The hard problem of consciousness is the most pressing unsolved mystery in both philosophy and science. To solve such a problem, we are going to need revolutionary ways of thinking. Philosopher of mind, Peter Sjöstedt-Hughes, argues higher spatial dimensions might hold the key to the hard problem.
(This is an abridged version of the chapter ‘Deeper than Depth’ in the book Modes of Sentience)
‘The waking have one common world [koinos kosmos], but the sleeping turn aside each into a world of their own [idios kosmos]’ 
The inquiry into the relation between ‘mind’ and ‘matter’ too often stops at the status of profound mystery because those very terms are poorly understood. What is understood is that the solution to the mystery requires revolutionary thought, since present concepts do not provide sufficient scope through which to see the end of this cosmic riddle. We should therefore extend our vision and experiment with ocular instruments beyond those found in the traditional philosopher’s observatory. To gain a greater gaze into this outer space we will analyse space itself – in its relation to sentience – fracturing it into three varieties and raising it beyond three dimensions. The mind-matter mystery beckons us to explore the relations between space, matter, and mind. What follows is a playful trip of radical speculation through hyperspace.
1. Mind, Matter, and Spac
2. The Varieties of Space
3.The Dimensions of Space and Sentience
1. Mind, Matter, and Space
From the Cartesian legacy, it is often believed that matter is spatial (extensive) and mind, or sentience,  is non-spatial (inextensive). 
This dichotomy of spatial-matter and non-spatial-mind conveys Descartes’ dualism, and is no doubt partly responsible for the ecologically-cataclysmic scientific view of nature as merely mechanical, insentient. William James criticized the Cartesian assumption:
‘Descartes for the first time defined thought as the absolutely unextended … . But to argue … that experience is absolutely inextensive seems to me little short of absurd.’ 
In an earlier paper, ‘The Spatial Quale’, James had grounded such a view, speaking of the existence of spatial properties not only without but within imagination:
‘The primary [visual] sensation is a simple vastness, a teeming muchness. The perception of positions within it results from sub-dividing it. The measurement of distances and directions comes later still.’ 
If we close our eyes and imagine two triangles next to each other, we can speak of their spatial properties – such as each having three sides, angles that sum 180°, that one triangle may be to the left of, and above, the other, and vice versa, their relative proximity, boundaries and topological features, their relative sizes, etc. These properties are perhaps often not as stable as would be two triangles perceived with eyes open, yet they are spatial properties regardless of such stable durability. 
Thus we have here two types of space, correlated yet not prima facie identical. Bertrand Russell expresses such a dichotomy when he writes that:
‘percepts are not identical with material objects, and the relation of perceptual to physical space is not identity.’ 
2. The Varieties of Space
Thus we are led to posit the reality of twofold space: physical space and sentient space. This pair can also, respectively, be referred to as extrinsic and intrinsic space, or objective and subjective space. Unfortunately all of these labels bear metaphysical connotations that can interfere with a proper understanding of their essences and relation – as we shall come to understand (e.g. physical space is perceived as such within our subjective space, and sentient space may not be merely intrinsic). But for now we note the relation of this general twofold space to the Herclitus epigraph above where sentient space refers to the idios kosmos, physical space to our shared koinos kosmos. Sentient space not only includes that of imagination, but also that of hypnagogia, psychedeliscapes, and the space of dreams. Russell’s teacher and later colleague and friend, Alfred North Whitehead, also spoke implicitly of this distinctive idios kosmos:
‘The distinction between the dream-world and nature is, that the space-time of the dream-world cannot conjoin with the scheme of the space-time of nature, as constituted by any part of nature. The dream-world is nowhere at no time, though it has a dream-time and dream-space of its own.’ 
We must distinguish then physical from sentient space. Sentient space, however, ramifies into more than visual space.  We also have the sense of space related to our bodies: somatic space. Maurice Merleau-Ponty distinguishes somatic sentient space from physical space, and argues that the distinction lies in the fact that the latter is indirectly intellectualized in terms of relations or positions to objective geometric coordinates whereas the former is directly felt in terms of a top-down ‘global awareness’ of the situationally-determined location of one’s bodily parts. 
