti-
3
 Alain Badiou, Logics of Worlds: Being and Event, 2, trans. A. Toscano, London, Continuum, 
2009, p. 234.
4
 Ibid.
5
 A theory that encompasses much more than simply economics. It is currently extensively 
used in sociology, the political sciences, and evolutionary theory.

76
uroš kranjc
mization functions of measured individual utilities (demand side) and marginal 
production costs (supply side) – all of which (conveniently) relies on pre-or-
dered or partly ordered sets and their formal conditions.
All of this has had significant repercussions. Once the “marginalist revolution” 
in economics had done away with the classics, i.e. with the labour theory of 
value of Smith, Ricardo, Malthus, and Marx’s critique of it, the neoclassical 
economics of Alfred Marshall and his disciples proceeded to introduce an in-
dividual preference ‘measurement’ of value, making no attempt to establish a 
clear link to the previous classical preconceptions. The maximizing preference 
function simply became a universally established law of axiomatically defined 
ordered sets. We argue that the inserted discontinuity between the classical and 
neoclassical approach in economics rests on a fallacy, since it failed to convinc-
ingly (in)validate any synthesis of the labour theory of value and the subjectivist 
valuations. We will examine the classical period in political economy through 
the lens of Badiou’s definition of a classical world – a world where a classical 
logic registers transcendental values, i.e. in the world of ontology. Put in terms 
of political economy, value must have some substance, or as Marx put it, there 
first has to be a value-forming substance – labour – in the form of “socially 
necessary labour-time” that has the capacity to create value. This fact (the ob-
jectification of human labour) is borne out each time merchandise arrives at 
the market for exchange in a Boolean-type logic evaluation. On the other hand, 
we assert that the current scheme for the subjectivist valuation of price-goods 
in terms of individual preference plans conforms to intuitionist logic – the log-
ic that also underpins Badiou’s theory of appearing. As this scheme is purely 
subjectivist, it has no ‘primordial substance’, meaning that its transcendental 
values, i.e. the ‘existence’ of goods on the market, represent an immediate func-
tion of appearing in a market-economy world. Put differently, the ‘price tag’ is 
the sole recognition of an object’s existence in a world of commodities and it 
obeys the same (mathematical) structural, i.e. algebraic, protocols as Badiou’s 
“minimum phenomenology of abstract appearing”.
Our investigation includes the following provisional thesis that will also act as 
the proximate aim of this article: The “marginalist revolution” does not rise to 
the scale of a Copernican-type revolution, i.e. an absolute modification of the 
scientific field. The switch from classical to marginalist economics is by far and 
away closer to a shift in doctrines – it is an intra-theoretical parallax view from 

77
magnitudes in badiou's objective phenomenology and economic consumer choice
production and distribution to exchange – and rooted in a formally dominant 
utilitarian philosophical worldview. According to our interpretation, Badiouʼs 
efforts offer a better and, more importantly, monolithic solution, precisely be-
cause of his upgrading from set-theory ontology to category theory. In doing 
so, he introduces a more robust logical framework that allows him to subsume 
the ʻprimitiveʼ belonging-to-situations multiplicities to a logical and relational 
framework of categories in a world. This gesture crucially also calls for a contin-
uous, unified, and coherent bottom-up framework – one with the potential of 
serving as a more suitable transitional scheme to a new economic doctrine, in-
stead of opting for an apologetic rendition of the discipline. Put more concretely 
in the context of our present discussion: Critical economic science should seek 
to reintroduce Marx’s “dialectics of the value form” when dealing with commod-
ity exchange. The approach should ideally also consider the subjective aspira-
tions and magnitudes of desired objects by means of an immanent structure 
capable of handling the categorical breadth.
In the introductory remarks, we touch on an array of different topics. To help es-
tablish some initial common ground, we list them below as either falling under 
classical or intuitionist logic: 
Characteristic
 property
Algebra
type
Truth 
values
Badiou Marx Economics
Classical 
Logic
Law of identity
Law of non-
contradiction
Law of excluded middle
Boolean
algebra
[0,1]
(binary)
World
of
ontology
Value 
Form
Labour
theory
of
value
Intuitionist 
Logic
Law of identity
Law of non-
contradiction
Heyting
algebra
Ω – set
(poset,  
pre-order)
World
of 
phenomena
Price 
Form
Marginal 
utility
Badiou's algebra of appearing: a brief overview
Before delving into our topic, let us briefly but no less thoroughly examine Ba-
diouʼs trajectory, connecting the first two volumes of the Being and Event pro-
ject. The first volume was entirely devoted to indifferent multiplicities without 
any particular designated qualities. It focused on the internal composition of 
belonging elements that come to count-as-one, consequently forming sets of 

