A Formalisation of Kant’s Theory of Space
MSc Thesis (Afstudeerscriptie)
written by Onindo Khan
(born 22.12.1988 in Dhaka, Bangladesh)
under the supervision of Michiel van Lambalgen and Riccardo Pinosio, and submitted to the Board of Examiners in partial fulfillment of
the requirements for the degree of
MSc in Logic
at the Universiteit van Amsterdam.
Date of the public defense: Members of the Thesis Committee:
29.06.2016 Floris Roelofsen (chair)
Dora Achourioti Benno van den Berg Michiel van Lambalgen Riccardo Pinosio
Abstract
This thesis provides formalisation and interpretation of the notion of space in Kant’s critical philosophy. The importance of space for the Kantian project is highlighted and through textual analysis, pointless topology and category theory a full interpretation and formalisation of the construction of space and its properties are provided.
Contents
1 Introduction 2
2 Kant and Space 7
2.1 The Role of Space in Kant’s Philosophy . . . 7
The Kantian Project . . . 7
Knowledge . . . 8
Representation . . . 11
Forms and Representation . . . 14
Synthesis and the Binding Problem . . . 16
Unity of Apperception and the Problem of Consciousness . 22 Space and Objectivity . . . 25
The Productive Synthesis of the Imagination . . . 30
2.2 Properties of Kantian Space . . . 36
Friedman’s reading . . . 36
Infinity . . . 38
Perspectives . . . 40
Metric . . . 44
Pointlessness . . . 47
Pointlessness and Perspectives . . . 50
2.3 Time and Space . . . 51
2.4 Transcendental Argument . . . 54
3 Formal treatment of Space 59
Introduction
As we promise in the title we shall deliver a formalisation of the Kantian notion of space. While space as an object of philosophical study falls into the purview of metaphysics1 the endeavour of this thesis is not one in meta-physics but one in the history of philosophy.
First of all this is an effort of translation. We shall translate Kant’s the-ory of space from his language into mathematical language. Objects, notions and processes figuring in Kant’s theory shall be translated into mathematical concepts.
A translation is usually performed to make an inaccessible text or speech accessible to a group of people. Kant’s writings have been translated into many different languages and many more people can access and read the originals than are familiar with the idiom of modern mathematics. Moreover a majority of people who understand the idiom of mathematics will be able to read Kant’s writing, as well as the extensive secondary literature, to come to an understanding themselves. The translation given here thus must serve a purpose beyond granting access to Kant’s philosophy to a different reader. Translation does not only widen the scope of possible readers but the trans-lation into another language changes the emphasis, can clarify distinctions and thus changes the text.
When translating from English to German for example the translator is forced to explicate the relationship between the speaker and the addressee from the English “you” to either the German familiar “du” or the polite “Sie.” Such distinctions are the centerpiece of translation from philosophy into mathe-matics.
Translating philosophy into mathematical language is very popular2 but has also been criticised.3
The mathematical language offers the user the comfort of being more precise
1[49]
2As not only the prefix in “formal epistemology” points to, but a great number of papers
relying on formal rules of proof and argumentation in analytic philosophy in general.
than natural language. As such a translation into mathematical language should give us many more of the kind of “du” and “Sie” distinctions when doing our work of translation. It should force us to clarify and introduce precision into our theory.
Rota4 argues that such precision is not to be conflated with rigour. Philos-ophy has its own rigour, and to identify this rigour with mathematical pre-cision would lead to the destruction of philosophy itself. Philosophy would become a different field, which would be, according to Rota, incapable of dealing adequately with the questions it has concerned itself with for the last millennia. Most prominently, the self destruction of the Tractatus5 ought to teach us to avoid using mathematical reasoning to soothe the philosophical mind.
A translation of philosophy into mathematics then seems like a pointless en-deavour. Following Rota such a translation cannot give us a good precise theory which solves the troubles that cloud interpretation of the primary text.
We grant Rota this point and agree that using formal methods to solve philo-sophical problems, or for that matter even positing philophilo-sophical questions in terms of problems, has significant limits. This does not mean that a trans-lation into mathematical language cannot be fruitful.
While Rota decries the impossibility of exchanging the philosophical method for the mathematical method our aim is to use the mathematical method to enrich, that is, add to the philosophical method.
This distinction becomes clear when we consider our understanding of trans-lation.
We noted that translation can provide additional precision and deeper in-sight. It can, however, also obscure and change the meaning.
Whilst we may gain a difference between “du” and “Sie” we may be faced with other words like “serendipity” for which there is no proper correlate in German. So while a translation may gain specificity in some cases, informa-tion may be lost in others. The project of translainforma-tion in this paper is thus not a way to “put Kant right.” Its aim is not to fill in words to give Kant’s theory the right kind of basis in the “true language” of mathematics, but rather to offer a new perspective on Kant’s philosophy through the lens of mathematical language.
This means that we do not believe the mathematical method to be the right and true method to do philosophy in, but that using such methods as an additional venue of inquiry should not be outright dismissed.
4[43] 5[52]
We understand a translation into mathematics in the same way as we would any secondary literature. The path which any secondary literature takes is a process of translation. It takes a primary text and recounts it in different words, to highlight certain features and then possibly criticise them. Usually the translation is not into mathematics but rather into a slightly different idiom particular to each writer. For the writer there is not much choice of the language to translate into, as she will always write in her voice. To con-sciously choose mathematics over your own idiom as a language to translate into needs a proper motivation.
The benefit of precision of mathematics is countered by the inability to for-mulate anything which is not definable in precise terms from the outset. Translating the late Wittgenstein from the Philosophical Investigations6 into mathematics, for instance, would be tantamount to going contrary to argu-ments in the book itself.7 In this case it seems that such a translation would not be very fruitful.
Indeed showing the “fruitfulness” of the translation of Kantian theory of space into mathematical terms is the central goal of this thesis. If it suc-ceeds, the fruitfulness gives a pragmatic justification for the truth of the interpretation.8
That a formal approach is fruitful is hardly surprising given the fact that Kant himself commits to a similar rigour as mathematicians do. His writings are ordered strictly in paragraphs, reflecting the argumentative structure. He aims to provide proof for all of his statements and tries to define most of his terms. As we shall see his paradigm of truth relies heavily on the sciences and as such using mathematics, the language of the sciences, to clarify his theory is well within Kantian convictions.
Thus we have two good reasons to choose mathematics as the language to translate Kantian thought into. First it is pragmatically sensible, second Kant himself is committed to the standards of mathematical rigour.