The haptic sense, touch, is a part of somatic space, and one that lends itself especially to the feeling of the real: ‘it is our sense of touch that gives our sense of “reality”’,  Russell claims. Moreover, the gustatory sense, i.e. taste, is also, arguably,  a type of touch (by the tongue) – one that yields more data (flavour) than does standard skin-based touch. As to whether the olfactory, auditory, and other senses,  involve spatiality is a question that I shall not address here (note that James argues that all senses are spatial).  It is sufficient for our purposes to show that there exist perceptual spaces distinct from extrinsic, physical space.
Further: there are spaces and spatial properties which we can conceptualize yet not visualize nor (at least directly) feel somatically. For instance, we can easily conceptualize four dimensional space by simply adding an axis w, to the traditional three – x, y, z – and thus create a hypothetical hyperspace  that can be developed through algebraic geometry. However, such a conceptual space is not identical to a visual space: though a fourth dimension orthogonal (right-angled) to our traditional three can be conceived, it is very difficult (to say the least) to visualize this directly.
Now, this threefold analysis raises a question: How are these varieties of space related? Let us here approach this question with a twofold space  – visual and physical space, idios and koinos kosmos – which will be a journey via hyperspace into the blackhole that is the mind-matter problem.
3. The Dimensions of Space and Sentience
‘[T]here can be infinitely many spaces and, hence, worlds, such that between them and ours there is no distance. ’ 
I close my eyes and imagine a blue equilateral triangle. This triangle exists as such, as an imagination, and has an endogenous correlate: certain activity in the occipital lobe of my brain, for instance. This latter is an example of the so-called neural correlates of consciousness.  The correlation presents the problem (not the answer) to the question of how visual and physical space relate. What are options  to resolve this relation?:
- Substance Dualism: the triangle and the neural correlates are separate substances, not dependent on each other.
- Emergentism: the triangle emerges from the neural correlates.
- Idealism: the neural correlates emerge from the mind.
- Psycho-neural Identity Theory: the triangle is the neural correlates.
- More-Broad-Smythies Theory: the triangle and the neural correlates are both cross-sections of a deeper hyperspace.
Let us here pass over the first four except to say in passing that innumerable criticisms have been levelled against each of them over the years. We have touched on one problem with substance dualism: the belief that mind is completely non-spatial. Emergentism cannot cope with upward (transordinal) and downward (mental) causation. Idealism begs the question by assuming perception is purely representational.  Psycho-neural identity theory assumes a neuroessentialism and cannot satisfactorily say why the spatial properties of mind and matter are identical despite their spatial differences. As neurophilosopher John R. Smythies put it:
‘Two groups of events arranged in a spatial order may not be said to be identical unless they are geometrically congruent. … [E]vents in the cerebral cortex … concerned in a particular perception are geometrically non-congruent with the sense-data that these events are alleged, under this [identity] theory, to be. … [This] can be used to refute with equal finality the theory of psycho-neural identity.’ 
An imagined triangle has its specific spatial properties (angles, boundaries, etc.) which are not identical to the spatial properties of the neuronal activity of which the triangle is the correlate. Therefore by the principle of the identity of indiscernibles, the psychological and neurological correlation cannot be a correlation indicating identity. 
A mental event then can have (at least) two simultaneous spatial configurations. If we reject dualism, idealism, emergentism, and psycho-neural identity theory, how can we explain this spatial simultaneity? This problem is a specific incidence of the more general mind-matter ‘explanatory gap’,  or ‘the hard problem of consciousness’.  It is a problem that has proved incendiary and intractable, and requires novel, radical approaches for its solution. One such solution expands the concept of space.
The More-Broad-Smythies Theory
‘[It] is impossible that sensa should literally occupy places in scientific space, though it may not, of course, be impossible to construct a space-like whole of more than three dimensions, in which sensa of all kinds, and scientific objects, literally have places. If so, I suppose that Scientific Space would be one kind of section of such a quasi-space, and e.g., a visual field would be another kind of section of the same quasi-space.’ 
