Exploring
Mathematics and the Noumenal Realm through Kant and Hegel
Abstract
This
paper discusses the philosophical basis of mathematics by examining the
perspectives of Kant and Hegel. It explores how Kant’s concept of the synthetic
a priori, grounded in the intuitions of space and time, serves as a foundation
for understanding mathematics. The paper then integrates Hegelian dialectics to
propose a broader conception of mathematics, suggesting that the relationship
between space and time is dialectically embedded in reality. By introducing the
idea of a hypothetical transcendental subject (HTS), the paper attempts to
overcome a potential limitation of Kant’s framework, particularly regarding the
application of mathematical truths to pre-human reality. This synthesis of
Kantian and Hegelian thoughts offers a lens through which the connection
between mathematics and reality can be understood, while also acknowledging
limitations in both philosophical systems.
Keywords:
dialectics; philosophy of mathematics; noumena; absolute spirit; transcendental
1. Introduction
There
are radical arguments that mathematics explains everything today. For instance,
Tegmark (2008) asserts that “our universe is
mathematics” (p. 1). His argument is debatable, but there is no denying
that mathematics matters enormously. But what is mathematics? Also, how to
explain its sheer efficiency in science?
When
discussing the foundations of mathematics, we typically encounter formalism,
logicism, and intuitionism. Formalism
was propounded by David Hilbert, who sought for “a formalization of all of
mathematics in axiomatic form” (Zach, 2023, Section 0). Under his formalist
program, “mathematical propositions and proofs … turn into formulas and
derivations from axioms according to strictly circumscribed rules of
derivation” (Section 1.3). His program sought to remove Kant’s intuitionistic
elements from mathematics. Logicism
was pursued by Bertrand Russell and Alfred Whitehead. Their school dictates
that mathematics can be reduced to logic and that all mathematical truths are
ultimately logical truths. “Russell ... emphasized that a chief aim of the
logicist project is to show that arithmetic and real analysis are not grounded
in Kantian intuition” (Goldfarb, 1982, p. 692). Intuitionists, on the other hand, hold that mathematics is
constructed by the human mind. They “hoped to found mathematics ... on our a
priori intuition of time” (Maddy, 2012, p. 486). They believed that
“mathematics is true of the world, and we can know this a priori, because the
world ... is partly shaped by our temporal form of intuition.”
Given
the influence of Kantian philosophy on the three schools, it is essential to
consider its ideas when researching the philosophical basis of mathematics. After
investigating them, the paper will present two brief philosophical theses for
mathematics under the Kantian context. Section 2 provides the first thesis that
defines mathematics. Section 3 presents the second one that explains the reason
for its efficiency. Section 4 discusses one worry about Kantian philosophy –
that is, how to understand the world before the era of the human species. To
address the worry, Section 5 examines Hegelian philosophy. In Section 6, we will
discuss Hegel’s conceptions of space and time. Building on these, Section 7
offers a renewed thesis for mathematics. Section 8 indicates issues with
Hegelian philosophy. Then, the paper will conclude that ultimate reality is not
fully knowable.
2. Kantian Philosophy of Mathematics
In
the Critique of Pure Reason (1998),
Kant asserts that “there are two pure forms of sensible intuition as principles
of a priori cognition, namely space
and time” (p. 157). “By means of outer sense ... we represent to ourselves
objects as outside us, and all as in space. In space their form, magnitude, and
relation to one another is determined.” Further, “[s]pace is a necessary representation,
a priori, which is the ground of all
outer intuitions” (p. 158). It is “in this a
priori necessity” that the “apodictic certainty of all geometrical
principles and the possibility of their a
priori construction are grounded.”
Meanwhile,
regarding “inner sense,” Kant notes that “everything that belongs to the inner
determinations is represented in relations of time” (p. 157). If the origin of
geometry can be attributed to our intuition of space, what branch of
mathematics could be said to be based upon the intuition of time? Kant states
that “arithmetic forms its concepts of numbers through successive addition of
units in time, but above all pure mechanics can form its concepts of motion
only by means of the representation of time” (Kant, 2004, p. 35). Regarding
this, Shabel (2021) argues that “a formal intuition of time is inadequate to
explain the general and abstract science of number” (Section 2.3). Accordingly,
Shabel believes that “Kant declares mechanics to be the mathematical science that
is to time what geometry is to space.”
