The Notion Of The Complex-Number System Governing The Laws Governing The Behaviour Of The World — Viewed Through Systemic Functional Linguistics

Penrose (2004: 73):

Over the four centuries that complex numbers have been known, a great many magical qualities have been gradually revealed. Yet this is a magic that had been perceived to lie within mathematics, and it indeed provided a utility and a depth of mathematical insight that could not be achieved by use of the reals alone. There had not been any reason to expect that the physical world should be concerned with it. And for some 350 years from the time that these numbers were introduced through the works of Cardano and Bombelli, it was purely through their mathematical role that the magic of the complex-number system was perceived. It would, no doubt, have come as a great surprise to all those who had voiced their suspicion of complex numbers to find that, according to the physics of the latter three-quarters of the 20th century, the laws governing the behaviour of the world, at its tiniest scales, is fundamentally governed by the complex-number system.

Blogger Comments:

To be clear, the complex-number system does not govern laws that govern the behaviour of the world, just as the quantitative data that is used to generate a map does not govern the map, and the map does not govern the territory it maps.

From the perspective of Systemic Functional Linguistic Theory, the behaviour of the world is first-order (phenomenal) meaning, and physical laws are reconstruals of the behaviour of the world as second-order (metaphenomenal) meaning that includes the complex-number system. Importantly, the laws of physics reconstrue the behaviour of the world in terms of probability (modalisation), not obligation (modulation).

The Invention Of Complex Numbers Viewed Through Systemic Functional Linguistics

Penrose (2004: 72-3):

Presumably this suspicion [of complex numbers] arose because people could not ‘see’ the complex numbers as being presented to them in any obvious way by the physical world. In the case of the real numbers, it had seemed that distances, times, and other physical quantities were providing the reality that such numbers required; yet the complex numbers had appeared to be merely invented entities, called forth from the imaginations of mathematicians who desired numbers with a greater scope than the ones that they had known before. 

But we should recall from §3.3 that the connection the mathematical real numbers have with those physical concepts of length or time is not as clear as we had imagined it to be. We cannot directly see the minute details of a Dedekind cut, nor is it clear that arbitrarily great or arbitrarily tiny times or lengths actually exist in nature. One could say that the so-called ‘real numbers’ are as much a product of mathematicians’ imaginations as are the complex numbers. Yet we shall find that complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. 

It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the invention of complex numbers by mathematicians is the logical expansion of the metaphenomenal system that realises the field of mathematics. This is achieved through the conscious processing of mathematicians: the verbal projection of texts that are instances of the system.

From this perspective, the 'unity' of complex numbers with Nature derives from the meanings of (metaphenomenal) mathematics being reconstruals of the meanings of (phenomenal) Nature.

To be clear, it is mathematicians, as the part of Nature that is conscious of the complex-number system, who are impressed by its scope and consistency.

‘Constructing’ Mathematical Notions Without Reference To The Physical World — Viewed Through Systemic Functional Linguistics

Penrose (2004: 64-5):

Moreover, it shows us, at least, that things like the natural numbers can be conjured literally out of nothing, merely by employing the abstract notion of ‘set’. We get an infinite sequence of abstract (Platonic) mathematical entities — sets containing, respectively, zero, one, two, three, etc., elements, one set for each of the natural numbers, quite independently of the actual physical nature of the universe. In Fig.1.3 we envisaged a kind of independent ‘existence’ for Platonic mathematical notions — in this case, the natural numbers themselves — yet this ‘existence’ can seemingly be conjured up by, and certainly accessed by, the mere exercise of our mental imaginations, without any reference to the details of the nature of the physical universe. Dedekind’s construction, moreover, shows how this ‘purely mental’ kind of procedure can be carried further, enabling us to ‘construct’ the entire system of real numbers, still without any reference to the actual physical nature of the world. Yet, as indicated above, ‘real numbers’ indeed seem to have a direct relevance to the real structure of the world — illustrating the very mysterious nature of the ‘first mystery’ depicted in Fig.1.3.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the 'physical nature of the world' is first-order meaning, construed of experience. The 'construction' of 'the entire system of real numbers' is the reconstrual of first-order meanings, cardinal numbers, as the second-order meanings of mathematics. This is why the real numbers of mathematics have 'direct relevance' to the structure of the physical world.