C. D. Broad here proposes that the triangle one imagines and the correlated physical, physiological patterns could relate not as identity, duality, emergence, or ideality but as real spaces within a greater space that encompasses them both. By ‘greater space’ is meant one with more dimensions than the three of width, height, and depth – i.e. a space deeper than depth, one that unifies the physical with the mental. Before we explore this theory of mind-matter relation via greater space, let us consider epistemic and ontic possibilities of such space (as illustrated by the hypercube in Figure 1, where the top-left-to-bottom-right diagonal represent the fourth dimension of space.
Is space not three dimensional by definition? In the fourth century BC, Aristotle asserted so in On the Heavens: ‘the three dimensions are all that there are’,  as did Ptolemy in the second century AD,  along with many others. In 1685 the English clergyman and mathematician John Wallis writes that extra dimensionality ‘is a Monster in Nature, and less possible than a Chimaera or Centaure. For Length, Breadth and Thick-ness, take up the whole of Space’.  This possibility of a space of more than three dimensions, a hyperspace, was denied up to modern times, with Hans Reichenbach in 1926 arguing against hyperspace by claiming that it ‘would destroy all existing causal laws’  (though his speculations on hypothetical hyperspatial phenomenology are phenomenal!). 
The first affirmation of the possibility of a fourth spatial dimension comes through the Cambridge Platonist Henry More in his book of 1659, The Immortality of the Soul, where he calls the fourth dimension spissitude. This rather spiritual apprehension of hyperspace was reflected in the twentieth century by certain writings  of the Welsh, Oxford philosopher H. H. Price – who, incidentally, was one of the first philosophers to write on the psychedelic (mescaline) experience.  In his later book of 1671, the Enchiridion Metaphysicum, More explicitly writes that ‘besides the three dimensions which are filled with all extended material things, a fourth must be admitted, with which coincides the spirit’.  A century later in 1746, in his very first publication, Immanuel Kant considers hyperspace as the condition of other universes:
‘If it is possible that there are extensions of different dimensions, then it is also very probable that God has really produced them somewhere. For his works have all the greatness and diversity that they can possibly contain. Spaces of this kind could not possibly stand in connection with those of an entirely different nature; hence such spaces would not belong to our world at all, but would constitute their own worlds. I showed above that, in a metaphysical sense, more worlds could exist together, but here is also the condition that, as it seems to me, is the only condition under which it might also be probable that many worlds really exist.’ 
In Kant’s later transcendental idealism, space is not taken as real but rather as a mere human mode of perception through which we frame the real, noumenal, world. Consequently, one can say, the three dimensions of space are but a human projection, not of necessity an actual reality. If space is subjective, then its observed three dimensions cannot be considered a necessarily objective limitation. One of the pioneers of Relativity, the great French mathematician and physicist Henri Poincaré was in agreement:
‘the characteristic property of space, that of having three dimensions, is only a property of our table of distribution, an internal property of the human intelligence … . [We] could conceive, living in our world, thinking beings whose table of distribution would be four dimensional and who consequently would think in hyperspace.’ 
It was, arguably, Kant’s conjectures that sparked the later interest in the fourth dimension, especially in the later nineteenth century. As one of the most prominent popularizers of hyperspace, the British mathematician Charles Hinton, expressed it in 1888:
'the exploration of the facts of higher [dimensional] space is the practical execution of the great vision of Kant’. 
We will leave to the side the controversial question as to whether time can properly be a dimension of space.  But looking back in time, we see that in the shadow of Kant, concepts pertaining to the fourth dimension were being considered in serious fashion by a series of first-rate mathematicians.  These mathematicians, first and foremost the German Georg Friedrich Bernhard Riemann, discovered that spaces of any number of dimensions, n-dimensional space, were not contradictory or paradoxical, but in fact intelligible and systematically congruent.
Riemann was the student of the equally great mathematician Carl Friedrich Gauss.
In the words of the prominent logical empiricist Hans Reichenbach, ‘[in] analogy to [Gauss'] auxiliary concept of the curvature of a surface … Riemann introduced the auxiliary concept of curvature of space’.  That is, the curvature of three-dimensional space itself into a fourth dimension, analogous to the curvature of a two-dimensional sheet into a third dimension. Riemann’s ultimate end was to simplify the laws of nature through his complexification of the laws of geometry – for instance by reducing “force” to curvature.