So far we have understood the act of formalisation as translation as merely
6[51]
7One of the central arguments being that the heart of meaning lies in the “Lebenswelt,”
that is, the (social) surroundings humans find themselves in, and not in definitions or formal rules.
8Pragmatically justified in the sense of American pragmatism [24]. The epistemology
in James’ writing explains the justification of truths through usefulness. True are the things which are expedient to believe. So one is justified to in calling a belief true if it facilitates doing the things one wishes to do. As such it is sufficient that our analysis be fruitful to justify the endeavour.
taking a given text to produce a new text using a different language. This is not totally accurate.
Kant’s writing in its difficulty is not always clear in its meaning.9 This lack of clarity means that the act of translation itself brings into being the meaning of Kant’s text.
This means that translating, or in our case formalising involves an interpre-tation of the primary text. Our reading of the text, that is, the meaning of the text, changes through the process of formalisation.
We obtain a dialogue between formalisation and reading in which new ideas for formalisation result in new readings, which again spawn new formalisa-tions. This feedback loop produces on the one hand the formalisation but on the other hand also a non-formal reading of Kant’s text. The circularity of this process should not be seen as a negative. Such loops can produce new insights regarding the original text, which are not visible when performing only one step of direct formalisation.
There is a risk that following these interpretation-formalisation loops will lead away from the text. It is important to be able to relate the formali-sation and the reading back to the original text. One must note, however, that it is one of the appeals of Kant’s text that different readings of it are possible.10
It follows that a reading of the primary text is as much the result of the formalisation as the presentation of the formal apparatus itself; The formal-isation does not begin with the introduction of the formal apparatus in the second part of this thesis, but with the reading of Kant that shall occupy the first half. This reading is not to be understood as the definite reading of Kantian theory of space, but one which conforms to the restrictions given through the formal approach.
The thesis will have two parts. In the first part, chapter 2, we will con-sider Kant’s text. In the second part, chapter 3, we are going to present and motivate the formalisation.
We will begin our analysis by investigating the aims of the Kantian program. A common thread throughout the work will be to show that the first step
9The huge, wildly disagreeing, two hundred years spanning, secondary literature is
ample witness of this fact.
10Historically there were significant changes in emphasis in Kant scholarship; While
his transcendental idealism was very important to German idealism (Most prominently Fichte, Schelling and Hegel.[18]), the first critique now has become part of an effort to explain cognition as a result of physical processes in the brain. (See for instance [31] or [36])
of embarking on the Kantian program must be to give a theory of space. Kant’s goal is, ultimately, that of developing an ethical theory. We will show the links which bring us from the question “what should I do” to the ques-tion “what structure does space have.” The second half of the first part will mostly be concerned with answering the second question (“what structure does space have”) while the first half of the first part will be dedicated to demonstrating the connection between the two questions. We will show that the first step towards a full ethical theory is an epistemic theory in section “The Kantian Project.” Kant’s notion of knowledge will be analysed and contrasted with Popper’s in the following section “Knowledge.” It will be-come apparent that pure intuition, and especially space, plays a central role in coming to know. To understand how space is essential to knowledge it is necessary to understand Kant’s notion of representation and how represen-tations are connected to space. These two questions are going to be dealt with in section “Representation” and “Forms and Representation.” After that it remains to be shown how the necessary truths which we find in space are applicable to the objective world. This feat shall be accomplished with the following three sections “Synthesis and the Binding Problem,” “Unity of Apperception and the Problem of Consciousness” and “Space and Objectiv-ity”. The first of these provides an analysis of the notion of synthesis in view of the modern binding problem. The second introduces the unity of apper-ception. Combining the general synthesis and the unity of apperception the third section (“Space and Objectivity”) shows how space is objective and thus how ethics, knowledge and space relate. This wraps up the first thread of the thesis.
The following sections from “The Productive Synthesis of the Imagination” through to the entire section 2.2 are dedicated to understanding properties of space. Friedman’s analysis of Kant’s notion of space will be critiqued and the exegetical foundation for the formalisation will be laid out.
In the last two sections (2.3, 2.4) of the first chapter the construction of space will be considered in relation to time and the structure of transcendental ar-guments.
The second part will be devoted to developing formal correlates for a selec-tion of the most important concepts and processes described in the first part. This project will provide us with a reading of Kant’s understanding of space as well as an understanding of the role of space in Kant’s philosophy.
Kant and Space
2.1
The Role of Space in Kant’s Philosophy
The Kantian Project
The Kantian program is perfectly described in a note Kant wrote in his own copy of “Observations on the Feeling of the Beautiful and Sublime.”
Everything goes past like a river and the changing taste and the various shapes of men make the whole game uncertain and delu-sive. Where do I find fixed points in nature, which can not be moved by man, and where I can indicate the markers by the shore to which we ought to adhere?1
The question Kant asks here has two sides. “Where do I find fixed points in nature, which can not be moved by man?” is a question that searches for the necessary and non-contingent in experience. To find any such thing, struc-ture or fact that is unchanging, and to also recognise that one has found such a thing, would constitute knowledge.2 Thus the question is directly linked to the first of Kant’s famous three questions posed in the doctrine of method: “What can I know?”3
Notably the knowledge considered here should not be about any (theoretical) concept or the mind as such but about nature. This means that Kant wants to find necessities which are in the world, not merely in theory.4
The second question: “where I can indicate the markers by the shore to which we ought to adhere?” also relates to one of the the three famous questions:
1[27] p.8
2Without committing Kant to the definition, we can assume knowledge to be justified
true belief to understand the connection between the questions.
3[A805/B833]
4The adjunct “in nature” is quite important as in formal logic it is trivial to provide
such necessities. Anything that follows from the chosen axioms is such a necessity. Finding such necessities in the world, in opposition to simply in a set of propositions with certain rules, is the difficult part.
“What should I do?”. The longer version in the note above has more presup-positions. Using the word “adhere” requires that ethics and moral conduct rely on rules (by which we adhere). Furthermore “markers” once “indicated” should hold timelessly and for different subjects. These presuppositions are closely related to the enlightened ideals of the human as universal5 and of the human as rational animal. Some universal rule should account for the good and bad in actions,6 while humans as such are rational in that they have a choice to adhere to such rules or deviate from them.
To highlight these commitments, the question can be reformulated as: What rules ought every human adhere to?
The two questions are not juxtaposed by accident. As Goldwaith notes in the translator’s introduction to the “Observations on the Feeling of the Beauti-ful and Sublime”[27] Kant already in pre-critical times thought knowledge and moral rules to be intimately connected, but only later in the first two Critiques does he manage to ground his moral rules within our reason (and thus our (capacity for) knowledge).