The expansion of the system of mathematics is achieved through instances of the system, which may be mentally projected as ideas, or verbally projected as locutions (spoken or written texts). It is only the latter that is shared between mathematicians, and which, therefore, expands the shared potential of the mathematical community.

The Independence Of Natural Numbers From The Physical World Viewed Through Systemic Functional Linguistics

Penrose (2004: 63-4):

We can, however, raise the question of whether the natural numbers themselves have a meaning or indeed existence independent of the actual nature of the physical world. Perhaps our notion of natural numbers depends upon there being, in our universe, reasonably well-defined discrete objects that persist in time. Natural numbers initially arise when we wish to count things, after all. But this seems to depend upon there actually being persistent distinguishable ‘things’ in the universe which are available to be ‘counted’. Suppose, on the other hand, our universe were such that numbers of objects had a tendency to keep changing. Would natural numbers actually be ‘natural’ concepts in such a universe? Moreover, perhaps the universe actually contains only a finite number of ‘things’, in which case the ‘natural’ numbers might themselves come to an end at some point! We can even envisage a universe which consists only of an amorphous featureless substance, for which the very notion of numerical quantification might seem intrinsically inappropriate. Would the notion of ‘natural number’ be at all relevant for the description of universes of this kind?

 

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, it is not that natural numbers 'have a meaning or existence independent of the actual nature of the physical world' but that the natural numbers of mathematics and the physical world constitute different orders of experience. The physical world is first-order meaning (phenomenal), construed of experience, whereas the natural numbers are second-order meaning (metaphenomenal), construed of first-order meaning (quantities).

To be clear, natural numbers arose in a universe in which experience came to be construed as countable things, through the linguistic systems that evolved in humans.

To be clear, in a universe where the numbers of objects keep changing, there are numbers — since it is these that are changing — and so the natural numbers of mathematics could be construed from them.

To be clear, in a universe that contains a finite number of things, it is the things that 'come to an end at some point', not the numbers used to count them.

The 'Momentum' Of Mathematical Notions Viewed Through Systemic Functional Linguistics

Penrose (2004: 59-60):

There is a profound issue that is being touched upon here. In the development of mathematical ideas, one important initial driving force has always been to find mathematical structures that accurately mirror the behaviour of the physical world. But it is normally not possible to examine the physical world itself in such precise detail that appropriately clear-cut mathematical notions can be abstracted directly from it. Instead, progress is made because mathematical notions tend to have a ‘momentum’ of their own that appears to spring almost entirely from within the subject itself. 

Mathematical ideas develop, and various kinds of problem seem to arise naturally. Some of these (as was the case with the problem of finding the length of the diagonal of a square) can lead to an essential extension of the original mathematical concepts in terms of which the problem had been formulated. Such extensions may seem to be forced upon us, or they may arise in ways that appear to be matters of convenience, consistency, or mathematical elegance. 

Accordingly, the development of mathematics may seem to diverge from what it had been set up to achieve, namely simply to reflect physical behaviour. Yet, in many instances, this drive for mathematical consistency and elegance takes us to mathematical structures and concepts which turn out to mirror the physical world in a much deeper and more broad-ranging way than those that we started with. It is as though Nature herself is guided by the same kind of criteria of consistency and elegance as those that guide human mathematical thought.

 

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, mathematics is the metaphenomenal reconstrual of quantitative phenomena.

The development of mathematical ideas is the building of mathematical potential through instances that establish new systems of relations that are interpersonally assessed as mathematically valid. This is the sense in which mathematical ideas 'have a momentum of their own'

From this perspective, the 'divergence' of mathematics from the physical world is the distinction between mathematics as second-order meaning and the physical world as first-order meaning, and the reason why mathematics models the physical world 'in a much deeper and broad-ranging way' is that it is a reconstrual of the quantitative meanings of the physical world in terms of strict logical relations.

It is not that 'Nature herself is guided by the same kind of criteria of consistency and elegance as those that guide human mathematical thought', but that Nature is first-order meaning and mathematical thought is the reconstrual of Nature as second-order meaning.