Whether
it is arithmetic or mechanics that naturally derives from the intuition of
time, it seems clear that Kant prioritizes geometry over arithmetic. In the Critique of Pure Reason, Kant does not
accord arithmetic the same level of explicit endorsement through the intuition
of time as that he provides for the “apodictic certainty” of geometry through
the intuition of space.[1] Moreover, Sutherland
(2004) argues that “magnitudes are at the heart of Kant’s theory of mathematical
cognition” and that Kant’s treatment of them is “strongly influenced by the
Greek mathematical tradition” (p. 157). As well known, the “Greek mathematical
tradition ... gave priority to geometry over arithmetic” (p. 158). For Kant,
“universal arithmetic (algebra) concerned the ratios between magnitudes” (p.
164). In his late times, Frege expressed a somewhat similar view that “the
concept of number requires pure temporal and spatial intuition in the Kantian sense” (emphasis added) (Hanna,
2001, p. 164). Meanwhile, Sir William Rowan Hamilton maintained that “since
geometry is a science of space, and since time and space are ‘pure sensuous
forms of intuition,’ algebra must be a science about time” (Burton, 2011, p.
634). Whatever the case, our Kantian thesis on mathematics can be briefly put
as follows:
Mathematics originates as a synthetic a priori[2]
discipline that deals with spatial constructs and numerical values, grounded in
our intuitions of space and time and developed in accordance with logic. (Thesis
1)
However,
there are several branches of mathematics that are seemingly independent of
arithmetic and geometry. To name just a few, there are combinatorics, graph
theory, set theory, etc. But none of them are entirely separable from the two
rudimentary concepts of mathematics: shapes and counting.[3]
First,
combinatorics is “concerned with arrangements of the objects of a set into
patterns satisfying specified rules” (Brualdi, 2010, p. 1). Although it has
nothing to do with, say, Euclidean geometry, it still relies on the abstract
spatial idea of “arrangements.” Moreover, the basic themes of combinatorics
such as permutations and combinations are closely related to the concept of
counting. Secondly, graph theory studies the properties and structures of
graphs, which are collections of vertices (or nodes) connected by edges (or
links).[4] One would not be able
to envision a concept of linking between vertices if she did not have an intuition
of space. Thirdly, set theory investigates “sets of mathematical objects, such as numbers, points of space, functions,
or sets” (Hrbacek, K., & Jech, T., 1999, p. 1). Set theory defines natural
numbers as follows:
1={∅}={0},
2={∅,{∅}}={0,1},
3={∅,{∅},{∅,{∅}}}={0,1,2},
and
so on.[5]
We
face one question: Does the above set-theoretic construction of natural
numbers, which causes no apparent conflict within Zermelo-Fraenkel set theory,
rightfully precede the intuitive concept of counting? According to Hanna
(2001), the late Frege’s view that “number requires pure temporal and spatial
intuition” could have “the same logical force as a well-founded or
non-paradoxical set theory -- only without the controversial axioms required
by, say, the Zemelo-Fraenkel theory” (p. 164). Therefore, despite the formalist
merits of the set-theoretic construction of numbers, the Kantian approach can
provide a more fundamental basis.
Given
this foundational understanding of mathematics, we will explore in the
subsequent section how these abstract, a priori principles could translate into
the effective applications of mathematics in the real world.
3. Kantian Explanation on Unreasonable Effectiveness
of Mathematics
Through
a Kantian framework, this section attempts to explain Wigner (1960)’s “unreasonable
effectiveness of mathematics in the natural sciences.” Wigner remarks that “the
enormous usefulness of mathematics in the natural sciences is something
bordering on the mysterious” (p. 2). Einstein’s general relativity provides a
good example. Wald (1984) comments that “general relativity has made a number
of strikingly successful predictions concerning the spacetime structure of our
universe” (p. 118). One of them is black holes. Given Einstein’s gravitational
field equations, Schwarzschild considered the case of spacetime surrounding a
spherically symmetric, non-rotating mass (e.g., a star) in the absence of any
other matter or energy. In this scenario, the “stress-energy tensor” in the
Einstein field equations is set to zero, which describes a “vacuum” solution.