78
uroš kranjc
consistent multiples in different situations. The set-theoretic connectors that 
constitute a non-relational framework, or rather situations and state-of-situ-
ations, are those of belonging ( ∈) and inclusion ( ⊆). For Badiou, this setting 
represents the entire world of ontology – everything proceeding/halting from/
on the null-set, the void ( ∅). The second volume moves from being to beings, 
i.e. being-there, their localized appearing in a world. Indeed, if multiples are 
different beings, they must appear differently – hence, we are no longer in the 
world of ontology of indifferent beings, but rather in a world or worlds where 
beings are relationally brought together. For Badiou, worlds are logical rep-
resentational spaces, or rather topological spaces, where different beings logi-
cally relate to one another based on their identities and differences expressed 
in transcendental degrees. Beingʼs appearing is sutured to a determinate logic 
of a particular world. If ontology was built from the two primitive connectors 
mentioned above, phenomenology now rests on just one ordering function, (≤). 
The step from ontology to phenomenology implies a tremendous upheaval in 
the underlying mathematical apparatus from axiomatic set-theory to concep-
tual category theory.
In simplified terms, appearance is a function determined according to a set of 
operations (an algebra) that constitute a transcendental. Say we have two mul-
tiples, a and b, in order for them to appear in a world, we also need a (differen-
tial) relation between them – so we can distinguish between them –, which also 
connects them to a third multiple (more on this multiple later on), the transcen-
dental (T), to formalize a world. “We will call ‘transcendental’ the operational set 
which allows us to make sense of the ‘more or less’ of identities and differences 
in a determinate world”
6
 Badiou says, and puts it formally as a function of ap-
pearance Id (a, b) = p; interpreted as multiplicities {a, b} ∈ A are to a p-degree 
identical, p ∈ T. The degree of identity, or inversely, difference, of two multiplic-
ities in a world is measured according to the degree of belonging to transcen-
dental algebra. It is worth noting here that the type of identity inferred here is 
a mathematical (CT) one – isomorphism – retaining both quantities in Set and 
the identity of structures (determinate algebraic operations). The transcenden-
tal algebra Badiou utilizes is the complete Heyting algebra that was designed as 
a model for intuitionistic logic (the law of excluded middle does not hold) and 
posits a specific type of Heyting-valued sets, known also as the Ω-valued sets. If 
6
 Ibid., p. 118.

79
magnitudes in badiou's objective phenomenology and economic consumer choice
we look closer, we see that Heyting-valued sets “regard an object in a topos as 
a ʻset-likeʼ entity consisting of potentially existing (partially defined) elements, 
only some of which possess actual existence (are totally defined).”
7
 Such a Ω-val-
ued set is then a set, say A, defined as a pair (A, Id), where, as we have seen, 
Id is a function of assigning degrees of identity between every pair of elements 
a, b ∈ A; Id(a, b) ∈ Ω. As stated, the Ω-set is a partially ordered set (poset), an 
order structure where the order connector ≤ organizes a transcendental scale 
(called a locale in mathematics) obeying certain ‘classical’ axioms, such as: x ≤ 
x reflexivity; [(x ≤ y) and (y ≤ z)] → (x ≤ z) transitivity; [(x ≤ y) and (y ≤ x)] → (x = y) 
antisymmetry. If it were simply a matter of Boolean algebra, we would encoun-
ter only non-existing or absolutely existing objects; however, dealing with an in-
tuitionistic logic, we get a minimum ( x) (µ ≤ x) or the greatest lower bound and 
a maximal element or envelope (∑), defined as a set of degrees B, where B ⊆ T –  
i.e., the smallest of all degrees that are greater than or equal to all the elements 
of B. Given the p and q degrees, we can always find a conjunction (∩), written p 
∩ q, so that it is the greatest of all those that are lesser than both p and q. Lastly, 
we have distributivity for attaining the completeness of a lattice, i.e. that all 
partial local conjunctions are in relation to the global envelope, making sure 
there are no gaps in the degrees scale via the infinite distributive law. Badiou 
puts it like this: d ∩ ∑B = ∑{d ∩ x / x ∈ B; take d as a degree in conjunction with 
the envelope of a subset B; you have an envelope of a given degree in regards 
to an element x in B to which that degree from the envelope of degrees is as-
signed. For a simple example of how the transcendent scaling/indexing works, 
let us imagine a topological space S and a set of elements (in CT, this can be any 
mathematical object or structure; for our purposes, let us say these are func-
tions) A, we can therefore determine to what extent a certain function exists in 
this space S. If we have two objects, say M, N ⊆ S, where f ∈ M and g ∈ N, we 
need a Ω-set to evaluate a pair of functions from subsets M, N, with the function 
Id(x, y) being the degree of their identity/difference.
8
 To recap the transcenden-
tal properties, we have an ordering structure obeying reflexivity, transitivity, 
antisymmetry, and distributivity; we then have the operations of conjunction, 
envelope, dependence, reverse and the minimal and maximal elements.
7
 Robert Goldblatt, Topoi: The Categorial Analysis of Logic, Mineola, NY , Dover Publications, 
2006, p. 274.
8
 Provided that they meet the two conditions: Id(x, y) = Id(y, x) (symmetry) and Id(x, y) ∩ 
Id(y,z) ≤ Id(x, z) (triangular inequality).