This approach to ethics as rule governed and reliant on knowledge requires Kant to explain what we can know before even starting to consider what we ought to do.
The “Critique of Pure Reason” is the attempt to demarcate the space of what we can know. Within the critical program and Kantian philosophy it is but merely the groundwork. The central question of the first Critique is “What can I know?”. To answer this question we should first turn to what knowledge means for Kant.
Knowledge
In this section we are going to investigate the notion of knowledge in the first Critique. Our goal is to understand first what shaped Kant’s paradigm of knowledge and second what site8 is appropriate to analyse the production of knowledge.
Kant’s paradigm of knowledge relies heavily on mathematics and the sci-ences, most prominently physics. In the preface to the second edition of the Critique of Pure Reason examples of knowledge creators are Baco of Verulam,
5See Meyer[37] for the role of universality.
6This is what he calls the Categorial Imperative in the “Grundlegung zu einer
Meta-physik der Sitten”7, or the fundamental law of pure (practical) reason in the second
Critique.[25]
8Following Latour[34] we use site broadly, allowing for the laboratory, the brain, the
Galileo, Torricelli9, Copernicus and Newton10 among scientists and Thales11 among mathematician.12
Kant highlights two features which he believes are central to the success in producing necessary truths. First the empirical method and second allow-ing for “representations a priori in intuition.”13
Let us turn to the empirical method first. Kant analyses the empirical method as consisting of two parts. First, reason has its principles14 and then with these it approaches observations15. Using the principles we come to see certain patterns in observations. The principles allow us to categorise different observations. This again is necessary for any law-like rules to order observations. Ordering observations into groups makes it possible to poten-tially discover rules that pertain to the relation between different classes of observations.
To discover these relations we conceive of experiments. We use the principles to design experiments and then order the observations of these experiments. In a second step we need to distinguish those relations which are random from those which are laws. This process is again a categorisation, now of relations of observations and not of observations themselves and is also given through the principles of reason.
Today’s understanding of the empirical method is rather different. While phi-losophy of science since Kuhn’s “The Structure of Scientific Revolution”16has turned more and more to sociology and ethnography,17 on the ground, within the sciences, many undergraduates still learn Poppers Falsificationism18 as foundation. Falsificationism differs from Kant’s view in several points.
9[BXII] 10[BXXIIn] 11[BXII]
12Note that all of these are (at least partially) physicists. 13[26]p.7
14The term “principles of reason” shall remain opaque. We shall not clarify it because
this would require a careful analysis of the table of categories, which goes beyond the scope of this thesis. Central to highlight the difference to Popper is that these principles are inner principles which spring from our mind.
15Kant specifies that we do not step towards observations, but rather to appearance,
but as this term has not been introduced, we shall (incorrectly) use the modern term observation.
16[33]
17Latour[34], Bloor[9] 18[41]
1. Popper believes that the scientist starts with a problem through which the observation is first afforded. For Kant on the other hand observa-tions are the result of principles of the human reason.
2. In Popper the law-like status of scientific theories lies in their potential to be falsified and their effectively not yet having been contradicted, that is, falsified by any experiment so far.
The possibility for failure is the central point for Popper, the more potential for falsification a theory has, the more information it carries,19 the better it is. For Kant the principles of reason make sure that our judgements about observations allow us to produce laws concerning these observations.
The central difference is that for Kant reason is the mediator between ob-servations and laws while for Popper problems produce obob-servations which then can be related to a theory.
In the social sciences probability theory and statistics have taken the cen-tral role of mediator. Instead of using reason as such to relate observations, formal methods of reasoning, namely statistics, are used to determine the relation between observation and theory.20 For Newton and the physicists of the time no such process is necessary. Reason on its own is capable of identifying connections of the kind Newton found between moving objects. Using statistical methods even introduces a new vagueness given through the choice of significance level and the statistical tools used.21
So while modern empiricism relies on outer objects or structures, like sta-tistical methods, ‘problems’ and the amount of information a theory carries, Kant relies on reason as a capacity given by our human mind.22
Human reason as invoked by Kant is unchangeable and necessary. Statistical methods may improve or change but human reason will not. Furthermore
19The amount of information could be measured in Dretzkes[13] sense. Using such a
treatment where more information equals more ruled out worlds makes immediately clear why a theory with more potential for falsification carries more information. Given we have a statement with 1 bit information there is a 50/50 chance that the experiment testing for this statement, will falsify the statement. Now if we have 2 bit we will get 4 worlds of which only one is the correct one. Thus the chance will be 75%. Each bit of information reduces the a priori likelihood of an experiment to go in favour of the piece of information.
20A succinct history of the move from determinist causality to statistical inference is
presented in Ian Hackings “The taming of chance”[19]
21See for instance Simpson’s paradox [35] for a case where statistical analysis contradicts
reason.
22Note that statistics is not such a capacity but an external tool to externally verify
causal connections. For Kant these causal connections can be understood without such tools merely by using our reason.
human reason is accessible to all of us, because we are all human reason-ers. As such the capacity for reasoning, that is, human reason not with its flaws and mistakes when it is used in every day life, but in its pure form, can be investigated in a non-empirical way.23 This means that there are things knowable independently of the empirical method. Such non-empirical inquiry is a priori. It is “completely independently of all experience”24 and is also called pure. Kant will insist that the type of representations which are studied a priori in physics and mathematics are of a certain kind, namely “in intuition.” This gives us the “representations a priori in intuition” as sec-ond object of study for the epistemologist. The other representations which are a priori, the concepts of the understanding, are fundamental to any kind of reasoning what so ever. We are going to focus on intuitions, however, as we have isolated the practices of mathematics and physics as most influential for Kant’s epistemology.
We can now answer the two questions we posed at the beginnings by not-ing first that mathematics and physics are paradigmatic for Kant, while the mind, as a capacity, is the site for inquiry.
The mind considered a priori deals (most importantly) in “representations a priori in intuition.” That means that if we understand what “representa-tions a priori in intuition” are we will have an understanding of where we are to find the source and location of knowledge about the world.
The next few sections are dedicated to lay out the capacity for “representa-tions a priori in intuition.”
Representation
To begin an inquiry into the pure human reason we need to consider what the objects of the human reasoning process are. For Kant these objects are representations. We need to understand what different types of representa-tions there are and how they are related.
For Kant there are different kinds of representations, which he lays out in a ladder-like structure in the passage known as ladder (or “stufenleiter”).