Schwarzschild derived the following solution to the Einstein field equations:[6]
ds²
= - (1 - 2GM / c²r) c² dt² + (1 - 2GM / c²r)⁻¹
dr²
+ r²(dθ²
+ sin²θ dφ²)
where:
ds
= spacetime interval
G
= gravitational constant
M
= mass of the non-rotating mass
c
= speed of light
r
= radius of the non-rotating mass
dt
= infinitesimal time interval
dr
= infinitesimal change in the radius
dθ = infinitesimal change in the polar angle (between the position
vector of a point and the z-axis in spherical coordinates)
dφ = infinitesimal change in the azimuthal angle θ (around the z-axis).
When
a particular mass satisfying the vacuum condition undergoes gravitational
collapse, its radius can decrease to equal the “Schwarzschild radius” (2GM/c²).
At this point, the time term “- (1 - 2GM / c²r) c² dt²” becomes zero (i.e.,
infinite time dilation), and the radial term “(1 - 2GM / c²r)⁻¹
dr²”
becomes infinite (i.e., infinite spatial stretching) for any finite “dr.” This is
known as the event horizon of a black hole. The existence of black holes has
been empirically confirmed through observations.[7] It is remarkable that
mathematical manipulation of the equations led to the prediction and subsequent
discovery of distant celestial bodies, whose existence is highly unintuitive.
What explains this unreasonable effectiveness of mathematics?
From
a Kantian perspective, mathematics corresponds to “reality” because it is
fundamentally rooted in our processes of imagination,[8] which preemptively
organizes the manifold[9] through schemata.
These schemata, as procedural rules, allow the imagination to construct mathematical
concepts that are linked to the spatio-temporal framework of our perception.
Since space and time are the forms through which we experience the world, and
mathematics operates within these forms, mathematical constructions are
inherently aligned with the way reality is perceived and understood.
In
a similar vein, Hall (2013) concludes that “imagination, via its operation of
schematising … could be aptly understood as the mathematiser of reality” (p.
244). Moreover, the manifold could be thought of as “a blank sheet of paper
covered with dots, in which case the (productive) imagination would be the
function or power connecting the dots according to a rule-governed procedure or
schema … allowing for the construction of, for example, geometric figures,
arithmetically countable strokes and algebraic symbols.” Thus, we can formulate
the following thesis:
Mathematics appears
to reflect reality because our imagination, at a preconscious level,
provides a mathematical rendering of the manifold for our intellectual faculty,
which operates within the intuitions of space and time. (Thesis 2)
However,
the thesis is not without its limitations. In the next section, we will briefly
discuss one worry about it.
4. One Worry about Kantian Philosophy
According
to Cummins (1968), “Kant asserted that neither the objects nor the forms of any
intuitions have transcendental reality; they merely have phenomenal (empirical) reality. However, although neither outer nor
inner intuition is apprehension of independent
(transcendental) reality, both are related to bodies of synthetic apriori
knowledge” (emphasis added) (p. 271). Indeed, Kant says:
“[I]f
we remove our own subject or even only the subjective constitution of the
senses in general, then all constitution, all relations of objects in space and
time, indeed space and time themselves would disappear, and as appearances they
cannot exist in themselves, but only in us” (Kant, 1998, p. 185).
In
other words, space and time as well as physical objects intuited in their
spatio-temporal form are not real independently of human cognition. This suggests
that the mathematically derived truths of natural science might not have
applied before the emergence of our species. However, it seems to go against
our empirical “intuition” (or gut feelings). For instance, Žižek
(2012) asks, “[I]s a dinosaur fossil proof that dinosaurs existed on Earth
independently of any human observer, whether empirical or transcendental? If we
can imagine transposing ourselves into the pre-historical past, would we
encounter dinosaurs the way we reconstruct them today?” (p. 647). This question
may be addressed in a simple way. In transcendental philosophy, it is nonsensical
to ask what would have happened before human beings, because there were no
cognitive agents who had intuitions of space and time. Similarly, Hanna (2001)
notes: “If creatures minded like us had not existed, then purely logically
speaking something might still have existed, but the assertion that it did
exist would have been at the very least empirically meaningless and without a
truth value” (p. 105).
In
other words, it is meaningless to ask what happened “before” the human species,
because time is meaningful only from a transcendental subject’s perspective.