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uroš kranjc
These are the transcendental operations of different elements related to their 
degrees of appearances when measured against each other. But how does an 
object actually come to fully exist? Or put differently, how does a phenomenon 
exist in a world? The answer is already partially contained in the preceding 
example above; once you have two elements, x and y, adding a third, z, for 
example in relation to y, you also get its relation to x, and so on, for all the el-
ements in a given set – that is to a subset of a given world. On the other hand, 
for an element to properly appear, it has to have self-identity in a given space; 
put in Badiou’s worlds of appearing, the function Id(x, x) does not necessarily 
return a Maximum value. As he says, the phenomenon is the set of the values 
in the function of appearing Id(a, x), for x that (co)appears along all the “a” in 
a set A. What does this operation accomplish? It ranks the ʻinternalʼ elements 
of a count-as-one set according to their individual transcendental degrees. So, 
if in the context of our function we substitute Id(x, a) for a an element x, we 
get Id(x, x), which measures the self-identity or the degree of existence of an 
element in a world. The more the element assumes the identity with the world, 
the more strongly it appears in it. We will not delve deeper into other algebraic 
operations of localizations, regionalizations, or atomic logic, as that would far 
exceed the scope of this paper and is not relevant in terms of the argument 
made in the present discussion. We do, however, have to modestly introduce 
certain key topological aspects of Badiouʼs objective phenomenology.
The first problem emerges directly from our preceding discussion: How does 
Badiou connect objects with the transcendental of a world? Specifically, how do 
all the elements as subsets come to be adjoined with their respective degrees? 
Here, Badiou introduces a CT concept called a functor – a transformational op-
eration preserving both elements and object-structures from one category to 
another. Badiou formally expresses it as: “The ‘transcendental functor’ of the 
object (A, Id), written FA, associates to every element p of T the part of A com-
posed of x’s such that Ex = p. That is, FA(p) = {x/x ∈ A and Ex = p}”
9
 To aide us 
in grasping the immense operationability of the functor, consider the following 
diagram:
9
 Badiou, Logics of Worlds, p. 289.

81
magnitudes in badiou's objective phenomenology and economic consumer choice
We have a category C: a → b with an arrow f. A functor is a function that assigns 
to objects a and b an operation F(a) and F(b), but also does the same for arrows, 
in our case F (f): F(a) → F(b), creating a category D. The remarkable thing about 
functors is that they also introduce natural transformations between the func-
tors themselves, creating yet again a new category with functors as objects as 
well. Say we now have two categories acting as objects – C and D – and a functor 
F between them, which is actually an arrow that maps every object of category 
C to some F(A) in D object. We now need an arrow to go from one functor to 
another, while retaining the structure and mapping of C in D. We need to assign 
an arrow to every object a of C that leads to D, incorporating the F mapping of a 
to G mapping of a. This is expressed as τ
a
 : F(a) → G(a) and τ
b
 : F(b) → G(b). This 
operation becomes the driving motor of ʻtranslatingʼ various degrees of appear-
ance, gathered into subsets, to correspond to different enveloped subobjects of 
phenomena and their atoms of appearing, now also connecting them at the on-
tological level. This is crucial, since it transforms the ontological and phenome-
nological into a unified categorical structure.
Category theory therefore allows for the retroactive positing of logical relations 
(appearing) onto indifferent multiplicities as objects (ontological strata) in the 
form of a real synthesis of atoms.
Badiou concludes his elaboration of the object in Logics of Worlds by stating 
that a structure of sheaves (a manifold of transcendental functors FA) coming 
from transcendental T to objects of the (A, Id) type is called a Grothendieck to-
pos
10
. The latter is also the topologically defined site of being-there, a world. In 
Mathematics of the Transcendental, he provides a formal definition of a topos, 
seeing it as a “possible mathematical universe, which is both ‘big’ (existence of 
10
 Ibid., p. 295. 

82
uroš kranjc
limits) and centred, and which presents its own internal logic.”
11
 We will, for our 
part, skip the segments dealing with the existence of limits and co-limits that 
determine the ‘adequate size’ of a category – an universal object, which is either 
itself visible from every object in a category (limit) or is the one you can see from 
the farthest distant objects (co-limit). Instead, we will focus on another point, 
i.e. its centeredness, meaning that every topos, or world as it were, needs to have 
a central object. In CT, this central object is known as the subobject classifier or 
truth-value object. Badiou’s version is presented in the diagram below: 
What we have is an object C which has an element (1 → C) marked T(rue) – what 
Badiou calls an evaluation procedure –, such that for every monomorphism  
f : a ↪ b there exists a unique characteristic arrow c(f) from b → C (centring) for 
the square to be a pullback. The centration (predicate characteristic) function 
c(f) is of prime importance for Badiou: in essence, it validates the values (de-
grees) found in C; p ∈ C for every subobject a of a set b. As already indicated for 
sets, the logic is Boolean (bivalent, pertaining to a well-pointed topos), C = [0,1]. 
At the level of ontology, it simply verifies whether an element belongs or does 
not belong to a subset. What makes toposes truly remarkable is their versatility, 
which Badiou uses to full effect. Within them, the central object C can modify 
its “own internal logic”; instead of being just a two-element set for [True, False], 
it can be a Ω-set, e.g. a Heyting lattice, obeying intuitionistic logic and having 
the range of intensities p between minimum to maximal elements. Given two 
elements of a (x,y), which are also the elements of b via the monic arrow f, they 
are identical in b inasmuch as in a to the degree assigned by being an element of 
c(f) in C, i.e. some p ∈ Ω. What we have arrived at is an evaluative structure for 
ʻmeasuringʼ appearance in a world through assigning truth-values to elements, 
acting as degrees or intensities of appearing in a strictly logical setting.
11
 Alain Badiou, Mathematics of the Transcendental, ed. and trans. A. J. Bartlett and A. Ling, 
London, Bloomsbury, 2014, p. 89. 

