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Theory of the Three Worlds

Fractions and Resonances

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Resonance works through shared overtones

Resonance occurs when two vibratory physical objects vibrate together. What matters is the two objects’ natural frequency:

  • 1st degree resonance: both objects vibrate in the same frequency (f2 = f1).
  • 2nd degree resonance: one object vibrates in an overtonefrequency of the other
    (f2 = n * f1).
  • 3rd degree resonance: both objects vibrate in a shared overtone frequency
    (f2 = n/m * f1).

The 3rd degree resonance reveals itself by the fact that the ratio of the two frequencies corresponds to a fraction of integers (n/m). This 3rd degree resonance is what interests us, for it has an effect on musical scales and chords rather than the 2nd degree frequency, as is often assumed (for this, cf. previous post).

Example: the fifth

One string a' has a basic frequency of 440 Hz and a string e" the frequency of 660 Hz. In this case, the second overtone on string a' vibrates with 3 × 440 = 1320 Hz and the first overtone on string e" vibrates with 2 × 660 = 1320 Hz. The overtones on the two strings have the same frequency and string a' can stimulate string e" through this common overtone. The ratio of the two basic frequencies is 3/2.

Proposition

For resonance to be generated, the frequencies must have a ratio of a fraction with integers.

 Why integers?

When a vibration medium (string, basilar membrane) vibrates, standing waves are produced. These are characterised by the fact that the string does not vibrate at either end, but only in the middle, with one or several antinodes (bellies). The number of the vibration bellies in the middle must be an integer since otherwise the standing wave would not be zero at either end.

Assessing musical scale tones with the help of resonances

 As we can hear, octaves and fifths can easily be made to resonate (fifth experiment) and are characterised by very simple resonance ratios (f2/f1), namely 2/1 for the octave and 3/2 for the fifth. With very few mathematical preconditions, we can now find further musical scale tones:

Criteria for the tones of a musical scale

In the cases below, we look at the interval between the tone and its fundamental tone.

  1. The interval must be situatedwithin an octave: this means that the fraction of the two frequencies (musical scale tone to fundamental tone) must be ≥ 1 and ≤ 2.
  2. The interval must be able to resonate: numerator and denominator must be integers.
  3. The resonance should be as vibrant as possible: the denominator of the fraction should be as small as possible.

The last two criteria are crucial but require some explanation, and I will illustrate the reasons for this later. For the time being, however, I will take all three mathematical criteria as given and see whether they enable us to find further well-known intervals.

Generation of a pool of possible musical scale tones with the help of the three criteria

 We begin with the denominators, starting with denominators 1 and 2. Our musical scale already includes:

All the other tones with denominators 1 or 2 are situated outside our octave range (1-2). Therefore we look whether further well-known intervals result from denominator 3:

All the other tones with denominator 3 are situated outside our octave range. We therefore continue with denominator 4:

All the other tones with denominator 4 are situated outside our octave range. We therefore continue with denominator 5:

Any further fractions with denominator 5 are situated outside our octave range.

A thought in between  

What is striking here is the fact that fractions with the numerator 7 do not exist as intervals here in Europe. My proposition is that this has something to do with the fact that seven is a prime number. This can explain why 8/5 and 9/5 are intervals that we are accustomed to although the denominator and numerators in these fractions are higher than with 7/5. Eight is 2×2×2 and nine is 3×3. We will see later that we can also compare several intervals with each other to assess the resonance. In frequency analysis, a comparison of two intervals means that the fraction of one interval is divided by the fraction of the other interval. In this situation it is of advantage that we can reduce the fraction. A prime number does not provide us with this option, but with figures such as 8 or 9, a reduction is often possible, particularly if, with fifths and fours, we have 2 or 3 in the denominator. Examples will follow.

In the case of our generation of resonant musical scale tones, however, it follows that we ignore 7/4 and 7/5 because the 7 in the numerator is too high a prime figure.

We continue with denominator 6:

Denominator 6 thus does not produce any new tones.