However, Meillassoux (2008) is not satisfied. He asks, “what is it exactly
astrophysicists, geologists, or paleontologists are talking about when they
discuss the age of the universe, the date of the accretion of the earth, the
date of the appearance of pre-human species, or the date of the emergence of
humanity itself? How are we to grasp the meaning of scientific statements
bearing explicitly upon a manifestation of the world that is posited as
anterior to the emergence of thought and even of life -- posited, that is, as anterior to every form of human relation to the
world?” (pp. 9-10).
To
answer the question, we can try assuming the legitimacy of establishing a
viewpoint held by a hypothetical
transcendental subject (or “HTS”) “before” humans came into existence. This
assumption is necessary when considering that the “thought that something might
have existed even if we had not already presupposes the cognitive capacities of
creatures minded like us” (Hanna, 2001, p. 105). By introducing the HTS, we can
retain the idea that the world in the “pre-historical past” might have existed
as it could be perceived by humans. So, to answer Žižek’s
question, yes, we would be able to discover dinosaurs as speculated today if we
time-traveled into the past. However, this scenario cannot escape from bringing
in the idea that time might be transcendentally real, as we should assume that
the world had been perceived through “virtual” intuitions of space and time.
But this conflicts with orthodox transcendental philosophy. Yet, adhering to
the existing doctrine of transcendentalism may not satisfy our growing sense
that scientific realism deserves more consideration, if not full acceptance.
To
reconcile between Kantianism and scientism, we must redefine the noumenal realm
by incorporating our intuitions. This redefinition can involve introducing the
HTS. If this existence can be justified, there is no reason to not suppose that
the noumenal realm shares a certain spatio-temporal structure with our
consciousness.[10] This issue will be
explored through a Hegelian framework in the next section.
5. Hegel’s Absolute Idea as Alternative to Noumenon
“[E]ternity
is not before or after time, not before the creation of the world, nor when it
perishes; rather is eternity the absolute present, the Now, without before and
after. The world is created, is now being created, and has eternally been
created; … Creating is the activity of the absolute
Idea; … the true universal is the Idea, which … is eternal. The finite,
however, is temporal” (emphasis added) (Hegel, 2004, 15).
Judging
from Hegel’s text, the “absolute Idea” (or simply “Idea”) can be described as super-temporal. Simply speaking, it is
beyond our temporal realm. Meanwhile, the Idea can originate time while the
Idea itself remains atemporal. From a standpoint located in the Idea,[11] the endless time span
of the world is graspable.[12] Meanwhile, when
confronted with the origin of the world, our finite consciousness can only
detect contradictions. This is evident in Kant’s first antinomy.[13] Let us say that such a
feature shows that our temporal realm is a sub-temporal
one. Unlike any hypothetical being that could have a super-temporal standpoint,
we cannot experience the “before” and “after” at the same time. On the other
hand, they can be seen simultaneously from “the absolute present.”
Then,
how could a source that is outside of time specifically create a phenomenal
world (or “Nature”) for the HTS? The answer lies in the dialectic formula:
thesis-antithesis-synthesis. According to Hegel, “Nature” is “the negative of
the Idea” (Hegel, 2004, p. 19). More specifically, Nature is an externalized
form of the Idea as its antithesis. Then what is the synthesis of the Idea and
Nature? It is the Absolute Spirit, which is the full realization and
self-manifestation of the Idea. Hegel says:
This
somewhat abstruse text can be understood as follows. While achieving absolute
self-identity throughout eternity, the Absolute Spirit can also transition into
something else and yet identify itself with this something. It is also
unchangeable in that it perennially (“unchangeably”) returns to itself after
the transition. Moreover, it represents the speculative movement[14] of consciousness
(which can be partially shared by finite consciousness). Specifically, from the
Absolute Spirit’s perspective, objective reality, which appears transient to
finite consciousness, appears permanent.[15] From our standpoint,
the sense of “now” persists only for a short time. Throughout this temporal
expanse, we are quite certain we are the same persons that we are. Although the
exact neural state of our consciousness at the beginning of this time must
differ from that at the end of this time, we maintain a sense of continuity
throughout. However, the Absolute Spirit differs from our finite consciousness
in that it retains this sense of “now” throughout eternity. Therefore, its
activity can be said to be pan-temporal.