83
magnitudes in badiou's objective phenomenology and economic consumer choice
With this demonstration, we conclude this very dense introduction to Badiou’s 
objective phenomenology, which we hope will provide a minimal basis for jux-
taposing this general outline of his calculable phenomenology with the prefer-
ence logic of consumer and rational choice theory, established in mainstream 
economic theory. The next section will introduce the main tenets of this theory.    
Logic of consumer preferences in economics
We are turning to a discipline that, when it came to measuring quantities and 
intensities, was very much confined to Berkeley and Hume – Economics. 
It was the empiricism of Bacon and Locke, the writings of the 3
rd
 Earl of Shaft-
esbury on morality, and Hobbes on human nature and egoism, coupled with 
Hume’s psychological accounts of the utility principle that, among others, left 
an indelible imprint on Jeremy Bentham’s utilitarianism. Rejecting any kind of 
idealism, Bentham claimed that matter is to be subjectively (and mathematical-
ly) quantifiable and therefore experienced based on direct pains and pleasures. 
Although Bentham’s account of pains and pleasures was dismissed precisely 
because it precluded any attempt at measurement, the future course of econom-
ic theory was already settled upon and well under way. Utility was amalgamated 
as a tendency of an object to increase or decrease the degrees of happiness or 
similar feelings. This simple proposition had an immense influence on succeed-
ing generations of economic thinkers, namely Bentham’s student John Stuart 
Mill and the subjectivist marginalist revolution representatives, such as H.H. 
Gossen and W . S. Jevons, all of whom later influenced the thinking of A. Marshall 
and F. Y. Edgeworth. When Jevons wrote the preface to the first edition (1871) of 
The Theory of Political Economy, he affirmed that “In this work I have attempted 
to treat Economy as a Calculus of Pleasure and Pain, […] I have endeavoured to 
arrive at accurate quantitative notions concerning Utility, Value, Labour, Cap-
ital, etc[…]”.
12
 After an enduring search and interplay of cardinal and ordinal 
measures of marginal and total utilities, of ratios between utilities and prices, 
of further axiomatization of sets and bundles of goods (by O. Morgenstern and 
J. von Neumann), the “utility calculus” finally became a universal law of text-
book economics in the 20
th
 century. In state-of-the-art research, it became the 
12
 William S. Jevons, The Theory of Political Economy, (5th ed.), New York, Augustus M. Kel -
ley [1871] 1965, pp. vi–vii. 

84
uroš kranjc
ever-present underlying assumption of mainstream economic modelling. Ra-
tional choice theory, premised on almost intact utilitarian axioms, continues to 
be used as a general setting for modelling economic objectivity.
Modern preference logic, however, does differentiate between intrinsic and ex-
trinsic preferences. They are distinguished in the following way: “A preference 
for x over y is extrinsic if a (non-circular) reason can be given for why x is pre-
ferred to y. Otherwise, the preference is intrinsic. That x is intrinsically preferred 
to y is sometimes expressed by saying that x is ‘in itself’ or ‘for its own sake’ 
preferred to y. Judgments of intrinsic preference, or at least many such judg-
ments, express our likings. To say that we intrinsically prefer x to y is often the 
same as saying that we like x better (more) than y.”
13
 We can clearly see here the 
linguistic operation of immediate propositional predication when it comes to 
intrinsic preferences. But, just immediately after, we learn that “It seems plau-
sible to think that all extrinsic preferences are ultimately founded on intrinsic 
ones.”
14
 Consequently, we get an assembled view of preferences, defined as a 
predicative evaluation of magnitudes experienced subjectively by individuals in 
a comparative manner.
Let us now see how preference logic is reproached by Gérard Debreu through his 
set-theory axiomatic analysis, presented in Theory of Value – the still seminal 
sourcebook for mathematical fundamentals of economic analysis. In chapter 4, 
which deals with “Consumers”,
15
 the task calls for preparing the foundations 
for drawing up a complete consumption plan. Instead of dwelling on budget-
ary constraints, we will instead concern ourselves with the formal depiction of 
these functions in relation to their logical assumptions on sets X
i
. We will write 
them in italics and comment successively: (a) X
i 
is closed (the axiom of continu-
ity); an infinite sequence of consumptions, (b) X
i 
has a lower bound for ≤, (c) X
i 
is 
connected (i.e. completeness or the axiom of order), (d) X
i 
is convex (A ~ B; the 
indifference relation). Another assumption, which is not on the immediate list, 
is the condition of transitivity. What these assumptions bring in terms of ordered 
relations are of course the characteristics of a pre-order (≾). The choice of pref-
13
 Georg H. von Wright, “The Logic of Preference Reconsidered”, Theory and Decision, 3 
(1972), p. 142. 
14
 Ibid.
15
 Gérard Debreu, Theory of Value an Axiomatic Analysis of Economic Equilibrium, New Ha-
ven, CT, Yale University Press, 1971, pp. 50–59. 