The genus is representation in general (repraesentatio). Under
23This distinction is similar to the competence-performance divide in linguistics.[11]
As our spoken language which can be observed through experimental methods contains irregularities and mistakes, so does our reasoning processes in the world. This does not mean that reason or language as such are irregular and inconsistent. There is an underlying structure on the competence level which is consistent and regular.
it stands the representation with consciousness (perceptio). A perception that refers to the subject as a modification of its state is a sensation (sensatio); an objective perception is a cognition (cognitio). The latter is either an intuition or a concept (intuitio vel conceptus). The former is immediately related to the object and is singular; the latter is mediate, by means of a mark, which can be common to several things.25
First of all it is presupposed that representation is a mental effort. Such effort may be with or without consciousness. If it goes with consciousness it is called perception. Such a perception again falls into two categories, the first is concerned with a change of the subject itself, that is, an inner change, which us called sensation. The second category, objective perception, is called cognition. Cognitions also fall into two slots, being either intuition or concept.
representation
perception
sensation cognition
intuition concept
Figure 2.1: The stepladder visualised.
Concepts are related to objects via marks or properties which are only me-diately attached to things. Concepts always have multiple objects in their sphere, that means that a property, to actually be a property, must apply to more than one thing.26
Intuitions are not as the Oxford dictionary defines them: “The ability to understand something immediately, without the need for conscious reason-ing.”27 While the immediacy of Kant’s notion of intuition is carried over
25[A320/B377]
26This does not exclude that in a certain instance of experience there is only one object
which instantiates the property. It is necessary though, that this property be in general applicable to more than one object.
to the modern term, intuition for Kant is by definition a conscious effort. Instead of being mediated by a mark the intuition is related directly to the object. It is furthermore singular because it cannot be common to many things. It does not have a sphere as concepts have.28
Concepts and intuitions must be combined to achieve cognition.
For two components belong to cognition: first, the concept, through which an object is thought at all (the category), and second, the intuition, through which it is given; for if an intuition correspond-ing to the concept could not be given at all, then it would be a thought as far as its form is concerned, but without any object, and by its means no cognition of anything at all would be pos-sible, since, as far as I would know, nothing would be given nor could be given to which my thought could be applied.29
Intuition takes the role of giving the concept something to be applied to. From a model theoretic perspective we could say the intuition is what gives us objects in the model, while the concepts give us the unary predicates.30 For example, the concept triangle contains in its intension as marks the con-cepts (properties) like having three lines, three angles, connectedness, cir-cumfencing an area etc. None of these properties, however, make sure that the triangle can really exist. We could make up a concept which contains in its intension as marks the concepts three lines and four angles. These properties are not obviously contradictory. There needs to be some process by which we can distinguish the ontological status of the objects which have three lines and three angles from those which have three lines and four angles. The first exist while the latter do not. In an axiomatic system one would show that the combined statement of having three sides and four angles is contradictory. For Kant the judgement which denies the existence of an ob-ject goes beyond concepts. It relies on the notion of intuition. There simply is no intuition which we can produce of an object with three sides and four angles.
We have now obtained a more detailed picture of what representations ac-tually are. They fall into different classes, most importantly intuitions and concepts.
28For further discussion of this passage and the concept/intuition distinction see
[40]p.4-7.
29[B146]
30This metaphor should not be taken too seriously, neither the ontological commitments
coming with objects in a model nor the simplicity of relating objects with statements in model theory should be assumed.
This classification is the first step towards understanding what “representa-tions a priori in intuition” are. We know what intui“representa-tions are, but we do not know yet what it means to “be in intuition.”
The central point, we will come to see, is that whatever may be “in intuition” has to follow the form of intuition. As such being in intuition is a certain restriction on the form of a representation.
One might think that sensations of “impregnability, hardness, color etc.”31 are the forms Kant means. This is not true as Kant holds that form has to be non-empirical. Of these pure, non-empirical forms there are, according to Kant, only two: Space and time.
The Transcendental Aesthetic32 is dedicated to proving that space and time are in fact such pure intuitions and also the only ones. Thus “representations a priori in intuition” are representations which consider only the spatiality and temporality of an object while leaving anything empirical aside.
This approach of seeing the spatial and temporal as fundamental properties (in Kantian terms forms and not properties) is closely related to physics. Newtonian physics is the science concerned with spatial change, that is, tem-poral processes in space. We know that Kant relies on physics as a paradigm for knowledge, so it is not surprising that our capacity to understand motion in space ought to be the process which is central to the production of knowl-edge. As such the importance of understanding spatial representations and the process of representing spatially becomes apparent.
Forms and Representations
Now that we know what “representations a priori in intuition” are the cen-tral question is, how can these be fundamental to mathematics and science and how can they constitute knowledge?
In the Aesthetic Kant argues that space and time are not relations of things independent of our cognition but rather the forms in which our experiences are structured. To be an experience a perception needs to be temporally and spatially located. As such space and time are not, as Leibniz believed, general relations of things but they are products of our way of cognising. The following citation highlights how this approach will allow us to find the “fixed points in nature,” that is, those points that we began our exposition with.
Space is nothing other than merely the form of all appearances
31[A20/B35]
of outer sense[...]. Now since the receptivity of a subject to be affected by objects necessarily precedes intuitions of these objects, it can be understood how the form of all appearances can be given in the mind prior to all actual perceptions, thus a priori, and how as a pure intuition, in which all objects must be determined, it can contain principles of their relations prior to all experience.33 As has been already stated, space is a form that our experiential episodes necessarily take. As the form of outer sense it is something which is bound up with our way of sensing. As such it is part of our receptivity as a sub-ject to be affected by obsub-jects. This is an essential part of the philosophy of transcendental idealism. As subjects we all rely on our cognitive functions to provide us with experience. According to Kant space is the form which is already given in our mind logically prior to any experience.34 By definition it then follows that we have the form of space a priori. Furthermore this form is not merely interesting in its own right but it shapes any experience to come. Knowledge about these pure forms thus results in knowledge about the world.
To be able to make statements about space, however, we need to represent it. Space needs to become an experience to us so we can make judgements about- and attain knowledge of it. Thus we need to consider how forms can be turned into representations. We will ignore this problem for now but come back to it below.
Kant identifies the science which describes the representation related to the form of space as geometry.
Metaphysics has to show how we can have a representation of space, while geometry teaches how we describe it, that is, not through drawing, but through a priori representation (Darstel-lung).35 36
So geometry is the science which describes the representation which somehow arises from our form of intuition. As we have noted the form of intuition is a
33[A26/B42]
34There is a caveat to be made here as from a transcendental viewpoint it is not prior,
the form emerges only in experience but it is also necessary condition for experience. We shall discuss the problem of which is prior in the section 2.4 after we have considered more instances of this structure.
