The Physical, Mental, And Platonic Mathematical Worlds Viewed Through Systemic Functional Linguistics

Penrose (2004: 17-18):

Thus, mathematical existence is different not only from physical existence but also from an existence that is assigned by our mental perceptions. Yet there is a deep and mysterious connection with each of those other two forms of existence: the physical and the mental. In Fig. 1.3, I have schematically indicated all of these three forms of existence — the physical, the mental, and the Platonic mathematical — as entities belonging to three separate ‘worlds’, drawn schematically as spheres. The mysterious connections between the worlds are also indicated, where in drawing the diagram I have imposed upon the reader some of my beliefs, or prejudices, concerning these mysteries.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, experience is construed as two distinct domains of first-order meaning (phenomena) (i) the material ('physical existence') and the relational, and (ii) the mental and the verbal (consciousness).

The latter domain constitutes the symbolic processing that creates second-order meaning (metaphenomena) through projection. From this perspective, the Platonic mathematical world is second-order meaning that symbolic processing creates through projection.

The Timeless Existence Of Mathematical Forms Viewed Through Systemic Functional Linguistics

Penrose (2004: 17):

I am aware that there will still be many readers who find difficulty with assigning any kind of actual existence to mathematical structures. Let me make the request of such readers that they merely broaden their notion of what the term ‘existence’ can mean to them. The mathematical forms of Plato’s world clearly do not have the same kind of existence as do ordinary physical objects such as tables and chairs. They do not have spatial locations; nor do they exist in time. Objective mathematical notions must be thought of as timeless entities and are not to be regarded as being conjured into existence at the moment that they are first humanly perceived. 

The particular swirls of the Mandelbrot set … did not attain their existence at the moment that they were first seen on a computer screen or printout. Nor did they come about when the general idea behind the Mandelbrot set was first humanly put forth — not actually first by Mandelbrot, as it happened, but by R. Brooks and J. P. Matelski, in 1981, or perhaps earlier. For certainly neither Brooks nor Matelski, nor initially even Mandelbrot himself, had any real conception of the elaborate detailed designs …. Those designs were already ‘in existence’ since the beginning of time, in the potential timeless sense that they would necessarily be revealed precisely in the form that we perceive them today, no matter at what time or in what location some perceiving being might have chosen to examine them.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the existence of 'objective mathematical notions/structures/forms' outside spacetime is their existence as potential meanings of mathematics. Instances of that potential are located in spacetime: the spacetime of the mental or verbal process through which that potential is actualised.

To be clear, logically, 'objective mathematical notions/structures/forms' cannot have existed, as potential, until mathematical systems of meaning existed, and that coming into existence occurred in spacetime.

The Platonic Existence Of The Mandelbrot Set Viewed Through Systemic Functional Linguistics

Penrose (2004: 15-7):

In Fig. 1.2, I have depicted various small portions of that famous mathematical entity known as the Mandelbrot set. The set has an extraordinarily elaborate structure, but it is not of any human design. Remarkably, this structure is defined by a mathematical rule of particular simplicity. … 

The point that I wish to make is that no one, not even Benoit Mandelbrot himself when he first caught sight of the incredible complications in the fine details of the set, had any real preconception of the set’s extraordinary richness. The Mandelbrot set was certainly no invention of any human mind. The set is just objectively there in the mathematics itself. 

If it has meaning to assign an actual existence to the Mandelbrot set, then that existence is not within our minds, for no one can fully comprehend the set’s endless variety and unlimited complication. 

Nor can its existence lie within the multitude of computer printouts that begin to capture some of its incredible sophistication and detail, for at best those printouts capture but a shadow of an approximation to the set itself. 

Yet it has a robustness that is beyond any doubt; for the same structure is revealed — in all its perceivable details, to greater and greater fineness the more closely it is examined — independently of the mathematician or computer that examines it. Its existence can only be within the Platonic world of mathematical forms.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the sense in which the Mandelbrot set is 'objectively there in the mathematics itself' is as the meaning potential of mathematics. As meaning potential, its instances can be realised materially, and as material realisations, they can be perceived mentally as metaphenomena (representations of meaning). None of this requires a distinct 'Platonic world of mathematical forms'.