85
magnitudes in badiou's objective phenomenology and economic consumer choice
erence algebra then comes down to the variant of a symmetry relation; if dealing 
with a symmetric relation A ~ B → B ~ A, we speak of the equivalence relation 
(the most usual preference relation); additionally, we either have asymmetry  
A > B → ¬(B > A) or antisymmetry A ≥ B  B ≥ A → A = B, where we consequently 
speak of a total preorder that is also a partial order (a quasi-preference rela-
tion). Total orders are mostly used in economic modelling when presupposing 
consumer functions, although different options of partial orders are becoming 
more and more common. The above relations therefore impose an algebra of a 
total or a partial order. Again, we encounter a truth-value object Ω for indexing 
preferences that could just as possibly obey, as with Badiou, a complete Heyting 
algebra.
16
 We interpret the basic preference order as follows: say we have a set 
X
i
 with the pair of elements a and b, both of which represent different combina-
tions of two goods – these two combinations are also quantitatively defined, say 
a = {1 apple, 2 pears} or b = {3 apples, 1 pear}. Applying preference relations, we 
can order different and mutually exclusive combinations of bundles of goods 
into a range between a minimal and a maximal element.
The above characteristics provide a general outline of preference logic. We are 
now able to link it with utility theory. We saw that different pairings in our X
i
 set 
combine the elements a, b, where each is initially endowed with a given quanti-
ty as different quantitative distributions obviously yield different utilities. Take 
our initial example: we had in element b a combination of goods (x, y), where 
the bundle implied a number of 3 apples and 1 pear. There is a utility function  
u : X → ℝ, for there $U(a, b) for X
i 
(a, b) ranking every bundle u: (x, y) → U(x, y), 
x ≾ y iff u(x) ≾ u(y), where U is a total utility in a consumption plan of ordered 
preferences. Decomposing the total derivative U into partial derivatives MU
x
 
= 
ΔU
ΔX
 
and MU
y
 = 
ΔU
ΔY
 
, we arrive at marginal utilities for each of the goods in a bundle – 
and to every economics student a very well known equation of: 
(Marginal rate of substitution) MRS =  
MU
x
MU
y
 ...
 
= 
 
P
x
P
y
  
...
16
 As is the case in the intuitionistic fuzzy sets application on decision theory. For more, 
see keywords intuitionistic fuzzy sets, decision theory, and preference logic. E.g. in Hülya 
Behret, “Group decision making with intuitionistic fuzzy preference relations”, Knowl-
edge-Based Systems, 70 (2014), pp. 33–43.

86
uroš kranjc
With most of the formalities introduced in this first analytical part, it is im-
portant to look at them in parallel to gain a broader view of our argument. The 
logical overlap between Badiou’s transcendental algebra and preference logic 
(in economics) is immediately apparent, resting on either a poset or a totally 
ordered set. These measure the degrees of appearing and existence in the tran-
scendental or ranking preferences of a particular bundle of goods – both of 
these measurements rely on an order-relation, either ≤ greater or equal or pre-
ferring or indifferent ≾, respectively. The only difference is that in economics, 
the symmetry/asymmetry/antisymmetry clause is invoked randomly, depend-
ing on the preference logic model employed, whereas Badiou strictly maintains 
the antisymmetric relation in the poset of the transcendental (neither x ≤ y or 
y ≤ x, they are incomparable). Both Badiou and preference logic, as used in 
economics, maintain conjunction, envelope, and distributivity operations. The 
relation A ≾ B, meaning that an individual wants B at least as much as A, as 
expressed in economics, corresponds to Badiou’s formulation p ∩ q = p. The 
envelope is, on the other hand, an entire bundle of goods chosen against all 
alternatives, i.e. all other subsets of combinatory bundles one can choose from, 
either a 2-element subset or a n-element subset bundle. But which one is cho -
sen? It stands to reason that it should be the intersection bringing the highest 
degree of utility/satisfaction.
As already pointed out, the transcendental functor plays a key role in Badiou’s 
theory of the object. Again briefly: observed from the point of view of an object, 
a functor takes the composite atomic logic of localizations and connects them 
with the transcendental indexation. It causes a decomposition of atoms (sub-
objects as count-as-ones, inferring they are singletons) in appearing in relation 
to the elements of the multiples on the ontological level, but as far as appear-
ing goes, they are determined by their mutual co-belonging to the function of 
‘phenomenal components’ – relating a given multiple A and the transcendental 
T. The atomic logic (localization, compatibility, ordering) is based on the exist-
ence of some x in conjunction with a degree p of the transcendental T, but more-
over, it is also a combination of elements that relate to one another; therein lies 
the phenomenal characteristic of Id(a, x) and existence in terms of self-identity, 
Id(x, x) = Ex. Without going into too much detail, we will state that every local-
ization of elements (atoms) in an object is subject to belonging to the phenome-
nal component (π), which can be distributed more widely through the compati-
bility of other elements of the same multiple, i.e. they share the same degree of 