35“ Die Metaphysik muss zeigen, wie man die Vorstellung des Raumes haben, die
Ge-ometrie aber lehrt, wie man einen beschreiben, d.i. nicht durch Zeichnung, sondern in der Darstellung a priori darstellen k¨onne.”[28]Bd.XX. p.419
priori. A priori is neatly defined in the introduction to the B edition of the first critique:
[W]e will understand by a priori cognitions not those that occur independently of this or that experience, but rather those that occur absolutely independently of all experience. Opposed to them are empirical cognitions, or those that are possible only a posteriori, i.e., through experience. Among a priori cognitions, however, those are called pure with which nothing empirical is intermixed.37
A cognition is thus non-empirical or pure when it does not rely on any spe-cific experience. The opposite, a posteriori is thus identified with empirical cognitions, which rely on experience. Although the two are opposed it seems there is a third class of a priori cognitions in which some empirical may be intermixed.38 Thus representation of the form into an object will, as pure, not rely on empirical data either. So results which are obtained in geometry are going to be a priori and as such necessary. We have found “fixed points” and connected physics with mathematics.
What we have not shown yet is that these “fixed points” are actually in na-ture. This is the goal that Kant outlines in the above quote for metaphysics. It has to show how form of space, space as object/representation and expe-rience are necessarily linked together in the human.
To do this we need to understand what processes actually happen in the human mind. The central process Kant considers is that of combination.
Synthesis and the Binding Problem
In this section we shall use the modern language of neuroscience to trace the Kantian problem of combination.
In modern neuroscience the binding problem has received much attention. The problem is that while one may be able to map out the neuronal pat-tern related to a certain representation (like redness or roundness) it remains unclear how these two representations are to be combined in one object. One needs to individuate the processes which link these two representations
37[B3-4]
38These are those which are considered in physics. Kant holds that the moveability of
an object is not known a priori but only through experience but the rules governing these kind of motions are still a priori. In the case of space we shall only be dealing with pure a priori cognitions.
and unify them in one object.39 For Kant the process of binding is called synthesis:
By synthesis in the most general sense, however, I understand the action of putting different representations together with each other and comprehending their manifoldness in one cognition.40 The binding process puts together two different representations and unifies them. The manifoldness of redness and roundness is combined into the single cognition of a red-and-round. The difference to the binding problem lies in the understanding of cognition. As we noted cognition has two representa-tional components - intuitions and concepts. Neuroscience on the other hand knows only one kind of representation - neural activity measurable through fMRI, EEG and mapped through PET. The binding problem asks how to combine two elements from the same class (neural representations) into one element of the same class. Kant on the other hand needs needs to supply a threefold synthesis, answering each of the following questions:
1. How are concepts bound to other concepts? 2. How are concepts bound to intuitions?
3. How are intuitions bound to other intuitions? Concepts and Concepts
The first, combining concepts with other concepts is judgement. Now the understanding can make no other use of concepts than that of judging by means of them. [...]
[I]n the judgement, e.g., “All bodies are divisible,” the concept of the divisible is related to various other concepts; among these, however, it is here particularly related to the concept of body.41 Understanding, the faculty which deals with concept-concept interactions, is limited to relate concepts by judgement. Such judgements look on the surface like standard logical propositions. “All bodies are divisible” would normally be translated into ∀x : B(x) ⇒ D(x). Such a sentence could be true or false in a model. For Kant a judgement, however, must mean more, since
39See Treisman’s work for an early formulation of the problem [48] and a modern solution
[47].
40[B103] 41[A68/B93]
in the evaluation of the proposition ∀x : B(x) ⇒ D(x) the manifoldness of B and D are not combined in any sense. Objects are separately checked for their B-ness and their D-ness but no new predicate of BD-ness is produced. For Kant when the judgement “All bodies are divisible” is made whatever “is cognize[d] under the predicates that together comprise the concept of a body, [...][must also be thought] through the predicate of divisibility.”42 So whenever an object is thought of as body, that is, an object is given as falling under B-ness, it must immediately also be thought in its D-ness. The predicate logic approach gives us the option to infer from B-ness to D-ness, but for Kant it becomes a necessity to think D-ness with B-ness if we are to be able to make the above judgement.
The important difference is that in predicate logic we can evaluate D-ness independently of B-ness even though we know that ∀x : B(x) ⇒ D(x). If we iteratively check for B-ness or D-ness we may observe that something is a B without committing to it being a D. An extra step of logical reasoning is necessary here. For Kant this extra step disappears. The judgement immediately entails that any objects that are B will immediately also be thought of as D’s. B-ness is replaced by BD-ness and no further step is required to move from B-ness to D-ness.
Thus judgments are bindings in the proper sense and produce new concepts and not merely new information about some object.
Concepts and Intuitions
The second synthesis relates concepts to intuition. Concepts are processed in the understanding while intuitions are processed by sensibility. It seems thus a hard problem to identify what form a combination of intuition and concept should take and how it should be mediated.
The central reference for this issue is the “The Schematism of the Pure Con-cepts of the Understanding.”43 In this section the connection between the categories, which are the pure concepts of understanding, and the objects which fall under them is explained. For our purposes we are not so much interested in the categories but in the relation between empirical concepts, (pure) intuitions and objects.44
Kant assumes that the concept and the intuition are in some sense homoge-neous.
42[28] Bd.17 p.616 43[A137/B176]
In all subsumptions of an object under a concept the represen-tation of the former must be homogeneous with the latter i.e. the concept must contain that which is represented in the object that is to be subsumed under it, for that is just what is meant by the expression “an object is contained under a concept.”45
There is a synthesis, called subsumption, which relates an object to the rep-resentation of the same object.
Somehow the content of the object and of its representation should make sure that only the right objects can be subsumed under the corresponding concept.
In the discussion Kant immediately jumps to the categories after saying this, implying that the homogeneity which was obvious in the relation between empirical concepts and objects is hard to fathom between pure concepts and objects. In this passage it thus seems as if being homogeneous is something which is primitive and in so far Kant sees no problem to be discussed. Later on, however, Kant discusses three empirical examples where the ques-tion of homogeneity is mediated by a further noques-tion, that of a schema.
[T]his representation of a general procedure of the imagination for providing a concept with its image is what I call the schema for this concept.46
A schema is a rule for a procedure which occurs in our imagination. It medi-ates between a concept and its image. This image is sensible, as such it must be processed by sensibility. We have noted that sensibility is concerned with intuitions and as such the image must be an intuition.