The 'endless variety', 'unlimited complication' and 'incredible sophistication and detail' of the Mandelbrot set are (nominalised) qualities attributed to the metaphenomenon, and as such, are not, in themselves, criterial to the location of the "existence" of the Mandelbrot set: mental, material or Platonic.

Objectivity As Platonic Existence Viewed Through Systemic Functional Linguistics

Penrose (2004: 15):

The mathematical assertions that can belong to Plato’s world are precisely those that are objectively true. Indeed, I would regard mathematical objectivity as really what mathematical Platonism is all about. To say that some mathematical assertion has a Platonic existence is merely to say that it is true in an objective sense. A similar comment applies to mathematical notions — such as the concept of the number 7, for example, or the rule of multiplication of integers, or the idea that some set contains infinitely many elements — all of which have a Platonic existence because they are objective notions. To my way of thinking, Platonic existence is simply a matter of objectivity and, accordingly, should certainly not be viewed as something ‘mystical’ or ‘unscientific’, despite the fact that some people regard it that way.

Blogger Comments:

To be clear, the intersubjective assessment of a mathematical proposition as 'objectively true' (i.e. valid) does not logically entail the existence of a Platonic world beyond the material and mental domains of meaning. From the perspective of Systemic Functional Linguistic Theory, mathematical propositions are projections of conscious processing: verbally projected locutions that realise mentally projected ideas.

On the other hand, the opposition of 'scientific' with 'mystical' betrays a common misunderstanding. Joseph Campbell:

There is no conflict between science and mysticism, but there is a conflict between the science of 2000 BC and the science of 2000 AD, and that's the mess in our religions.

Weak And Stronger Platonists Viewed Through Systemic Functional Linguistics

Penrose (2004: 14-5):

It should perhaps be mentioned that, from the point of view of mathematical logic, the Fermat assertion is actually a mathematical statement of a particularly simple kind, whose objectivity is especially apparent. Only a tiny minority of mathematicians would regard the truth of such assertions as being in any way ‘subjective’ — although there might be some subjectivity about the types of argument that would be regarded as being convincing. 

However, there are other kinds of mathematical assertion whose truth could plausibly be regarded as being a ‘matter of opinion’. Perhaps the best known of such assertions is the axiom of choice. … Most mathematicians would probably regard the axiom of choice as ‘obviously true’, while others may regard it as a somewhat questionable assertion which might even be false (and I am myself inclined, to some extent, towards this second viewpoint). Still others would take it as an assertion whose ‘truth’ is a mere matter of opinion or, rather, as something which can be taken one way or the other, depending upon which system of axioms and rules of procedure (a ‘formal system’) one chooses to adhere to.

Mathematicians who support this final viewpoint (but who accept the objectivity of the truth of particularly clear-cut mathematical statements, like the Fermat assertion discussed above) would be relatively weak Platonists. Those who adhere to objectivity with regard to the truth of the axiom of choice would be stronger Platonists.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, on this view, the objectivity or subjectivity of mathematical statements is a matter of intersubjective (interpersonal) assessment. 'Stronger Platonists' are those who assess mathematical statements as unconditionally 'objective', whereas 'weak Platonists' are those who assess mathematical statements as 'objective', but with validity dependent on the starting assumptions and formal systems that are assessed as criterial for assessing validity.

Proof Of The Validity Of ‘Fermat’s Last Theorem’ Viewed Through Systemic Functional Linguistics

Penrose (2004: 13-4):

Let me illustrate this issue by considering one famous example of a mathematical truth, and relate it to the question of ‘objectivity’. In 1637, Pierre de Fermat made his famous assertion now known as ‘Fermat’s Last Theorem’ (that no positive nth power of an integer, i.e. of a whole number, can be the sum of two other positive nth powers if n is an integer greater than 2), which he wrote down in the margin of his copy of the Arithmetica, a book written by the 3rd-century Greek mathematician Diophantos. In this margin, Fermat also noted: ‘I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.’