That is, the Absolute Spirit is a pan-temporal consciousness.
Adopting
this principle, we can connect the sub-temporal dimension of our realm with the
higher source of all things (i.e., the Idea). Since we are a part of the
Absolute Spirit, which manifests the Idea and looks over Nature in a
pan-temporal manner, we can indirectly relate to the Idea even though we stay
in a sub-temporal domain. Under this scheme, space and time can be treated as
more “real” than they were under the Kantian scheme. The super-temporal Idea carries
within it the seed of the pan-temporal Spirit. The Absolute Spirit mediates
between the Idea and the phenomenal, thereby ensuring the “realness” of space
and time throughout eternity.[16] This suggests that
mathematics can be partially structurally embedded within reality even without
our existence.[17] Thus, mathematics can
explain reality. As such, Hegel’s speculative philosophy bridges between the
Kantian construction of reality and contemporary scientific endeavors. Based on
this renewed understanding, we can establish a new thesis for mathematics.
However, before we do that, we need to understand Hegel’s discussion on space
and time.
6. Hegelian Approach to Space and Time
(1)
Hegel on Space
“The
first or immediate determination of Nature is Space, the abstract universality of Nature's self-externality,
self-externality’s mediationless indifference. It is a wholly ideal side-by-sideness because it is
self-externality; and it is absolutely continuous,
because this asunderness is still
quite abstract, and contains no specific difference within itself” (Hegel,
2004, p. 29-30).
Hegel’s
text suggests space is the most basic and immediate way that Nature takes form.
However, it is unclear how space can be regarded as the “abstract universality”
of Nature’s self-externality. Hegel likely would have meant that space is an
externalized form of Nature. But how could it be “abstract universality”? This could
suggest that space is a universal concept that applies everywhere in its pure
form. At the same time, it has “mediationless indifference,” meaning that space
lacks internal distinctions. Moreover, space is simply a continuous
“side-by-sidedness” where parts exist next to each other without any deeper
mediation. Now, let us examine what Hegel says about time.
(2)
Hegel on Time
“Time,
as the negative unity of self-externality, is similarly an out-and-out abstract,
ideal being. It is that being which, inasmuch as it is, is not, and inasmuch as
it is not, is: it is Becoming directly intuited; this means that differences,
which admittedly are purely momentary, i.e. directly self-sublating, are
determined as external, i.e. as external to themselves” (Hegel, 2004, p. 34).
The
“negative unity of self-externality” likely indicates that time is the
antithesis of space. However, even as time exists, simultaneously it does not
exist. Or even as time does not exist, it exists. Here, Hegel seems to
anticipate dialetheism. Further, “[time] is Becoming directly intuited.” In
other words, a temporal transition is something that is immediately perceived
or understood by us without reasoning or mediation. In the previous section, we
contrasted the short time span of the “now” that we experience with the eternal
time span of the “Now” occupied by the Absolute Spirit. The time discussed in
the above text pertains to the former and not the latter.
(3)
Space and Time
Regarding
the conceptual sequence of space and time, Hegel says:
“The
truth of space is time, and thus space becomes time; the transition to time is
not made subjectively by us, but made by space itself. In pictorial thought,
space and time are taken to be quite separate: we have space and also time” (Hegel, 2004, p. 34).
In
the previous section, we saw how the phenomenal world (“Nature”) came into
being through the thread of dialectics within a three-tiered temporal
hierarchy: Super-temporal (Idea) – pan-temporal (Absolute Spirit) – sub-temporal
(phenomenal realm). Does this contradict what Hegel said above? No. Again, the
“time” in Hegel’s remarks relates to our phenomenal realm. Thus, there is no
inconsistency.
(4)
Motion in Space and Time
“In
motion, space posits itself temporally and time posits itself spatially ...
Motion falls into the Zenonian antinomy, insoluble if the places are isolated
as points of space, and the time-moments as points of time; and the solution of
the antinomy, i.e. motion, is reached only when space and time are grasped as
in themselves continuous,[18] and the moving body as
being at once in and not in the same place, i.e. as being at once in another
place; just as the same point of time at once is and is not, i.e. is at once
another point of time” (Hegel, 2004, pp. 134-135).