87
magnitudes in badiou's objective phenomenology and economic consumer choice
appearing (p) and order that ranks all the combinations according to ≤. All this 
culminates in the transcendental functor operation “which associates to every 
degree of the transcendental the set of elements of the object whose common 
characteristic is that their existence is measured by this degree.”
17
 So we have 
the grouping of degrees (ʻdegrees of the transcendentalʼ) associated with (ʻset of 
elementsʼ) subsets that are parts of a phenomenal component (ʻcommon char-
acteristicʼ). 
We have thus far been concentrating on the structural correspondence between 
preference logic and transcendental algebra – in essence, they are both ordering 
relations, resting on total and partial ordering. However, more can be said about 
how the logic of preference formally relates to utility theory. It does so in much 
the same way through a functor relation, transcribing the category of preferenc-
es onto the category of utilities, and going even further by transforming utilities 
and reflecting them in prices. Say we have a set X
i 
of elements (a, b) that are first 
evaluated against their quantitative determination, (a, x) and (y, b), and find 
those combinations such that (a, x) ∩ (y, b) ≤ U
a,b
, so that by applying the utility 
function u, we get a determinate utility for this particular bundle. By consider-
ing the whole range of X
i
 bundles, we can construct an entire consumption plan 
for all the combinations of these two elements. What does constructing an en-
tire plan mean? It involves the decomposition of all the possible degrees of utili-
ties by combining the two elements, or conversely, assigning to every individual 
combination a specific degree of utility. In other words, it takes each sub-part of 
the whole universe of bundles X
i
, collecting them into new objects and subsum-
ing them to the function with the corresponding degree of utility. In this sense, 
the utility function (or rather monotonic transformation
18
) has the same rank as 
the transcendental functor. Bearing all this in mind, one could facetiously say 
that Badiou’s analysis of the columns in Robert Hubert’s painting The Bathing 
Pool is an excellent example of what the microeconomists would enthusiastical-
ly analyse with their indifference curves.
17
 Badiou, Logics of Worlds, p. 278. 
18
 In utility theory, a monotonic transformation is presupposed as a way of transforming one 
set of numbers into another set of numbers in a way that preserves the order of the num-
bers. The operation is homologous to the natural transformation in functors.

88
uroš kranjc
We can now diagrammatically represent the transformation of preferences to 
utilities: 
Badiou’s objects can therefore be said to be composed of atoms, i.e. subobjects, 
one by one linked to their phenomenal appearing through the “phenomenal 
component” function. The parts, assembled according to their degrees of ap-
pearing, are evaluated against the transcendental. The objects of a determinate 
consumer choice are similarly decomposed into bundles of elements to which 
we attribute different magnitudes of utility. The function that orders all the util-
ities is a partial derivative of the assigned utility to each element of the bundle 
against all assemblages with other elements. However, what is most striking in 
the above diagram is the categorical setting and structural correspondence be-
tween Badiou’s theory of appearing and consumer choice theory. This is in effect 
because both degrees of appearance and ranking of preferences rely on the same 
mathematical ordering operations. At the same time, they also both operate on 
set-theoretical fundamentals when it comes to defining their objective disposi-
tion. Afterwards, they both face the identical problem of linking or ʻtranscodingʼ 
the ʻobjectiveʼ part to the ʻtranscendentalʼ one – Badiou resolves it by using the 
concept of the atomic structure and phenomenal components to transcendental 
indexing, while consumer choice theory relies on consumer plans and marginal 
propensities for each object in a bundle to complete the consumer plan.
World of commodities – a possible synthesis?
We are now at a point where we can address our ulterior motive for imagining a 
coherent economic structure, capable of subsuming both the classical tradition 
and the subjectivist turn. This was one of the irritating prospective tendencies of 
political economy that Marx identified and sought to solve. The critique of polit-
ical economy and its categories had much to say about the inadequate concep-
tual unfolding at the time and Benthamite utilitarianism was one of the driving 