We omit denominator 7 as a high prime number and go directly to denominator  8:

Thus denominator 8 produces two new intervals, namely the major second (9/8 and the major seventh (15/8). The numerator and the denominator may be quite high in both fractions, but they can be divided by 2, 3 and 5. Thus the fractions of the intervals in combination with musical scales and chords become reducible, and the intervals prove to be resonant.

This is the end of our search for tones, and we can list our findings in the order of ascending frequencies:

Pool of the strongly resonating intervals

The following is striking in this pool:

a) Most intervals used by us have very simple frequency ratios.

b) The fourth, with a low denominator and numerator, is the “fourth most logical” musical scale tone. However, this tone is never an overtone. Nonetheless it makes sense, both in the mathematical world (simple fraction), in the physical world (simple resonance ratios) and mentally (subjective musical hearing experience).

c) The minor second and the tritone are missing from our pool. Mathematically speaking, these tones are not simple fractions and sound correspondingly sharper.

In music, such sharpness is interesting, because it brings change and diversity, but these intervals definitively fail to reach the ideal of a smooth resonance.

We will see later how the tritone nonetheless fits well into the resonances in certain conditions, whereas the minor second always sounds sharp and thus becomes the actual leading tone in European music.

For the time being, however, we are leaving it at our pool of ten tones. This is sufficient for many musical scales, particularly those which are used most frequently.

Further thoughts in between

As is well known, mathematicians like prime numbers. In this respect, we can define the above-mentioned criterion 3 (as small as possible numbers for numerator and denominator) more precisely:

The 3rd criterion in more precise terms:

For an interval to be able to resonate well in combination with other intervals, the prime factorisation of the numerator and the denominator should result in the smallest prime numbers possible:

2 is better than 3
3 is better than 5
5 is better than 7
7 is already too high for practical purposes.

Intervals and rhythms

This is different from rhythms, where fraction ratios also play a part. Why this is different there, and why rhythms with 7 or 11 beats sound good, can be plausibly explained with the theory of the three worlds. More about this later.

From a pool to musical scales

We now look for further criteria for attractive musical scales. The pool we have found is not a musical scale yet, but merely our starting point from which tones for different musical scales can be compiled. Here, further criteria are applicable.

Further criteria

4th criterion: preferably, the musical scale tones should also to be able to enter into resonances with each other.

5th criterion: the musical scale has a fundamental tone (a so-called tonality), which has a very special function in the musical scale.

6th criterion: the musical scale tones must not be too close to each other; if they are, we as human beings (lay people) cannot distinguish between them any longer. This is a practical constraint from the mental world. In a purely mathematical world, any number of differentiations would be conceivable; in a real situation, this is not the case. More about the criteria for musicl scales on the following page.

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Theory of the Three Worlds

“Breaking down” the Fifth

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The fifth

Let us first have a look at the fifth. It is a feature of  practically all the musical scales of human cultures. Musical scales without this pure fifth do exist, but these musical scales strike me either as artificial and deliberately designed like the whole tone scales or rather uncommon like the Locrian mode. The blues scale makes use of the “blues note” – the “flat five”, a note close to the fifth known as the diminished fifth – but also uses the perfectly normal fifth. After the octave, the fifth is certainly the interval that occurs most frequently in all the thousands of musical scales on this earth.

Fifth and twelfth

Can this normal fifth be generated by resonance like the octave? Although it is not an overtone, it can still be reached through the overtones. Below, I will show how this works, namely with a short detour through the twelfth, the third overtone.

The graph with the vibrating overtones will serve to illustrate this:

Fig. 1: A vibrating string with the fundamental tone and the first four harmonics (overtones)

In Fig. 1 I have even described the third overtone as a fifth, which is actually wrong, for in reality it is a twelfth. Nonetheless, this tone immediately strikes us as a fifth when we hear it. In Fig. 2, you can see an example of fifths and twelfths on a piano keyboard:

Fig. 2: Octave and twelfth on the piano

In our example, the fundamental tone is a (capital) C. The first overtone, the octave, is a (small) c and the second overtone, the twelfth, a (small) g. In relation to the fundamental tone C, the small g is a twelfth, but in relation to the first overtone, i.e. the small c, the g is a fifth.