The
dialectical relationship between space and time is evident in the statement
that “space posits itself temporally and time posits itself spatially.”
However, this relationship is explained with respect to a moving body. Under
the context of motion, they are not isolated entities; they are interwoven.
When the object moves, its position in space changes as time goes on. This
temporal passage can also be delineated through the object’s motion.
Furthermore,
whereas time is dynamic, space is static. In this regard, a spatial segment can
be understood as a static, simultaneous representation of
successive units in time. Conversely, time can be represented as dynamic, successive changes throughout space (via motion). In the next
section, these notions will be taken into consideration when formulating a new
thesis for the philosophy of mathematics.
7. New Thesis for
Philosophy of Mathematics
Based on everything that we have
discussed, we deduce:
Mathematics is a synthetic a priori discipline that
logically engages with spatial constructs and numerical values, grounded in our
intuitions of space and time, which are objectively embedded as a dialectical
pair[19]
in the manifest world. (Thesis 3)
Arguably,
Kant’s greatest contributions to the philosophy of mathematics are (1) his clarification
of mathematics as a synthetic a priori discipline and (2) demonstration of its
connection to the phenomenal world through the intuitions of space and time. However,
his philosophy was limited in scope due to its exclusive focus on explaining
everything through a transcendental lens. By replacing Kant’s noumenon with
Hegel’s Idea, we can now justify our mathematization of reality as that of a
substantially real outside world. This carries an important implication that
our mathematical construction is at least partially embedded in the universe.[20] This is enabled by
justification of the HTS’s ontological status. Then what guarantees its
existence? It is none other than the pan-temporal Spirit.
However,
Hegelian philosophy, which the above thesis is based upon, is not without its
limitations, either. We will investigate this in the following section.
8.
Limitations of Hegelian Philosophy
According to Taylor
(1975), “Absolute spirit is … higher than Spirit’s realization in objective
reality which has not yet come to full self-consciousness” (p. 466). However,
it is questionable whether the Absolute Spirit ever will come. One cannot help
but feel that Hegel’s system of philosophy is overly ideal.
First, even if a state
of absolute knowledge were achievable, it is difficult to imagine what state of
mind we would actually experience in such a condition. Suppose nevertheless
that our world did enter the stage of absolute knowledge. Even in this stage,
we would be the finite components of the Absolute Spirit. Therefore, absolute
knowledge would be still unattainable on an individual level. One could say
that all the truths of the universe could be potentially knowable because
finite consciousness partakes in the Absolute Spirit. But this reminds us of a
democracy where every citizen could potentially become president but only a
select few ever do. Further, while citizens may actively care about how their
state is run by their government, most of them are completely in the dark about
what it is doing on their behalf.
Second, Hegel’s
philosophy might be too teleological. His system implies that history and
reality are progressing toward a specific, rational end. While this provides a
coherent picture of the world, it may not be realistic. For instance,
evolutionary biology teaches us that life is not driven by any intrinsic
purpose but is instead the product of random mutations and natural selection.
Evolution is contingent and unpredictable, with no predetermined direction. Also,
considering that we are potentially on the brink of nuclear annihilation, it is
hard to believe that there is some teleological force driving our
civilizations. Although this critical view might seem irrelevant to the topic
of our discussion, we must note that teleology plays an important role in
Hegel’s dialectics. In Hegel’s view, it is necessary that finite spirit emerge[21] due to the
“requirement that Geist be embodied”
(Taylor, 1975, p. 89).
In the author’s humble
opinion, the only way for Hegel’s philosophy to make sense is to assume that
the Absolute Spirit has always existed and will continue to exist, regardless of our knowing it. This way,
we can ensure the realness of space and time. Or we should rid of “Absolute
Spirit” and instead assume that the “absolute Idea” possesses the dual feature
of super-temporality and pan-temporality. But this approach would weaken the authority
of the dialectical system as established by Hegel. In Hegel’s system, the Idea
is atemporal but carries within it the potentiality to generate temporality, precisely because of the ontological principle
of dialectics.[22] That is,
pan-temporality should be achieved as a synthesis of super-temporality and an
externalization of the super-temporality. Therefore, the proposed “dual feature”
is not a good enough solution.
9.
Conclusion
This paper’s ideas can
be briefly summarized as follows.