Fermat’s mathematical assertion remained unconfirmed for over 350 years, despite concerted efforts by numerous outstanding mathematicians. A proof was finally published in 1995 by Andrew Wiles (depending on the earlier work of various other mathematicians), and this proof has now been accepted as a valid argument by the mathematical community. 

Now, do we take the view that Fermat’s assertion was always true, long before Fermat actually made it, or is its validity a purely cultural matter, dependent upon whatever might be the subjective standards of the community of human mathematicians?

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the proof of the validity of Fermat's Last Theorem was mathematical potential only until it was finally instantiated by Andrew Wiles in 1995.

Platonic Existence As The Objectivity Of Mathematical Truth — Viewed Through Systemic Functional Linguistics

Penrose (2004: 13):

It may be helpful if I put the case for the actual existence of the Platonic world in a different form. What I mean by this ‘existence’ is really just the objectivity of mathematical truth. Platonic existence, as I see it, refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture. Such ‘existence’ could also refer to things other than mathematics, such as to morality or aesthetics, but I am here concerned just with mathematical objectivity, which seems to be a much clearer issue.

Blogger Comments:

To be clear:

In its most basic fundamentals, platonism affirms the existence of abstract objects, which are asserted to exist in a third realm distinct from both the sensible external world and from the internal world of consciousness

Clearly, the mere objectivity of mathematics does not logically entail the existence of a world that is distinct from both 'the sensible external world' and 'the internal world of consciousness'.

From the perspective of Systemic Functional Linguistic Theory, 'the sensible external world' is the construal of experience as material and relational processes, and 'the internal world of consciousness' is the construal of experience as mental and verbal processes. As a semiotic system, mathematics is the content of consciousness that is projected by mental and verbal processes. Mathematical objectivity simply refers to the internal consistency of a semiotic system that is concerned with relations between quantities.

The Objectivity Of The Mathematical World Viewed Through Systemic Functional Linguistics

Penrose (2004: 12-3):

Nevertheless, one might still take the alternative view that the mathematical world has no independent existence, and consists merely of certain ideas which have been distilled from our various minds and which have been found to be totally trustworthy and are agreed by all. Yet even this viewpoint seems to leave us far short of what is required. Do we mean ‘agreed by all’, for example, or ‘agreed by those who are in their right minds’, or ‘agreed by all those who have a Ph.D. in mathematics’ (not much use in Plato’s day) and who have a right to venture an ‘authoritative’ opinion? There seems to be a danger of circularity here; for to judge whether or not someone is ‘in his or her right mind’ requires some external standard. So also does the meaning of ‘authoritative’, unless some standard of an unscientific nature such as ‘majority opinion’ were to be adopted (and it should be made clear that majority opinion, no matter how important it may be for democratic government, should in no way be used as the criterion for scientific acceptability). Mathematics itself indeed seems to have a robustness that goes far beyond what any individual mathematician is capable of perceiving. Those who work in this subject, whether they are actively engaged in mathematical research or just using results that have been obtained by others, usually feel that they are merely explorers in a world that lies far beyond themselves — a world which possesses an objectivity that transcends mere opinion, be that opinion their own or the surmise of others, no matter how expert those others might be.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, 'agreed by all' means that the validity of mathematical propositions is intersubjectively defined in terms of the internal relations of mathematical semiotic systems, regardless of the identity or personal opinion of any individual mathematician. It is because of the strict internal logicality of mathematical relations that mathematical systems can be intersubjectively agreed to 'possess an objectivity that transcends mere opinion'.

By the same token, the exploration by mathematicians of 'a world that lies far beyond themselves' is the instantiation of hitherto uninstantiated mathematical potential.