But the physics of Riemann’s age was behind the mathematics, and so his endeavour to explain natural law through geometry was unfulfilled. But his geometry did enable the new physics to come: the theories of Relativity. As physicist and a co-founder of string theory Michio Kaku puts it, ‘Einstein fulfilled the program initiated by Riemann 60 years earlier, to use higher dimensions to simplify the laws of nature.’  The well-known instance of this is the reduction of the “force” of gravity to spacetime curvature. As Bertrand Russell puts it:
‘the sun exerts no force on the planets whatever. Just as geometry has become physics, so, in a sense, physics has become geometry. The law of gravitation has become the geometrical law that every body pursues the easiest course from place to place, but this course is affected by the hills and valleys that are encountered on the road.’ 
The notion that imperceptible spatial curvature is perceived through forced feeling rather than vision is one that was brought out through the English translator of Riemann’s aforementioned paper, the great mathematician and philosopher William Kingdon Clifford.  In the 1870s Clifford wrote of a hypothetical one-dimensional worm (AB) that lived in a thin oval tube, endlessly circling it clockwise, without any degree of freedom to go counter-clockwise let alone escape “up” or “down” (which would be useless concepts or intuitions to the worm). The worm itself would not even see the second dimension, that is, the oval-like shape in which it lives its life. However it would perceive differences in extra-dimensional curvature (i.e. two-dimensional curvature) as bodily feelings, because its body would curve more at points of acute curvature (viz. H, E, F, and G in Figure 2). 
Figure 2: Clifford’s one-dimensional worm
Clifford writes that:
‘a being existing in these [<3] dimensions would most probably attribute the effects of curvature to changes in its own physical constitution in nowise connected with the geometrical character of its space. … [If we consider ourselves,] changes in shape may be either imperceptible … or if they do take place we may attribute them to “physical causes” – to heat, light, or magnetism – which may be mere names for variations in the curvature of our space. … [We may be] treating merely as physical variations effects which are really due to changes in the curvature of our space; … some or all of those causes which we term physical may … be due to the geometrical construction of our space … variation in the curvature of our space…’ 
Following Einstein’s revelations  we see how advanced Clifford was, at least with regard to the feeling of gravity. Yet there are perhaps further developments to be made in this field relating extra-dimensional curvature to qualia  – thereby correlating not just force to geometry but qualia too. That is to say that a relation of (n-dimensional) space and sentience is here suggested.
Mathematicians and physicists, then, have given feasibility to the idea of n-dimensional space.  We have seen how Clifford relates such space to sentience, let us augment this relation by looking at the ideas of John R. Smythies (1922 – 2019), a neurophilosopher and associate of psychedelic cognoscenti Aldous Huxley and Humphrey Osmond. Smythies provides two sub-theories through which we can understand the relation of space to sentience:
Theory I: ‘Sense-data ... are spatial entities distinct from physical objects and bear temporal and causal relations but no spatial relations to physical objects.’ – i.e. an exclusive theory.
Theory II: ‘Sense data … are spatial entities distinct from physical objects and bear both temporal and causal relations and higher-dimensional spatial relations to physical objects.’  – i.e. an inclusive theory.
Theory I is taken by certain figures such as H. H. Price and Bertrand Russell,  but Smythies considers Theory II preferable as it is more parsimonious and offers a contiguous spatial connection between mind and matter; mind-matter spatial relations that would be lacking in Theory I (which would then only have temporal (i.e. successive) and causal (i.e. transordinal) relations between physical space (PS) and visual space (VS).
Theory I advances that all the three-dimensional spaces of all beings’ sense data, and the one three-dimensional space of physicality are a multiplicity of separate spaces. In emergentism, each VS would ‘emerge’ from sections (such as those within brains) of the singular PS. We have already hinted at the inadequacy of this mysterious transordinal upward transition. Theory I would require causal rather than spatial relations between all myriad spaces, and thus would be an emergentism, and thus the mystery of transordinal nomology emerges once more. Thus we reject Theory I.