At first glance there seems to be no need for an extra process to connect concept and image; the concept five should directly be connected to five points simply by seeing the points. But if we consider a bigger multitude like thousand the answer becomes apparent. Thinking thousand is “the rep-resentation of a method for representing a multitude (e.g., a thousand) in accordance with a certain concept [rather] than the [only having the] image itself, which in this case I could survey and compare with the concept only with difficulty.”[A140/B179] So we can only connect the thousand points (as much as the five points) to the concept thousand by understanding the con-cept thousand as the result of a method. This method is the schema.
For the triangle the issue is even worse, as no image of a triangle can ex-actly cover all possible triangles. Each image is always limited and only
45[A137/B176] 46[A149/B179]
because we may say that the sum of the internal angles of this triangle is 180◦ it will not follow that another one, of which we do not have an image, will also have an internal angle sum of 180◦. “No image of a triangle would ever be adequate to the concept of it.”47 Again the schema solves the prob-lem as it “signifies a rule of the synthesis of the imagination with regard to pure shapes in space.”48.
What was method above is now rule, this rule allows for the generality of the triangle. We will see below how exactly this works in the case of geometry and arithmetic. To be clear this kind of mediation is also at work for con-cepts like “dog,” where at least to a mathematically trained it seems much harder to give a rule than for multitudes49 or shapes in space. Nonetheless Kant holds that:
The concept of a dog signifies a rule in accordance with which my imagination can specify the shape of a four-footed animal in general, without being restricted to any single particular shape that experience offers me or any possible image that I can exhibit in concreto.50
So again the concept relates to a schema which is a rule for the imagination. Note that the concept through the schema specifies the shape of the object. Kant specifically emphasises the shape of the dog as well as its number of legs. It seems that the concept of dog here is not much different from the concept of triangle or five.
There are two possible readings of the fact that Kant neglects other proper-ties (the brown hair, the soft fur etc.).
First one may believe that because these are not pure the homogeneity be-tween the intuition of soft fur and the concept of soft fur is obvious. The problem only arises where pure intuition is required, as is in shape (geometry and as such space) or number (arithmetic and thus time).
Second it is possible that softness or brownness works in exactly the same way. There is a rule, a schema which mediates between concept “brown” and intuition “brown” because no brown of experience can give the generality of the concept brown. In this reading Kant believes that these rules are so obvious that discussing them is not necessary.
Whichever reading one accepts does not interfere with our concerns of un-derstanding the relation of space to experience.
47[A141/B180] 48[A141/B180]
49Hilberts rule to construct natural numbers comes to mind.[23] 50[A141/B180]
To come back to the question how space and experience relate we need to consider shape and, potentially, multitude. For these cases we have seen that the schema, which is a rule of construction, allows for the combination of a concept with an intuition, given that the concept and the intuition are homogeneous.
Intuitions and Intuitions
Finally the combination of intuitions with other intuitions is closest to the binding problem. Treisman51 proposes spatial maps as points to bind differ-ent intuitions to. The idea is that we first map our surroundings and then representations for redness or roundness are linked to points on the spatial mapping we have created. Using such a map different representations can be attached to different objects.
For Kant on the other hand the combination does not rely on a third point of reference. The redness and the roundness are both immediate and as such also immediately combined.
In the case that the redness and the roundness are not properties of the same object they could not be unified into a single cognition. Why this is so we will see below when we discuss the special unity that is necessary for such synthesis to be possible.
For modern neuroscience with its different ontology52 there is no binding problem which asks how properties of the same type can be held together within one cognition.
An object is always either brown or red, not both (or if it is it needs to be cognised separately), a triangle is thought of as one representation and not as a representation which is made up of further line representations. Thus there is no need for binding together properties or objects of one kind. For Kant on the other hand the constructive procedure is central for shapes and as such intuitions often are a combination of two intuitions of the same class. We see that the modern binding problem is concerned with the question how different properties can be cognised for different objects. For Kant the question is how different properties can be thought in the same object. This difference in problem springs immediately from the different ontologies
51[47]
52Ontology is here a little stronger than in Kant’s sense. For Kant the things that
appear in the word exist. Representations, whether intuitions or concepts, do not exist for Kant in the same way stones do. To be able to discuss the difference between neuroscience and Kant it is necessary to expand the term existence to cover those objects which are posited within the mind.
that neuroscience and Kant commit to.
Instead of explaining how it is possible that the representation of redness is bound to a representation of an object whilst a representation of brownness is attached to another representation of an object, Kant needs to explain what kind of process is involved in putting together different intuitions. This process is the synthesis of imagination. We have seen in the section on the combination of intuition and concept that spatial and temporal in-tuitions are paradigm cases for the synthesis discussed in this section.53 We shall trace the role of this synthesis on spatial intuitions below and provide the answer to how intuitions are combined with other intuitions.
Before we can consider the synthesis we need to discuss the special unity that makes such a synthesis possible.
Unity of Apperception and the Problems of
Conscious-ness
After using the binding problem from neuroscience as a stepping stone we shall now come to a better understanding of Kant’s notion of the unity of apperception through considering the (neurophilosophical) problem of con-sciousness.
Chalmers54identifies a couple of “easy” problems of consciousness. The easy problems of consciousness deal with the functional role of our mind. Some other underlying, or rather overarching, structure should explain the func-tional roles. For Chalmers this inquiry is in the purview of science, or more specifically, neuroscience. This project amounts to finding an explanation for questions like “how does the cognitive system integrate information?,” “how can a system access its own mental states?” etc. which rely on the ontology of biology.
For Kant these questions are not to be answered from a biological but rather from a transcendental standpoint. Our capacity to reason, together with our knowledge of how it is to experience, should lead us to the answers to these questions.
For our purposes the analysis of the first question regarding integration is of interest. Kant answers this question by explaining what synthesis is and how it works. Both in Kant and in some streams of modern analytic philosophy,
53The examples of triangle, number and even dog were fashioned around spatial and
temporal intuitions.
unity is of central importance to consciousness.55
Kant argues that the power of integrating information, or synthesis requires something which goes beyond the appearances and also beyond the rules which were discussed above. Given that we have a rule to combine a “red” with a “hairy” (it is not important whether these are concepts or intuitions), and we also have the representations “red” and “hairy” it is still not obvious that the two are given in such a way that they may put in combination. To use a metaphor from computer science we could imagine the information “red” to be written on a piece of paper while the representation “hairy” is given through a painting. A CPU with assembler code will not be able to integrate the two representations into one piece of information. Both the writing and the painting need to be put into binary form and fed into a memory which is connected to the CPU to be combinable.
In Kant, the CPU is the self, the I.