Importantly, the use of the words 'mirror' and 'reflect' here misrepresent mathematics and the physical world as being of the same level of abstraction (order of experience).

Physical Space Vs The Mathematical Notion Of Number Viewed Through Systemic Functional Linguistics

 Penrose (2004: 58):

By the 19th and 20th centuries, however, the view had emerged that the mathematical notion of number should stand separately from the nature of physical space. Since mathematically consistent geometries other than that of Euclid had been shown to exist, this rendered it inappropriate to insist that the mathematical notion of ‘geometry’ should be necessarily extracted from the supposed nature of ‘actual’ physical space. 

Moreover, it could be very difficult, if not impossible, to ascertain the detailed nature of this supposed underlying ‘Platonic physical geometry’ in terms of the behaviour of imperfect physical objects. In order to know the nature of the numbers according to which ‘geometrical distance’ is to be defined, for example, it would be necessary to know what happens both at indefinitely tiny and indefinitely large distances. Even today, these questions are without clearcut resolution. Thus, it was far more appropriate to develop the nature of number in a way that does not directly refer to physical measures.

 

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the view that 'the mathematical notion of number should stand separately from the nature of physical space' was a recognition that mathematical notions and physical space are of different orders of experience.

Physical Vs Geometrical Objects Viewed Through Systemic Functional Linguistics

 Penrose (2004: 58):

A physical object such as a square drawn in the sand or a cube hewn from marble might have been regarded by the ancient Greeks as a reasonable or sometimes an excellent approximation to the Platonic geometrical ideal. Yet any such object would nevertheless provide a mere approximation. 

Lying behind such approximations to the Platonic forms — so it would have appeared — would be space itself: an entity of such abstract or notional existence that it could well have been regarded as a direct realisation of a Platonic reality. 

The measure of distance in this ideal geometry would be something to ascertain; accordingly, it would be appropriate to try to extract this ideal notion of real number from a geometry of a Euclidean space that was assumed to be given.

 

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, a square drawn in the sand or a cube hewn from marble in physical space is phenomenal, whereas a geometrical square or cube in Euclidean space ('Platonic ideal') is metaphenomenal. Phenomena are first-order meanings construed of experience, whereas metaphenomena, such as the notion of real number, are second-order meanings construed of first-order meanings.

Importantly, the metaphenomenal (Platonic geometrical ideal) also differs from the phenomenal in being a construal only of the numerical relations of the physical object: the relative lengths of its sides, the relative angles of intersecting sides or planes, etc.

Physical Vs Geometrical Space Viewed Through Systemic Functional Linguistics

Penrose (2004: 58):

There was a basic difference in viewpoint, however, between the Greek notion of a real number and the modern one, because the Greeks regarded the number system as basically ‘given’ to us, in terms of the notion of distance in physical space, so the problem was to try to ascertain how these ‘distance’ measures actually behaved. For ‘space’ may well have had the appearance of being itself a Platonic absolute even though actual physical objects existing in this space would inevitably fall short of the Platonic ideal.

 

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the ancient Greeks began the reconstrual of phenomena, first-order meanings such as the distance between physical objects, as metaphenomena, the second-order meanings that realise the field of mathematics. Thus, the space of physical objects is phenomenal, whereas the 'Platonic' space of mathematics is metaphenomenal.

The Hyperbolic Geometry Of Velocities Viewed Through Systemic Functional Linguistics

Penrose (2004: 48):

For the space of velocities, according to modern relativity theory, is certainly a three-dimensional hyperbolic geometry, rather than the Euclidean one that would hold in the older Newtonian theory. This helps us to understand some of the puzzles of relativity. For example, imagine a projectile hurled forward, with near light speed, from a vehicle that also moves forwards with comparable speed past a building. Yet, relative to that building, the projectile can never exceed light speed. Though this seems impossible, we shall see that it finds a direct explanation in terms of hyperbolic geometry.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the hyperbolic geometry of velocities is not the geometry of space, but the geometry of motion through space.