Theory II then advances the actuality of a unified space of multiple dimensions (= n-dimensional space) in which all of VS and PS are cross-sections. Moreover, Smythies agrees with psychiatrist Paul Schilder that the perception of PS is VS. He quotes Schilder thus: ‘The space in which objects are perceived and the space in which they are imaged, are one and the same.’  This in turn implies, Smythies writes, that ‘[in] this n-dimensional space Scientific Space [PS] and a visual field [VS] would not be two different kinds of section but would merely be two different sections.’ 
This is not to say that PS is not real but rather to say that our access to it is through VS (plus other senses) which is prosaically three-dimensional. Thus the reality of physical space as more than three-dimensional is not falsified by our common perception of it as three-dimensional. I write ‘prosaically’ because it may be possible to visualize objects of more than three spatial dimensions – Smythies does suggest that ‘[t]here is no a priori reason why we should not develop the ability to appreciate directly an n-dimensional spatial system’, and there are reports of such vision.  Indirectly, we can easily conceptualize and work with more than three dimensions of space through algebraic topology using the Cartesian coördinate system where points, areas and volumes, etc., can be located by numeric variables of each dimension’s axis, e.g. point h: (x1, y2, z3). To locate a point in a four-dimensional space, one simply adds an axis and its variable, e.g. point h: (x1, y2, z3, w4). Ad infinitum. Alternatively, one can visually represent (though not prosaically present)  four-dimensional space through for instance a four-dimensional cube, or tesseract (hypercube) – see Figure 2.
The word tesseract was coined by the aforementioned mathematician and author Charles Howard Hinton,  whose work on the fourth dimension can be used to our ends. In his essay of 1880, ‘What is the fourth dimension?’ – published four years prior to the related book Flatland by Edwin A. Abbott – Hinton employs analogy to lower dimensional worlds to elucidate a speculated four-dimensional world. I shall briefly explain it, then connect this four dimensional world to the n-dimensional world of Broad and Smythies, so to entertain a theory of the relation between space and sentience. Note that by four dimensions, we are speaking of four spatial dimensions, not a fourth temporal dimension in addition to three spatial dimensions. 
Let us imagine a two-dimensional world, a plane, or a Flatland as Abbott calls it, like a sheet of paper. Any beings therein would only be aware of two dimensions, and would only be aware of borders describable with two axes (x,y). Thus they would be unaware of the existence (as we perceive it from our three-dimensional perspective) of the top and bottom faces of their plane that is also contiguous, that borders, their two-dimensional world. Now, we three-dimensional observers could see a multiplicity of such planes, sheets, each floating one above the other. Although each entity of the flatland could not perceive the other flatlands (just as in our world we cannot perceive other entities’ experienced three-dimensional spaces), as they were not contiguous at the x and y axes, we could perceive the multitude of flatlands, or worlds, from our higher-dimensional space – and we could perceive the spatial contiguity (i.e. fundamental unity) of two-dimensional worlds in a three-dimensional space. Thus though each such two-dimensional world would not be contiguous with another two-dimensional world,  each two-dimensional world would be contiguous with, i.e. within the same space as, all the other two-dimensional worlds via the intervening three-dimensional space. Thus the relationship between such flatlands would be spatial rather than merely causal, under the perspective of a world with a higher dimensionality than that of each two-dimensional world. The nomology would be of one order rather than transordinal, because the levels would be unified here. Rather than one world emerging from another (as in emergentism), they would each be equally fundamental and unified. Now, let me allow Hinton, 1880, to shift the argument up a dimension:
‘Take now the case of four dimensions. Instead of bringing before the mind a sheet of paper conceive a solid of three dimensions. If this solid were to become infinite it would fill up the whole of three-dimensional space. But it would not fill up the whole of four-dimensional space. It would be to four-dimensional space what an infinite plane is to three-dimensional space. There could be in four-dimensional space an infinite number of such solids, just as in three-dimensional space there could be an infinite number of infinite planes.
Thus, lying alongside our space, there can be conceived a space also infinite in all three directions. To pass from one to the other a movement has to be made in the fourth dimension, just as to pass from one infinite plane to another a motion has to be made in the third dimension.’ 
Thus we place Smythies’ n-dimensional spaces (i.e. PS with a multitude of beings’ VSs) within the Hintonian four-dimensional space so to render intelligible the Theory II relation between VS and PS.