The I think must be able to accompany all my representations; for otherwise something would be represented in me that could not be thought at all, which is as much as to say that the rep-resentation would either be impossible or else at least would be nothing for me.56
The representation of “I think” must be able to accompany any other repre-sentations, as otherwise the representation would not be a thought. As such it would, to go back to the metaphor, not be a valid input. Kant calls the act of representing the “I think,” the pure apperception or original apperception; its unity is the transcendental unity of apperception. Kant insists that to be able to think something we need to be aware that this is actually a thought of ours.
We need to understand what kind of representation the “I think” is and how it relates to other representations.
It is impossible that the “I think” comes from the objects, because then “I would have as multicolored, diverse a self as I have representations of which I am conscious.”57 Consequently a scattered consciousness, which is not the experimenting master in its world, but rather play-ball to its representations. It follows that the seat of the “I think” must be internal and as such it must be either part of intuition or of the concepts.
Recall that intuitions relate to sensibility and concepts to the understanding.
55We are not going to consider the modern counterparts closely. For a good overview
see the introduction[7]. For an in-depth look and defence of its importance see [6].
56[B131-132] 57[B134]
Now the act of representing “I think” is not an act of sensibility, it requires spontaneity (because if it were merely receptive we would fall into the prob-lem of scattered consciousness) and as such is part of the understanding. Still every representation needs to be accompanied by that “I think,” also intuitions even though they are not even cognised in the same faculty. Considering the discussion of the previous chapter it seems that an act of synthesis should be able to add the “I think” to other representations. Thus we would have overlooked a very specific kind of synthesis which attaches “I think” to intuitions and concepts. This is not the case. It is impossible to think of the addition of “I think” to another representation as a synthesis because these other representations only become representations (to me) by already having the “I think” attached to them. To again rely on the metaphor adding an initial zero (the representation of “I think”) to a painting with a CPU is impossible and wouldn’t make that painting binary or accessible to the CPU.
So how is the “I think” related to our representations?
Kant argues that the “I think” is part of the process of combination itself. The [...] relation [between identity of the subject and different representations] therefore does not yet come about my accom-panying each representation with consciousness, but rather my adding one representation to the other and being conscious of their synthesis.58
There is no way to become aware of the “I think”, that is, to attain access to the pure apperception, other than attending to the way one representation is synthesised with another. Hence there is no additional synthesis which combines concepts and pure apperception. We should rather think of pure apperception as a form which only emerges in the act of actually combining something.
Therefore it is only because I can combine a given representations in one consciousness that it is possible for me to represent the identity of the consciousness in these representations itself.59 While the possibility of combination relies on the unity which gives identity of consciousness, the possibility of this combination produces the awareness of the identity and unity of consciousness. That means that the possibility for combination is prior to the unity of consciousness but that the unity of
58[B134] 59[B133]
consciousness is only accessible through observing an actual combination.60 So to sum up there is an act of representation which is called pure apper-ception, the act of thinking “I think”. This act goes with all representations and is a necessary condition for the representation of anything. Further the representation of “I think” is the result of our adding one representation to another and being aware of their synthesis.
This means that the unity of apperception is essentially synthetic, as it relies on our combining (or adding up) of representations. Furthermore it means that spontaneity, as the capacity to synthesise, underlies all representation. This discussion has led us seemingly far astray from the first questions of knowledge and how space ought to provide us with “fixed points in nature.” But the insight we have gained through analysing the process of representa-tion has led us to an important finding. In our search for “fixed points” we have come to see that one capacity for synthesis is common to all represen-tations we have. We understand now that all knowledge is funnelled through our synthesis. The synthesis thus must be the place where the necessity of knowledge emanates from.
To understand mathematical and physical knowledge, which is paradigmatic for knowledge as such, we need to understand which synthesis to study. For physics (in Kant’s and Newton’s time) motion is the most central notion. Motion is change of space through time. Thus the synthesis to consider is one which gives representations of space. That means that to study the role of knowledge in conjunction with the synthesis of imagination we should turn to the combination of spatial intuitions.
Space and Objectivity
We have come to see that the unity of apperception is the form which all our representation (explicitly also intuitions) need to have. We have also noted that this unity is a synthetic one. Any synthesis has to conform to, or fall under, this unity. All synthesis has the form that allows us to be attentive to it and recognise ourself thinking.
We noted above that space and time are also forms. They are forms of how certain representations (namely intuitions) are given to us.61
60We have seen this reciprocal priority earlier and shall return to it as promised. 61“The supreme principle of the possibility of intuition in relation to sensibility was,
All the manifold representations of intuition stand under the [..] [formal conditions of space and time] as they are given to us, and under [...] [the original synthetic unity of apperception] insofar as they are combined in one consciousness.62
Intuitions can thus not only be seen as given but also as synthesised.
If one sees them as given, they fall under the form of time and space but if one considers them as the result of a synthesis, they fall under the unity of apperception.
In the Transcendental Aesthetic Kant understood intuitions merely as given. In the Transcendental Deduction space and time appear as representations. Earlier we asked how a representation of space might be possible. Being a representation means that space also needs to be thought in accordance with the unity of apperception. But for the unity of apperception to apply we need a synthesis as mediating operation. If space falls under the unity of apperception, which it does because it is a representation, it must also be the result of a synthesis. This seems to lead to a rift; first there is space which is a formal condition, which does not need to be represented but which given manifolds adhere to and second there is space as a representation which itself must be the result of a synthesis. Space is at once conditioning (the manifold) and conditioned (by the unity of apperception).
Kant solves this issue with positing two different kind of spaces. One as form and one as representation.
The first is called form of space, the second formal space, space represented or space thought as an object.
Space, represented as object (as is really required in geometry), contains more than the mere form of intuition, namely the com-prehension of the manifold given in accordance with the form of sensibility in an intuitive representation, so that the form of in-tuition merely gives the manifold, but the formal inin-tuition gives unity of the representation.63
Space as a representation is richer, because it is accompanied by the “I think” which allows for the unity of the representation. So the manifold which is given according to the form of space is not yet space without it being unified.
the formal conditions of space and time. The supreme principle of all intuition in relation to the understanding is that all the manifold of intuition stand under conditions of the original synthetic unity of apperception.”[B136]
62[B136] 63[B161n]
To understand the spatiality of spatial information its status as information needs to be realised first. Such a realisation (the “I think”) can only come about if space is the result of a synthesis.