The Large-Scale Spatial Geometry Of The Universe Viewed Through Systemic Functional Linguistics

Penrose (2004: 46, 48):

What about our actual universe on cosmological scales? Do we expect that its spatial geometry is Euclidean, or might it accord more closely with some other geometry, such as the remarkable hyperbolic geometry (but in three dimensions). This is indeed a serious question. We know from Einstein’s general relativity that Euclid’s geometry is only an (extraordinarily accurate) approximation to the actual geometry of physical space. This physical geometry is not even exactly uniform, having small ripples of irregularity owing to the presence of matter density. Yet, strikingly, according to the best observational evidence available to cosmologists today, these ripples appear to average out, on cosmological scales, to a remarkably exact degree, and the spatial geometry of the actual universe seems to accord with a uniform (homogeneous and isotropic) geometry extraordinarily closely. Euclid’s first four postulates, at least, would seem to have stood the test of time impressively well. …

What, then, is the observational status of the large-scale spatial geometry of the universe? It is only fair to say that we do not yet know, although there have been recent widely publicised claims that Euclid was right all along, and his fifth postulate holds true also, so the averaged spatial geometry is indeed what we call ‘Euclidean’. On the other hand, there is also evidence (some of it coming from the same experiments) that seems to point fairly firmly to a hyperbolic overall geometry for the spatial universe. Moreover, some theoreticians have long argued for the elliptic case, and this is certainly not ruled out by that same evidence that is argued to support the Euclidean case.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, there is an important distinction between space (location and extent) and the processes that unfold in space. From this it follows that there is an important distinction between the geometry of space and the geometry of processes that unfold in space. As a consequence, while the geometry of space may be Euclidean, the geometry of motion through space may be otherwise.

This confusion of the geometry of motion through space with the geometry of space leads to the misunderstanding of General Relativity in which curved trajectories (geodesics) are misunderstood as curved space(time).

What Hyperbolic Geometry Actually Is — Viewed Through Systemic Functional Linguistics

Penrose (2004: 40, 43):

The existence of various different models of hyperbolic geometry, expressed in terms of Euclidean space, serves to emphasise the fact that these are, indeed, merely ‘Euclidean models’ of hyperbolic geometry and are not to be taken as telling us what hyperbolic geometry actually is. Hyperbolic geometry has its own ‘Platonic existence’, just as does Euclidean geometry.

… hyperbolic geometry does actually exist, in the mathematical sense that there is such a consistent structure. In the terminology of §1.3, hyperbolic geometry inhabits Plato’s world of mathematical forms.

 

Blogger Comments:

To be clear, in the very simplest of terms, 'hyperbolic' geometry replaces the flat plane of Euclidean geometry with curved plane of the inside of a sphere. (Likewise, 'elliptical' geometry replaces the flat plane of Euclidean geometry with curved plane of the outside of a sphere.)

From the perspective of Systemic Functional Linguistic Theory, hyperbolic geometry, like all mathematics, is second-order meaning (metaphenomenal), and like all meaning, forms part of the content of the collective consciousness of its community.

From this perspective, the expression of hyperbolic geometry in terms of Euclidean space is the use of the expression plane of one geometric system to realise the content plane of another.

The Assessment Of Mathematical Validity Viewed Through Systemic Functional Linguistics

Penrose (2004: 24n):

There is perhaps an irony here that a fully fledged anti-Platonist, who believes that mathematics is ‘all in the mind’ must also believe — so it seems — that there are no true mathematical statements that are in principle beyond reason. For example, if Fermat’s Last Theorem had been inaccessible (in principle) to reason, then this anti-Platonist view would allow no validity either to its truth or to its falsity, such validity coming only through the mental act of perceiving some proof or disproof.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, as a semiotic system, mathematics is understood as the content of consciousness, which, through verbal processes, may be instantiated in texts that are realised materially.

Since mathematics is concerned with logical relations between quantities, no mathematical statements are 'beyond reason', though determining the validity of a specific statement may be beyond the reasoning ability of individual mathematicians.

To be clear, if 'Fermat's Last Theorem had been inaccessible (in principle) to reason', then 'no validity either to its truth or to its falsity' could reasoned by anyone: whether Platonist, anti-Platonist, or neither. Mathematical validity is assessed through the 'mental act' of reasoning on the basis of agreed mathematical principles.