The frequency of the fifth

What about this c-g interval in terms of frequency? To find out, let us compare Figures 1 and 2: Tone 3 (g) is three times the fundamental vibration (C) and tone 2 (c) twice the fundamental vibration. Thus tone 3 (g) vibrates 3/2 times as fast as tone 2 (c). If we take the small c rather than the capital C as the fundamental tone, then the small g is the fifth. And in the fifth, the upper tone (g) vibrates 3/2 times as fast as the lower tone (c). This is generally applicable:

A tone which vibrates 3/2 times as fast as another sounds a fifth higher to us.

The three worlds in the fifth

The fraction 3/2 is the mathematical aspect of the fifth. We have deduced it through the physics of the string vibrations. At the same time, we complied with the previously mentioned conditions (constraints) from the mental world: if the quint is meant to be a tone of a musical scale, it must not be too far away from the fundamental tone. This applies to every tone of a musical scale; it must be within an octave. In mathematical terms, this means that the ratio between its frequency and the frequency of the fundamental tone must be between 1 (= fundamental tone) and 2 (= octave). The fifth satisfies this requirement with the frequency ratio of 3/2 = 1.5. In the case of the twelfth, the frequency ratio is 3, i.e. greater than 2, and thus the twelfth is not a note of the musical scale. We perceive it as a fifth, but as mentioned before, in the mental world we perceive the octave as the “same” tone.

The resonance experiment for the fifth

To illustrate the relationship between the fundamental tone, the fifth and the twelfth, I propose a further resonance experiment on the piano:

Fig. 3: Resonance experiment for the fifth. In comparison to Fig. 2, the key on which we investigate the resonance is now the fifth – the capital G – and not the small g.

To begin with, we again test the twelfth and press the twelfth (the small g) with the right hand like in the octave experiment . The string should not make a sound, but the key must be kept depressed. With the left hand, we briefly and vigorously hit the C, i.e. the fundamental tone. Like in the octave experiment, the depressed string (g) should make a sound although it was not hit. This is pure resonance; the string makes a sound because it was stimulated by sound waves. This works because the small g is an overtone of the capital C.

But what about the capital G, i.e. the fifth? To test this, keep the capital G mutely depressed while vigorously hitting the fundamental tone, i.e. the capital C. You can now hear a high tone. If you listen carefully, you will notice that this is not the fifth, i.e. the capital G, but the twelfth, i.e. the small g. How come this tone sounds although you don’t keep the key of the small g depressed at all?

In fact the sound of the small g is produced on the string of the capital G! This means that the string does not vibrate in its fundamental vibration but in its first upper vibration, the octave. This works well because the string can vibrate almost as well with two antinodes (bellies) as with one. This is the so-called flageolet-note.

In other words: you have stimulated a harmonic on the string whose frequency is twice the speed of the fundamental frequency of the string. But where did the stimulus for this frequency come from? – It is the capital C string which initiated the overtone. The vibration of this C string contains the small g as a harmonic, namely the second overtone. This second overtone stimulates the (capital) G string to resonate, but not with its fundamental vibration (G), but with its first overtone, the small g, for only this can be stimulated as a harmonic of the (capital) G string. You can hear this tone (g) on the G string as long as you keep the G key depressed.

Table 1: The resonance of the fifth

Thus the resonance of the fifth is generated by the detour of the overtones. Neither string is involved in its fundamental vibration, but both strings are involved through their harmonics. The fact that this works has been demonstrated with the fifth experiment.

The fifth, a simple fraction

In Table 1, the fifth is represented as a fraction: 3/2.

As we have seen, all the tones of a musical scale must be within the range of an octave, i.e. their frequency must be between the single value and the double value of the frequency of the fundamental note. This has been achieved with the frequency ratio of 3/2 = 1.5. We have thus found the first tone within the octave range which has a very simple interval relation to the fundamental note. Whereas the octave vibrates at twice the speed of the fundamental note, the fifth vibrates 3/2 times as fast.