This synthesis is called synthesis speciosa or figurative synthesis. Kant de-scribes its action in the following way:
But since in us a certain form of sensible intuition a priori is fundamental, which rests on the receptivity of the capacity for representation (sensibility), the understanding, as spontaneity, can determine the manifold of given representations in accord with the synthetic unity of apperception, and thus think a priori synthetic unity of the apperception of the manifold of sensible intuition, as the condition under which all objects of our (human) intuition must necessarily stand.64
This repeats the argument made above while taking it a step further: The premises of Kant’s argument are:
1. Space and time are a priori and fundamental.
2. Space and time (as form) rely on sensibility, which is a capacity for representation.
3. Only intuition and concepts together can form experience. 4. Representations fall under the synthetic unity of apperception.
Combining the premises 2 and 4 seems to just repeat the argument explained above. This argument ought to do more though, and although it is beyond the scope of this work to discuss the two step readings in detail,65 we can say that this second argument goes a step further in insisting that when we add premise 1 we obtain the result that not only time and space as objects fall under the unity of apperception, but that any objects given within time and space will fall under the unity as well.
After this argument Kant continues by linking the unity of apperception to the concepts of the understanding, or the categories. He argues there that the application of the understanding to objects given in intuition, that is given through sensibility, is secured by the fact that the synthesis of pure
64[B150]
65The fundamental problem is that it remains unclear why the extra step through time
and space is actually necessary, any representation should fall under the unity of apper-ception without mediation, simply because it is a representation. For an overview of the literature concerning the two step readings see [38].
concepts (the categories) falls under the same unity as the synthesis of pure intuitions.
The synthesis of pure intuitions is the synthesis speciosa or productive syn-thesis. This synthesis which combines intuitions and puts them under the rule of the understanding falls under the imagination: “Imagination is the faculty for representing an object even without its presence in intuition.”66 This faculty can be spontaneous, as it does not rely on the senses to give any input, but is “determining.” It “determines the form of sense a priori in accordance with the unity of apperception.”67 Form of sense is of course space and time. Thus space and time which appeared to be merely given have found their determining a priori synthesis in the synthesis speciosa. This theoretical framework is explained with a phenomenological insight:
We cannot think of a line without drawing it in thought, we cannot think of a circle without describing it, we cannot represent the three dimensions of space at all without placing three lines perpendicular to each other at the same point.68
The act of drawing in thought, of describing the circle or placing lines perpen-dicular is an act of the productive synthesis. Without it no representation of spatial (or temporal) information is possible. Even stronger, no true objec-tive representation (which is experience) is possible without it, as we could not represent intuitions without them being produced through a synthesis. There would be nothing binding together the intuitions given in the form of space and time. Thus the synthesis speciosa encountered here is fundamental to the synthesis of intuitions.
We have gained an understanding of the relationship between the intuitions and the understanding as mediated through synthesis. Furthermore this pro-vides an analysis of how the categories can be applied to objects given in time and space. All of these relations are thoroughly pure, however, and it is un-clear how they relate to our actual empirical cognition (as for instance in seeing a table).
Such empirical cognitions are generated by a synthesis as well. In opposition to the synthesis of the apperception that synthesis is called the synthesis of apprehension. The synthesis of apprehension furnishes us with perceptions and as such with empirical appearances.
First of all I remark that by the synthesis of apprehension I under-stand the composition of the manifold in an empirical intuition
66[B151] 67[B152] 68[B154]
through which perception, i.e. empirical consciousness of it (as appearance), becomes possible.69
Perceiving is not thought as a passive state in which impressions are made upon us and we then process these impressions to give us cognition. The perception itself is thought of as an act. This means that instead of think-ing of the cognitive process as a filter on the world we make a Copernican turn and the world (of our experience) becomes constructed by our mind.70 This constructive process is the synthesis of apprehension. The synthesis of apprehension follows the rules of time, space and the categories.
We have forms of outer as well as inner sensible intuition a priori in the representations of space and time, and the synthesis of the apprehension of the manifold of appearance must always be in agreement with the latter, since it can only occur in accordance with this form.71
So the apprehension, because it produces perceptions and thus intuitions, has to be such that the intuitions actually have the form of space and time. That is, they must be temporal and spatial in order to obtain the status of perceptions. This is only possible if the synthesis which creates these per-ceptions conforms to time and space.
The process of a synthesis conforming to a form can maybe be illustrated by a comparison to industrial production. In a rubber duck production for in-stance a mass of plastic granulate is filled into a mould, the top of the mould hits the bottom and through heat and pressure the granulate is combined into a rubber body.
This process is like the synthesis of apprehension. A manifold (plastic granu-late) is combined (melted and pressed) in accord with some form (the rubber duck mould).
For the synthesis of apprehension the form is not rubber duck but time and space.
Given all we have learned we can now understand how objectivity is achieved. First it follows from the all encompassing unity of apperception which is part
69[B160]
70This is not a solipsistic idealism but follows the rules of time, space, the concepts
of pure understanding and the concepts which are presented to us by this experience. Our experience is thus not random, and purely subjective but there are objective facts given by common cognitive processes. For Kant being human means having such cognitive processes.
of the understanding and is thus in accord with the pure concepts.
Second our representations of space and time (which are unified through the understanding) are the ground for the unity of the synthesis of apprehen-sion. These representations fall under the unity of apperception through the productive synthesis of imagination. As the unity is a result of applying the synthesis speciosa it follows that for any apprehension the synthesis speciosa has to come first.
Consequently that synthesis which constructs our representation of space is also involved in apprehending any appearance.
So now finally the question which we posed in the beginning is answered. The fixed points in nature are accessible through the synthesis needed for the representation of space.
If we understand how space (and time) is constructed we will have gained insight into how any of our experiences come about. Understanding how space is constituted effectively tells us something about the structure of how experience is constituted.
The structure of space is pure or a priori, thus whatever we come to know about space has the necessity of the “fixed points” Kant set out to find. Fur-thermore we have shown how this pure space is linked to experience. The world is what appears to us in experience, so the structure of experience is the structure of the world. Nature is just the world and so the “fixed points” in nature are the structure of space and time, as well as the categories. This means that our experience in physical experiments is fundamentally un-derstandable through geometrical considerations. When we understand our pure intuition we produce physical scientific knowledge, which then, through the second critique should lead us to just action. This means we have man-aged to expose the thread from physics to the world.
Now we need to take a close look at the structure of space to obtain the connection between geometry and physics. To do so we need to understand how space itself is created through the productive synthesis of imagination.
The Productive Synthesis of Imagination
In the last section we discussed the role the productive synthesis of imagina-tion plays in relaimagina-tion to experience, space and other syntheses. This secimagina-tion is dedicated to fleshing out what kind of operations the productive synthesis performs and how they relate to geometry as a science.