(1) Hypothetical
Transcendental Subject (HTS)
: A conceptual entity
proposed to bridge the gap between Kantian philosophy and the reality of the
pre-human world, which enables the application of mathematical truths in a
context independent of human cognition.
(2) Three-tiered
Temporal Structure
l Super-temporal
(Idea): Associated with Hegel’s absolute Idea, existing beyond our temporal
realm and outside of time.
l Pan-temporal
(Absolute Spirit): A mediating level where the Absolute Spirit retains a sense
of "now" throughout eternity, thereby connecting the super-temporal
and sub-temporal.
l Sub-temporal
(Phenomenal): The temporal realm of human experience and perception.
(3)
Mathematics
:
A synthetic a priori discipline that logically engages with spatial constructs
and numerical values within the context of human intuitions of space and time,
which are objectively embedded as a dialectical pair in the structure of the
universe.
As
well as the above ideas, the paper discussed both the strengths and limitations
of Kant’s and Hegel’s approaches. Kant's framework, with its emphasis on
synthetic a priori knowledge rooted in the intuitions of space and time,
provides a strong foundation for understanding how mathematics applies to the
phenomenal world. Building upon this, the paper sought to extend the foundation
by legitimizing the realness of the outside world by using Hegel’s ideas.
As
noted in the previous section, the Hegelian system also carries limitations. It
is perhaps more realistic to say that reality will largely remain beyond our
grasp. The principles of reality in its entirety, if they could be viewed from
the standpoint in the Absolute Idea, would not be expressible in linguistic
form. Any propositional judgment requires that it be established in a
sub-temporal realm -- that is, within the confines of time and the finite
perspectives that we inhabit. Thus, the entirety of our reality cannot be
comprehended from the viewpoint of a finite being, whose understanding is
limited to propositional judgments. In Kantian terms, it is beyond our judgment
or sensibility. A corollary of this view is that even if mathematics could
perfectly describe the physics of our reality, it would not be able to explain
everything.[23]
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[1] This view is shared
by Cummins (1968): “Geometry Kant straightforwardly associated with intuition.
In contrast, …, indeed nowhere in the Critique
of Pure Reason, did Kant shed much light on the basis of arithmetic or the
nature of the truths associated with inner theory of intuition” (p. 271).
[2] A synthetic a priori
judgment produces a proposition where the content of the predicate is not
deducible from that of the subject, yet it can be established independently of
empirical investigation. For details, see Kant (1998, pp. 143-144).
[3] This paper interprets
Kantian philosophy of mathematics to embrace every branch of mathematics as
long as it involves the concepts of shapes and counting. Therefore, even the
theorems of non-Euclidean geometry or Cantor’s set theory can be considered to
be synthetic a priori if they meet the following conditions in Hanna (2001, p.
279): (i) semantic experience-independence; (ii) intuition-dependence; (iii) consistent
deniability; and (iv) restricted necessity in the special sense that a
proposition is true in every humanly objectively experienceable world and lacks
a classical truth value in any other possible world. Regarding (ii),
non-Euclidean geometry relies on spatial visualization to some extent (though
not in the strong sense that Kant suggested), and Cantor’s theory relies on the
concept of counting, even though it is somewhat removed from our ordinary sense
of time.
[4] See Wilson (1996, p.
8).
[5] See Hrbacek, K.,
& Jech, T. (1999, p. 39).
[6] For details, see Wald
(1984, pp. 118-124).
[7] See Abbott et al.
(2016); Event Horizon Telescope Collaboration (2019); and Genzel et al. (2010).
[8] Imagination
“functions automatically, without being self-consciously willed into action --
not in the sense that it is a mere mechanism” (Hanna, 2001, p. 39). Moreover,
it is “essentially spontaneous, goal oriented, and vital.” Also, “as the engine
of synthesis, it is also the very seat or ground of all consciousness and hence
properly speaking preconscious.”
[9] Per Oxford Reference
(2024), “the manifold is the unorganized flux presented to the senses, but not
experienced, since experience results from the mind structuring the manifold by
means of concepts. The nature of the unstructured manifold is unknowable
(transcendental).”
[10] Hanna (2006) agrees
that “actual space and actual time must be … structurally identical (that is,
isomorphic) with our pure or formal intuitions of space and time” (p. 302).