So: through this approach, we exhibit the possibility that though visual spaces and physical space are not strictly identical, refuting the Psycho-neural Identity Theory, they neither need be strictly distinct, as in Substance Dualism. Neither need one (VS) emerge from the other (PS). Through a four-dimensional perspective, we can see that the mental (all of which for James is necessarily spatial)  and the physical can be both fundamental and unified, i.e. a mind-matter monism. The imagined triangle and the physical correlates thereof are both part of one n-dimensional space rather than members of distinct categories. This is all to say that the More-Broad-Smythies Theory (Theory II) is one, albeit radical, way to respond to the mind-matter mystery. It is a radical monism of space and sentience.
Whether we can call such a monism an identity theory is merely a matter of definition. Spinoza’s system, for instance, is certainly a monism and has certainly been classified as an identity theory. In this regard, it is interesting to note that Hinton, in the above-quoted 1880 essay, also writes that:
‘In the [four-dimensional manifold] which we have traced out, much that philosophers have written finds adequate representation. Much of Spinoza’s Ethics, for example, could be symbolized from the preceding pages.’ 
It is also interesting to note here that Hinton corresponded with William James on the subject of four-dimensional consciousness. Both Spinoza and James were, in the end, panpsychists, and the full extent of the relationship between higher-dimensionality and panpsychism – or more broadly, between n-dimensional space and sentience – is a woefully underexplored world,  a world where one may find idios kosmos within koinos kosmos, thought within extension.
(This is an abridged version of the chapter ‘Deeper than Depth’ in the book Modes of Sentience)
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 In this chapter I use as synonyms 'mind', 'sentience', 'the mental', and 'experience'.
 Descartes, 1641/1996, p. 100.
 James, 1904, pp. 488–489.
 James, 1879, p. 75.
 For further analysis of such sentient space and its distinction as such, see Bradley, 1895; Russell 1948/2009 (pp. 192–7, ch. 6); Smythies, 1958a and 1958b; French, 1987.
 Russell, 1948/2009, pp. 192 … 196.
 Whitehead, 1922–3, p. 5.
 For a good analysis of this distinction and the question concerning the type of space that visual space is, see Rosar, 2016.
 Merleau-Ponty, 1945/2014, pp. 102–103.
 Russell, 1925/2009, p. 2.
 As argued, e.g., by C. A. Strong, 1918, p. 301.
 In addition to the traditional five senses, I am sympathetic to the additional existence of a primal, causal sense which A. N. Whitehead calls ‘perception in the mode of causal efficiency’ (see, e.g. Whitehead, 1927/1985, Ch. 2, §4.)
 James, 1879, p. 70: ‘all our sensations, without exception, have this spatial quale.’
 ‘Hyperspace’ refers to space of more than three dimensions.
 The ontological status of conceptual space is discussed in my book, Modes of Sentience.
 This is from a text written in 1676 in Paris, and reproduced in part in Rescher, 1981, p. 90.
 The neural correlates of consciousness are a subset of the physical correlates of consciousness.
 These options are not exhaustive.
 Against such representationalism, A. N. Whitehead offers us an 'organic realism' where the 'object' perceived becomes part of the 'subject'.
 Smythies, 1956, p. 16.
 Though note that it could be identity if the ‘matter’ is an abstraction which in its actual totality can carry more than one spatial configuration. But accepting this view of matter would nonetheless be contrary to the materialism of the psychoneural identity theory (see Smart, 1963) and thus would be too much of a qualification to maintain psychoneural identity theory as such.
 Levine, 1983.
 Chalmers, 1995. The problem ultimately goes back millennia – the new names merely indicate revived interest.
 Broad, 1923/1927, pp. 392–3. Note that the inconsistent capitalization of ‘scientific space’ is as on the original.
 Aristotle, 1984, p. 447 (228a:9).
 See Cajori, 1926, p. 397.
 Ibid., p. 401.
 Reichenbach, 1926/1957, p. 274.
 Ibid., pp. 280–283.
 Notably Price, 1953.
 Price, 1963. Price was given mescaline by John R. Smythies (1922–2019 [Jan 28th]), as the latter told me in private correspondence (28 November 2018). Smythies also told me that he gave mescaline to C. D. Broad and R. C. Zaehner.