89
magnitudes in badiou's objective phenomenology and economic consumer choice
forces leading to the bourgeois delusion. Marx saw economic subjects standing 
before a “topsy-turvy” world of economic objectivity, bestowed with the reifica-
tion process. Qualitatively determined subjects overturned into individualized 
quantifiable objects of commodity-like exchange and vice versa. Reification 
was notably scrutinized in the work of György Lukács and the proponents of 
the Frankfurt School, Adorno and Horkheimer, in Dialectic of Enlightenment, 
as well as Herbert Marcuse. He made it clear that in order to critically assess the 
actual dimension of the reigning mode of production, we have to disentangle 
the actual social relations “as a totality of objective relations”,
19
 i.e. we have to 
decipher wages in relation to the value of labour, or prices and utilities in rela-
tion to production factors. 
Economic relations only seem to be objective because of the character of com-
modity production. As soon as one delves beneath this mode of production, and 
analyzes its origin, one can see that its natural objectivity is mere semblance while 
in reality it is a specific historical form of existence that man has given himself. 
Moreover, once this content comes to the lore, economic theory would turn into 
a critical theory.
20
If economic science is itself caught in an estranged form – as the young Marx 
suggests
21
 –, there has to be an alternative, critical science capable of demys-
tifying this topsy-turvy world. To construct such a science is to answer Marx’s 
call for the handling of a “true materialism [wahren Materialismus]” with a 
“real science [reelle Wissenschaft]”. It must be able to deconstruct the structur-
al relations of the apparent objectivity, decompose its laws and propositions, 
and, even more importantly, lay the foundations for a theoretical continuum –  
a cumulative hierarchy of theoretical bodies. It is here that Badiou’s objective 
phenomenology can play a decisive role in the critical re-examination of current 
economic objectivity. Our introduction also implied a question: In a reified so-
cial exchange of commodities, how does one ʻvaluateʼ the objects brought for 
exchange? Put another way – If commodity fetishism is a mode of appearance, 
19
 Herbert Marcuse, Reason and Revolution: Hegel and the Ruse of Social Theory, (2nd ed.), 
London, Routledge & Kegan Paul, 1955, p. 280.
20
 Ibid., p. 281. 
21
 Karl Marx, Economic and Philosophic Manuscripts of 1844 / and the Communist Manifesto; 
Karl Marx and Frederick Engels, Buffalo, NY, Prometheus Books, 1988, p. 44. 

90
uroš kranjc
what is the transcendental structure of these objects, how do they come to exist 
on the market, in a world?
When Badiou introduced his theory of object, and appearing of objects in a 
world that-of, he asked them to be thought in “a world, as a site of being-there, 
is a Grothendieck topos.”
22
 Let us return once more to the Mathematics of the 
Transcendental and more formally say that a topos is a category which admits 
limits and co-limits for every finite diagram, i.e. that a diagram admits a univer-
sal property object, called a cone or co-cone; it has an initial object (0), a termi-
nal object (1), admits pullbacks and pushouts for objects, as well as equalizers 
and co-equalizers for arrows. There must exist a power object, and lastly, there 
must also exist a Central Object C (of which we have already spoken, it being 
the subobject classifier). We have shown elsewhere that Marx’s value form can 
be categorically and structurally represented as a configuration of a subobject 
classifier
23
. Before going into the structure itself, let us first see how Marx in 
Capital relates value, its magnitude as a function of money and transformation 
into prices through the lens of the value form:
The magnitude of the value of a commodity therefore expresses a necessary re-
lation to social labour-time which is inherent in the process by which its value 
is created. With the transformation of the magnitude of value into the price this 
necessary relation appears as the exchange-ratio between a single commodity 
and the money commodity which exists outside it. […] The price-form, however, 
is not only compatible with the possibility of a quantitative incongruity between 
magnitude of value and price, i.e. between the magnitude of value and its own 
expression in money, but it may also harbour a qualitative contradiction, with the 
result that price ceases altogether to express value, despite the fact that money is 
nothing but the value-form of commodities. Things which in and for themselves 
are not commodities, things such as conscience, honour, etc., can be offered for 
sale by their holders, and thus acquire the form of commodities through their 
price. Hence a thing can, formally speaking, have a price without having a value. 
The expression of price is in this case imaginary, like certain quantities in mathe-
matics. On the other hand, the imaginary price-form may also conceal a real val-
22
 Badiou, Logics of Worlds, p. 295.
23
 On CT analysis of the value form, see Uroš Kranjc, “Logics of the Value Form – Marx with 
Badiou” Cosmos and History, 14 (3/2018), pp. 72–102.

91
magnitudes in badiou's objective phenomenology and economic consumer choice
ue-relation or one derived from it, as for instance the price of uncultivated land, 
which is without value because no human labour is objectified in it.
24
 
Hans-Georg Backhaus posits: “The content of Marx’s form analysis is the genesis of 
price as price.”
25
 Michael Heinrich further succinctly points out that the relation 
between value and price is that “the magnitude of value of a commodity and its 
price are categories pertaining to different abstraction levels, so strictly speaking 
it makes no sense to posit their direct concurrence or divergence.”
26
 Marx has it 
all already there: the necessary social labour-time and the relation from labour 
to (the magnitude of) value (i.e. the so-called law of value) with the insurrection 
of the universal – money commodity, the contradictory transformation of value 
to price, and finally their objective and subjective character; all immanent to 
the value form unfolding. Marx, ever the ‘dialectician’, handles the conceptual 
interchange and unfolding with considerable ease, while the formal logic needs 
to be somewhat more imaginative when trying to reconstruct and accommodate 
the inner logical structure of the dialectic method. How does one start?
In our exposition of the value form
27
, we propose a bivalent logic of value when it 
comes to the immediate relation between a commodity and the universal object. 
The inner dialectical move of “doubling” [Verdopplung] of commodity into com-
modity and money, or put differently, the “value-forming substance”, labour, 
that comes to be expressed with a third object, in this categorical interpretation 
represents precisely the same function of evaluation as analysed above. What 
the products of labour achieve when entering the market exchange is precisely 
the binary evaluation, the Ω-valued set = (0, 1) of being, a) a commodity for ex-
change or b) refused for exchange by market, hence no (exchange) value. This 
is the proposition on which the whole of classical economics and labour theory 
of value depend. Once this was done away with as part of the “revolutionary” 
shift from objectivist to subjectivist, or rather parallax view from production and 
distribution to exchange, the neoclassical school had simply altered the ʻfunc-
24
 Karl Marx, Capital: A Critique of Political Economy, Volume One, London, Penguin books 
1993, pp. 196–197. 
25
 Hans-Georg Backhaus, “On the Dialectics of the Value-Form”, Thesis Eleven, 1 (1/1980), 
p. 105. 
26
 Michael Heinrich, Die Wissenschaft vom Wert, Münster, Westfälisches Dampfboot, 2006, 
p. 243. 
27
 Cf., Kranjc, “Logics of the Value Form”, pp. 84–86.