More specifically, they are “intrinsically
the objectively real objects that actually satisfy our pure or formal
intuitions of space and time” (pp. 302-303). This can support the view that
we share a common spatio-temporal feature with the noumenal realm.
[11] Perhaps,
Kant’s “positive” noumenon is the closest one to the absolute Idea. Kant
considers the noumenon in both a positive and negative sense (Kant, 1998, pp.
360-361). Per Hanna (2001),
“negative noumena fall outside our sensibility” (p. 106). But this “leaves open
the possibility … that they are cognitively accessible to alien creatures with
different forms of sensibility.” Meanwhile, “a noumenon in the positive sense”
is “any fully thinkable yet non-sensible object, in so far as it could be
cognized by a being possessing a faculty of intellectual intuition, or divine
cognition, yet could not be cognized by a being possessing a finite sensory
cognitive capacity like ours nor indeed by any sort of sensible cognizer, human
or non-human.”
[12] But it is only
graspable for the Absolute Spirit.
[13] See Kant (1998, pp.
470-475)
[14] This speculative
movement is also reflected in the Liar Paradox.
[15] McTaggart (1908) argues
that “time is unreal” (p. 457). This is based on the notion that the two times
series -- i.e., “A series” and “B series” -- are incompatible. Specifically,
the A series describes time as a “series of positions running from the far past
… to the present, and then … to the far future.” The B series describes it as a
“series of positions which runs from earlier to later” (p. 458). Today, we
correlate the A series with “presentism” (i.e., time flows as events come to
exist and vanish) and the B series with “eternalism” (i.e., all moments of time
exist equally). Under our Hegelian scheme, both series can coexist. The A
series corresponds to the sub-temporal, while the B series corresponds to the
pan-temporal.
[16] Per Hanna (2006),
Kant’s “Transcendental Aesthetic ultimately implies that objectively real space
and time are neither wholly ontologically dependent
on minds like ours, nor wholly ontologically independent of minds like ours” (pp. 303-304). This view
anticipates our Hegelian approach. Specifically, we do not have to exist for
space and time to be ontologically real, because their existence is guaranteed
by the Absolute Spirit. Meanwhile, provided that finite consciousness should
necessarily emerge under the Hegelian system, the connection between us and
space/time is also inevitable.
[17] This view can
complement the Platonic view of mathematics. Mathematics is ontologically real
even without us. However, it does not exist in the super-temporal realm. Its
ontology relies on the Absolute Spirit.
[18] Hegel’s notion of
continuity differs from that of contemporary mathematics. In Hegelian
scholarship, continuity is synonymous with oneness or unity. For details, see
Pinkard (1981, p. 459).
[19] Minkowski was perhaps
the first to scientifically formulate the interconnectedness of space and time.
He famously stated: “Henceforth space by itself, and time by itself, are doomed
to fade away into mere shadows, and only a kind of union of the two will
preserve an independent reality” (Minkowski, 1952, p. 75).
[20] “Whatever is
mathematizable can be posited hypothetically as an ontologically perishable
fact existing independently of us. In other words, modern science uncovers the speculative but hypothetical import
of every mathematical reformulation of our world” (Meillassoux, 2008, p. 117).
[21] This teleological
view is also implicitly shared by Hanna’s (2006) weak transcendental idealism:
“Things can exist without existing human persons, and in fact did so for
millions of years before we came along. But things
could not have existed unless it were really possible for us to come along”
(emphasis added) (p. 32). If this teleological view is correct, then even if
humanity were to vanish from the face of the earth, “creatures minded like us”
would arise again, much like in a situation where, if the tip of an iceberg is
cut off, another portion would emerge as a new tip.
[22] This dialectical
principle is contradictory in that while it works as an inter-level mediator
between the super-temporal realm of the Idea and the sub-temporal realm of the
phenomenal, it can be also referred to by a finite being within the
sub-temporal realm. Simply put, dialectics surpasses us and yet is discernible
for us.
[23] Similarly, Lee (2024)
states: “Even if a [scientist] had all the information regarding her mind/body
as well as [the universe] from a materialistic viewpoint, she might still fail
to explain how her bodily composition gives rise to consciousness. Even a
complete mathematical formulation of the neural correlates of consciousness might
not fully elucidate its nature” (p. 25).
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