 More, 1671, ch. 28, part 1, §7 (via Cajori, 1926).
 Kant, 1747/2012 , p. 28 (§11, I:25).
 Poincaré, 1906/1913, pp. 177 … 179. In the same text, Poincaré speaks of a type of Japanese mouse that perceives in two dimensions (p. 178).
 In the essay ‘Many Dimensions’, 1888, in Hinton, 1896/2008, p. 35.
 See for instance Čapek, 1955, and Henri Bergson’s works generally.
 Notably Carl Friedrich Gauss (1777–1855), Nikolai Lobachevsky (1793–1856; 1829: non-Euclidean geometry), János Bolyai (1802–1860; 1832: non-Euclidean geometry), Hermann G, Grassmann (1809–1877; 1844: Extension Theory), Georg F. B. Riemann (1826–1866), William Kingdon Cliffford (1845–1879), Charles Howard Hinton (1853–1907), Jules Henri Poincaré (1854–1912), David Hilbert (1862–1943), Hermann Minkowski (1864–1909), and Theodor F. E. Kaluza (1885–1954).
 Reichenbach, 1926/1958, p. 10.
 Kaku, 1994/2016, p. 98.
 Russell, 1925/2009, p. 80.
 Clifford was a Cambridge Apostle – see the chapter Concrescence of Dissent in this volume.
 This is Clifford’s own diagram (1885/1904, p. 218).
 Clifford, 1885/1904, p. 222…223fn…224. Note that Clifford’s book is posthumous. He wrote (and dictated) the book in the 1870s (he died in 1879), but it was not published till 1885.
 As well as the associated four-dimensional Kaluza-Klein theory (1926) that precipitated the n-dimensional String Theories of the 1980s and the M-Theory of the 1990s.
 ‘Qualia’ means pre-conceptualized experience – such as colours, sounds, scents, feelings, etc. (See Lewis, 1929.)
 Bernard Carr is a contemporary professor of mathematics and astronomy who has offered his own theories relating n-dimensional space and sentience in line with the More-Broad-Smythies Theory (see, e.g. Carr 2015).
 By ‘sense-data’ Smythies refers to visual images.
 Smythies, 1956, p. 27.
 Ibid., p. 28.
 Price, 1953.
 Russell, 2009/1948.
 Smythies, 1956, p. 47. From Schilder, 1953, p. 41.
 Smythies, 1956, p. 49.
 Ibid., p. 124. n>3. Though this is in direct contradiction to the seminal figure of hyperspace, mathematician C. H. Hinton who wrote, after years of attempting to achieve such perception, that ‘all attempts to visualise a fourth dimension are futile. It must be connected with a time experience in three space’ (1904, p. 207). However, fellow hyperspatial mathematician Rudy Rucker claims that in the fifteen years of trying, ‘I’ve enjoyed a grand total of perhaps fifteen minutes’ worth of direct vision into four-dimensional space’ (Rucker, 1984/6, p. 8). The Dutch orientalist Johan van Manen also writes of experiencing direct vision of four-dimensional space, specifically a four-dimensional globe (1913, pp. 58–61 [Case 15]). There are also reports of psychedelic-induced visions of n-dimensional space. In relation to this, see my review of Andrew Gallimore's book, Alien Information Theory.
 This use was established with Riemann’s 1854 thesis, Über die Hypothesen welche der Geometrie zu Grunde liegen.
 Analogously one can represent a 3D cube on a 2D surface (e.g. computer screen), which is not equivalent to seeing a 3D cube as such (due in part to stereopsis).
 In The Fourth Dimension, 1904.
 To consider time as a fourth spatial dimension, see the thought of Minkowski, 1918, and Ouspensky, 1912/22.
 If ‘two’ such worlds met at the x and y axes, this would actually only represent one two-dimensional world – just as if we met another world on another planet in our three-dimensional physical world.
 Hinton, 1880, p. 20.
 James, 1879, 1904.
 E.g. Della Rocca, 1993.
 Hinton, 1884, p. 22.
 See Throesch, 2017.
 For an analysis of the relation of hyperspace to a