Mathematics As 'Something Outside Ourselves' Viewed Through Systemic Functional Linguistics

Penrose (2004: 12):

If the model itself is to be assigned any kind of ‘existence’, then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato’s world to be in any sense absolute and ‘real’. Yet, there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise, unreliable, and inconsistent in their judgements. The precision, reliability, and consistency that are required by our scientific theories demand something beyond any one of our individual (untrustworthy) minds. In mathematics, we find a far greater robustness than can be located in any particular mind. Does this not point to something outside ourselves, with a reality that lies beyond what each individual can achieve?

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, mathematical models "exist" as semiotic systems, potential and instance, that are projections of mental and verbal processes. From this perspective, 'real' refers to the ideational meanings of semiotic systems, and so 'real' does not transcend semiotic systems. Reality is the meaning we construe of experience.

Mathematical models do have 'a reality of their own' in the sense that mathematics is a distinct semiotic system.

The opposition of 'imprecise judgements of individual minds' and the 'precision found in mathematics' is, in this context, the opposition of mentally projecting meanings of non-mathematical systems and of mentally projecting meanings of mathematical systems.

The sense in which mathematics is 'something that is outside ourselves' lies in it being an evolved social semiotic system.

The Existence Of The Platonic World Of Mathematical Forms Viewed Through Systemic Functional Linguistics

Penrose (2004: 12):

This was an extraordinary idea for its time, and it has turned out to be a very powerful one. But does the Platonic mathematical world actually exist, in any meaningful sense? Many people, including philosophers, might regard such a ‘world’ as a complete fiction — a product merely of our unrestrained imaginations. Yet the Platonic viewpoint is indeed an immensely valuable one. It tells us to be careful to distinguish the precise mathematical entities from the approximations that we see around us in the world of physical things. Moreover, it provides us with the blueprint according to which modern science has proceeded ever since. Scientists will put forward models of the world — or, rather, of certain aspects of the world — and these models may be tested against previous observation and against the results of carefully designed experiment. The models are deemed to be appropriate if they survive such rigorous examination and if, in addition, they are internally consistent structures. The important point about these models, for our present discussion, is that they are basically purely abstract mathematical models. The very question of the internal consistency of a scientific model, in particular, is one that requires that the model be precisely specified. The required precision demands that the model be a mathematical one, for otherwise one cannot be sure that these questions have well-defined answers. 

If the model itself is to be assigned any kind of ‘existence’, then this existence is located within the Platonic world of mathematical forms.

Blogger Comments:

To be clear, the notion and value of a 'purely mathematical model' does not logically entail that they exist in a Platonic world. From the perspective of Systemic Functional Linguistic Theory, the existence of mathematical models is within the semiotic order, whereas the existence of 'the world of physical things' is within the material order. Mathematical models exist as metaphenomenal (second-order) meaning, whereas physical things exist as phenomenal (first-order) meaning. Mathematical models are reconstruals of things — or more precisely: of processes, participants and circumstances — in terms of quantification.

The Platonic World Of Mathematical Forms Viewed Through Systemic Functional Linguistics

Penrose (2004: 11-2):

Thus, we must be careful, when considering geometrical assertions, whether to trust the ‘axioms’ as being, in any sense, actually true.

But what does ‘true’ mean, in this context? The difficulty was well appreciated by the great ancient Greek philosopher Plato, who lived in Athens from c.429 to 347 BC, about a century after Pythagoras. Plato made it clear that the mathematical propositions — the things that could be regarded as unassailably true — referred not to actual physical objects (like the approximate squares, triangles, circles, spheres, and cubes that might be constructed from marks in the sand, or from wood or stone) but to certain idealised entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world. Today, we might refer to this world as the Platonic world of mathematical forms. Physical structures, such as squares, circles, or triangles cut from papyrus, or marked on a flat surface, or perhaps cubes, tetrahedra, or spheres carved from marble, might conform to these ideals very closely, but only approximately. The actual mathematical squares, cubes, circles, spheres, triangles, etc., would not be part of the physical world, but would be inhabitants of Plato’s idealised mathematical world of forms.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, physical structures and mathematical structures differ in terms of orders of experience. Physical structures are first-order meanings, phenomena, whereas mathematical structures are reconstruals of these as second-order meanings, metaphenomena.