The "Actual" Nature Of Mental Processes Viewed Through Systemic Functional Linguistics

Penrose (2004: 21):

Perhaps one comment will not be amiss here, however. This is that, in my own opinion, there is little chance that any deep understanding of the nature of the mind can come about without our first learning much more about the very basis of physical reality. As will become clear from the discussions that will be presented in later chapters, I believe that major revolutions are required in our physical understanding. Until these revolutions have come to pass, it is, in my view, greatly optimistic to expect that much real progress can be made in understanding the actual nature of mental processes.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the material and the mental are different domains of meaning, and consciousness (the mind) involves verbal as well as mental processes, and through these, the interpersonal enactment of the self.

In terms of a hierarchy of systems, the material basis of the mind is a matter for biology (neuroscience), not physics, since it is biological organisation that gives rise to consciousness, as demonstrated by the absence of consciousness in material phenomena that are not biological, such as rocks or cyclones.

The notion that the "actual" nature of mental processes equates with their lowest level of organisation, that described by physics, is an example of reductionism.

Why Mathematical Laws Apply To The World With Such Phenomenal Precision

Penrose (2004: 20-1):

For it remains a deep puzzle why mathematical laws should apply to the world with such phenomenal precision. … Moreover, it is not just the precision but also the subtle sophistication and mathematical beauty of these successful theories that is profoundly mysterious. 

There is also an undoubted deep mystery in how it can come to pass that appropriately organised physical material — and here I refer specifically to living human (or animal) brains—can somehow conjure up the mental quality of conscious awareness. 

Finally, there is also a mystery about how it is that we perceive mathematical truth. It is not just that our brains are programmed to ‘calculate’ in reliable ways. There is something much more profound than that in the insights that even the humblest among us possess when we appreciate, for example, the actual meanings of the terms ‘zero’, ‘one’, ‘two’, ‘three’, ‘four’, etc.

Blogger Comments:

To be clear, the reason why mathematical laws apply to the world 'with phenomenal precision' is that the former are quantifications of the latter. From the perspective of Systemic Functional Linguistic Theory, mathematical laws are metaphenomenal meanings that quantitatively model phenomenal meanings construed of experience.

From the same perspective, 'the mental quality of conscious awareness' is understood as mental processes ranging over phenomena that are meanings construed of experience. The material basis of conscious processes is modelled, in a biologically plausible and epistemologically enlightened way, by Edelman's (1992) Theory of Neuronal Group Selection.

From the same perspective, 'perceiving mathematical truth' is determining the validity of mathematical propositions on the basis of the consistency of their logical relations.

The Physical World As Governed By Mathematical Laws — Viewed Through Systemic Functional Linguistics

Penrose (2004: 18-9):

Thus, according to Fig. 1.3, the entire physical world is depicted as being governed according to mathematical laws. We shall be seeing in later chapters that there is powerful (but incomplete) evidence in support of this contention. On this view, everything in the physical universe is indeed governed in completely precise detail by mathematical principles — perhaps by equations, such as those we shall be learning about in chapters to follow, or perhaps by some future mathematical notions fundamentally different from those which we would today label by the term ‘equations’. If this is right, then even our own physical actions would be entirely subject to such ultimate mathematical control, where ‘control’ might still allow for some random behaviour governed by strict probabilistic principles.

Many people feel uncomfortable with contentions of this kind, and I must confess to having some unease with it myself. Nonetheless, my personal prejudices are indeed to favour a viewpoint of this general nature, since it is hard to see how any line can be drawn to separate physical actions under mathematical control from those which might lie beyond it. In my own view, the unease that many readers may share with me on this issue partly arises from a very limited notion of what ‘mathematical control’ might entail. Part of the purpose of this book is to touch upon, and to reveal to the reader, some of the extraordinary richness, power, and beauty that can spring forth once the right mathematical notions are hit upon.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the physical universe is experience construed as first-order meanings (phenomena) and mathematical laws are experience construed as second-order meanings (metaphenomena). On the one hand, mathematical laws are laws in the sense of modalisation (probability), not modulation (obligation), and on the other hand, metaphenomena (e.g. maps) cannot govern or control phenomena (e.g. territories).