92
uroš kranjc
tion of phenomenal appearingʼ, from average labour-time to marginal utilities 
as degrees of value. This process resembles Badiouʼs differentiation between 
classical (ontological) and non-classical (appearing) worlds. Whereas in the 
classical economical conception, the centrality of value is attested exclusively 
on the predicate is/is not (an exchanged product of labour – everything further 
is treated as an attribution to this basic principle), the neoclassical and main-
stream orientations conform to the absolutization of values between the min-
imum µ and maximum element M, Ω = µ ≤ x ≤ M. Such interchangeability is a 
truly remarkable achievement of Badiouʼs philosophical project, for it opens up 
a new mode of thinking, capable of merging classical and subjective economics 
through its critical dispositive. If we look at the subobject classifier, which we 
have elsewhere called the “classification schema of exchange”, one last time:  
Without going into too much detail, we have the following objects: L (labour) 
as an universal subobject of elements x and y composing commodities A
1
 
and 
A
2
 … A
n
,
 
respectively. Let T be a validation function pertaining to all subobjects: 
V[1] → Ω, which outputs to a given object from the co-domain Ω the value {true} 
for all those subobjects that meet the conditional evaluation v
1
. The L-A
1
-A
2
 
tri-
angle represents what Badiou calls a triangular inequality, where the elements, 
say x and y, get expressed by a third (value-substance coming under elements 
v
1
 
and v
2
 so that the triangle diagram commutes). The Ω-set is treated here as 
an algebra of the universal equivalent, to which all objects relate their mode of 
ʻexistence in exchangeʼ and pulled back via v
1
 and v
2
 
in degrees, i.e. going from 
value – magnitudes of value – prices.
Does Badiou offer a final and decisive answer to our problem, namely a trans-
formation from value to price that would sublate the semblance of a break in 
economics? Indeed, he does! It is his theory of points, which in our case rep-

93
magnitudes in badiou's objective phenomenology and economic consumer choice
resents the model for a synthesis of the classical labour theory with the neo-
classical subjectivist orientation. Badiou’s point of view here is the following: 
“This triple determination of the concept of transcendental is what allows it 
to regulate appearing as localization (being-there), as cohesion (logical form of 
a being), and as situation (underlying multiple-being of being-there). There is 
an immanent onto-topo-logical (or ‘ontopological’) regulation.”
28
 Here, Badiou 
employs yet another CT operation – homomorphism (a structure-preserving 
mapping), which he uses to formalize “a structural homomorphism between 
the initial transcendental form T and the binary transcendental T
0
”,
29
 where T 
is the non-classical transcendental of appearing [µ ≤ x ≤ M] and T
0
 a binary one 
[0, 1]. We will not go into details
30
 for wont of space and will take Badiou for 
granted when he states that homomorphism Ø indeed associates (is a surjective 
function) to every point p’ ∈ T’, whenever there exists q ∈ T such that Ø(q) = p’. 
The homomorphism therefore enables the continuous ascending and descend-
ing transformation from the ontological to the phenomenological plane and vice 
versa. This fact is simultaneously of crucial importance for our discussion of the 
commensurability between classical labour theory and marginalist subjectivist 
theory of value, as it implies congruence with economic objectivity, instead of a 
historically experienced break.
The notion of magnitude is, interestingly, one of those notions that concern 
practically all philosophical traditions of the modern period and beyond. Al-
though they are casually employed in different natural and applied sciences, 
their philosophical grounding still remains an ambiguous matter. As for our 
part, we sided with Badiou against Kant in arguing for the confluence of exten-
sive and intensive magnitudes when it comes to economic analysis. We believed 
it to be a productive starting point for our investigation into economic consumer 
choice theory. Hereon, we can also provide a generalised answer to the initial 
speculation of whether our juxtaposition of Badiou’s logics of worlds and the 
logics of classical and neoclassical economics can be examined within a single 
framework, particularly the one underlying Marx’s critique of economic cate-
gories. The thesis seems justifiably viable and in need of further analysis if any 
28
 Badiou, Mathematics of the Transcendental, p. 209.
29
 Badiou, Logics of Worlds, p. 438.
30
 For the complete formal algebra of points, see Badiou, Mathematics of the Transcendental, 
pp. 209–216.  

94
uroš kranjc
serious attempt at bringing new critical insights into contemporary economic 
thought is to be anticipated.  
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