Specifically, these metaphenomenal 'inhabitants of Plato's idealised mathematical world of forms' are construals of quantifying Numeratives of Things, and the relations between them, rather than construals of the Things (physical structures) that they quantify. Mathematical structures are construals of the dimensions of physical structures.

The 'Truly Timeless' Nature Of Mathematics Viewed Through Systemic Functional Linguistics

Penrose (2004: 10):

In the long run, the influence of the Pythagoreans on the progress of human thought has been enormous. For the first time, with mathematical proof, it was possible to make significant assertions of an unassailable nature, so that they would hold just as true even today as at the time that they were made, no matter how our knowledge of the world has progressed since then. The truly timeless nature of mathematics was beginning to be revealed.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the validity of propositions ('significant assertions of an unassailable nature') is a matter of interpersonal meaning. Mathematics does not exclude interpersonal meaning, but it does provide a means of determining validity, given an agreed set of assumptions.

From this theoretical perspective, one reason why the validity of mathematical assertions does not change with changing 'knowledge of the world' is that mathematical assertions are concerned only with intrinsic logical relations between quantifying Numeratives, whereas changing 'knowledge of the world' is more broadly concerned with the changing construal of experience as Things, and their participations in processes.

From Gods To Mathematical Laws — Through Systemic Functional Linguistics

 Penrose (2004: 7-8):

What laws govern our universe? How shall we know them? How may this knowledge help us to comprehend the world and hence guide its actions to our advantage?

Since the dawn of humanity, people have been deeply concerned by questions like these. At first, they had tried to make sense of those influences that do control the world by referring to the kind of understanding that was available from their own lives. … Accordingly, the course of natural events — such as sunshine, rain, storms, famine, illness, or pestilence — was to be understood in terms of the whims of gods or goddesses motivated by such human urges. And the only action perceived as influencing these events would be appeasement of the god-figures.

But gradually patterns of a different kind began to establish their reliability. The precision of the Sun’s motion through the sky and its clear relation to the alternation of day with night provided the most obvious example; but also the Sun’s positioning in relation to the heavenly orb of stars was seen to be closely associated with the change and relentless regularity of the seasons, and with the attendant clear-cut influence on the weather, and consequently on vegetation and animal behaviour. The motion of the Moon, also, appeared to be tightly controlled, and its phases determined by its geometrical relation to the Sun. … If the heavens were indeed controlled by the whims of gods, then these gods themselves seemed under the spell of exact mathematical laws.

Likewise, the laws controlling earthly phenomena — such as the daily and yearly changes in temperature, the ebb and flow of the oceans, and the growth of plants — being seen to be influenced by the heavens in this respect at least, shared the mathematical regularity that appeared to guide the gods. … It took many centuries before the rigour of scientific understanding enabled the true influences of the heavens to be disentangled from purely suppositional and mystical ones. Yet it had been clear from the earliest times that such influences did indeed exist and that, accordingly, the mathematical laws of the heavens must have relevance also here on Earth.

Blogger Comments:

To be clear, this presents the history of natural science as the erection of a control hierarchy. First, gods are construed as controlling natural events, and later, mathematical laws are construed as controlling the gods that control natural events.

From the perspective of Systemic Functional Linguistic Theory, this is a hierarchy with metaphenomena (mathematical laws) at the highest level, and phenomena (natural events) at the lowest level. But interestingly, the middle level of the hierarchy (gods) is ambiguous in terms of these two orders of phenomena, since gods acting on nature construes gods as phenomena, whereas gods as explanations of natural events construes them as metaphenomena.

Mathematics As More Than Just A Human Cultural Activity — Through Systemic Functional Linguistics

Penrose (2004: xix):

To mathematicians (at least to most of them, as far as I can make out), mathematics is not just a cultural activity that we have ourselves created, but it has a life of its own, and much of it finds an amazing harmony with the physical universe. We cannot get any deep understanding of the laws that govern the physical world without entering the world of mathematics.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, mathematics — reasoning with quantities — is a trans-cultural semiotic system that is realised by a highly designed and restricted register of language, with its own specialised graphological system. As such, it is the creation of language users, and depends on language users for its continued existence.

The 'amazing harmony' of much of mathematics with the physical universe is the relation of construals of first-order material-order phenomena (the physical universe) to their reconstruals as semiotic-order phenomena (mathematical physics).

As previously explained, there are no laws that govern the physical world, just as there are no maps that govern the territory they map.

The Platonic Existence Of Mathematical Entities Viewed Through Systemic Functional Linguistics

Penrose (2004: xviii):

It is better to think of 3/8 as being an entity with some kind of (Platonic) existence of its own…

Blogger Comments:

To be clear, the claim here is that the fraction 3/8 is an abstract object that does not exist in space or time and is neither physical nor mental.

From the perspective of Systemic Functional Linguistic Theory, on the other hand, the fraction 3/8 is meaning realised by the realisation of wording in sounding or writing. In the first instance, it may be a participant or projection of a mental process mediated by a senser, or of a verbal process mediated by a sayer, and it exists semiotically at the time and place of the unfolding of the mental or verbal process. As a projection, it is prototypically a participant in an elaborated identifying process (the solving of a mathematical equation).

As meaning potential, its "existence" depends on its being instantiated by language users.

All The Underlying Principles Of Physics That Govern The Behaviour Of Our Universe — Through Systemic Functional Linguistics

Penrose (2004: xv):

The purpose of this book is to convey to the reader some feeling for what is surely one of the most important and exciting voyages of discovery that humanity has embarked upon. This is the search for the underlying principles that govern the behaviour of our universe. It is a voyage that has lasted for more than two-and-a-half millennia, so it should not surprise us that substantial progress has at last been made. But this journey has proved to be a profoundly difficult one, and real understanding has, for the most part, come but slowly. This inherent difficulty has led us in many false directions; hence we should learn caution. Yet the 20th century has delivered us extraordinary new insights — some so impressive that many scientists of today have voiced the opinion that we may be close to a basic understanding of all the underlying principles of physics.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, 'the behaviour of our universe' is not governed by 'underlying principles'. There are two reasons for this. Firstly, the two differ in terms of order of experience: 'the behaviour of our universe' is first-order meanings (phenomena), whereas 'the underlying principles' are second-order meanings (metaphenomena), which are reconstruals of first-order phenomena. The claim that metaphenomena govern phenomena is the claim that a map governs the territory it represents.

Secondly, to the extent that 'the underlying principles' are the laws of physics, the notion of them governing misconstrues 'law' in the sense of modalisation (probability/usuality) as 'law' in the sense of modulation (obligation/inclination). The laws of physics are probabilistic statements about 'the behaviour of our universe', not obligatory commands that 'the behaviour of our universe' obeys.

From the perspective of Systemic Functional Linguistic Theory, a scientific understanding is a reconstrual of data (phenomena) as scientific theory (metaphenomena), and a 'real' scientific understanding is one that is consistent with scientific principles and validated by the data.

From the perspective of Systemic Functional Linguistic Theory, the notion that 'all the underlying principles of physics' are there to be eventually understood betrays a 'transcendent' view of meaning: that meaning transcends semiotic systems. In the opposing 'immanent' view, meaning is solely a property of semiotic systems, such that phenomena (data) are meanings construed of experience, and metaphenomena (scientific theories) are meanings construed of phenomena. Crucially, the findings of Quantum physics validate the 'immanent' view, and invalidate the 'transcendent' view, since they demonstrate, in the words of John Wheeler, that 'no phenomenon is a real phenomenon until it is an observed phenomenon'.

In the 'immanent' view, the history of science is not a progress to a 'real' understanding of pre-existing principles, but the evolution of semiotic systems which adapt to changes in the environment in which they function — both phenomenal (e.g. the data provided by new technologies) and metaphenomenal (e.g. new theories).