With the exception of the octave, the overtones do not fulfil the conditions for musical scales. Nonetheless, they play a role as transmitters of resonance. We have produced the fifth by simply breaking down the twelfth (the second overtone) by one octave. This breaking down by an octave manifests itself as the 2 in the denominator. The 3 in the numerator is the “inheritance” of the second overtone, the twelfth, which vibrates at three times the rate of the fundamental tone.

Summary

With the fraction of 3/2, which defines the fifth, we have obtained a strikingly simple ratio. This is no accident. We will see how these simple ratios (ideal world!) also play a part for the other notes of a musical scale.

The fact, however, that we are no longer able to use ratios of single integers, like with the overtones, for the musical scales and that instead, fractions (of two integers) are used, is due to the constraint of the octave limitation, which is a constraint of the physical/mental world.

The fifth is not the sole fraction among our musical scale intervals. Integer fractions define the most important tone intervals. We can find them with a simple design rule. On the following page, I will show how this design rule works.

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Theory of the Three Worlds

The overtone series is not a muscal scale

The series of overtones

As we know, the octave is the first overtone. The physical phenomenon of resonance has helped us understand the role played by the overtones: in the physical world, vibrating media such as strings or pipes can vibrate with their basic frequency, but also with an integral multiple of this frequency. In this way the octave results as the first integral multiple of the basic frequency – i.e. by the fact that its frequency is doubled.

It would now be obvious to add the further overtones, which are also integral multiples of the basic frequency, and to explain the single notes of the musical scales in this way.

Fig. 1: The fundamental tone and the first four harmonics

Fig. 1 shows how a string vibrates and how the octave and further overtones are added. Whereas the fundamental tone vibrates with precisely one “belly”, the overtones vibrate with two, three, four and more bellies. If we start with the fundamental tone C, this will result in the following series of overtones:

Fig. 2: Overtone series ranging from the fundamental tone C (note 1) to c“‘ (note 16)

The musical scale works in a closed frequency range

Resonance explains the overtone series, including the octave. How, though, does a musical scale come about, which is meant to fill the narrow range between the fundamental tone and the first overtone, i.e. the octave? As mentioned in the previous post, the notes of the musical scales are subject to this constraint whereby they all have to be within the range of an octave. The overtone series, however, leads far beyond an octave.

In addition, Fig 2 also reveals that the distances between the overtones vary a great deal. Whereas initially, they are very far apart, they get closer and closer as the series proceeds. This would be highly impractical for a musical scale to be applied in reality.

We see, however, that the overtones from note 4 onwards, and even more between notes 8 and 16, constitute something like a kind of major scale: c, d, e, (f), g (a), (h), h, c: nearly our major scale, but not exactly, since the tones in brackets (11, 13, 14) don’t quite fit.

The natural “musical scale” of the alphorn

In fact, the musical scale between notes 8 and 16 corresponds to the natural musical scale of the alphorn – including, however, the “off” notes 11, 13 and 14, and without the actually important fourth, namely the f. Nonetheless, the natural tone series of the alphorn – but also only between notes 8 and 16 – is almost something like a natural musical scale in that it packs a reasonable number of notes into the range of an octave, and this even on the basis of resonance.

Yet the overtone series that can be played on the alphorn is still not a real musical scale. Only the musical scale between notes 8 and 16 can be played on this instrument; most notes of the musical scale below this range are missing while above it, there is a confusing array of many more tones, which are increasingly closer to each other. This is not tantamount to a musical scale which repeats itself octave after octave. Additionally, the instrument is somewhat impractical. In order for the notes to be played at a normal pitch, the horn must of necessity be very long. This is different with a violin or a flute, i.e. substantially more practical. With these smaller instruments, however, we cannot achieve the tones of the musical scale as overtones (which would be flageolet tones with the violin, and with the flute pure overblows like with the alphorn), but by means of deliberate mechanical manipulation of the physical object that vibrates, namely the strings of the violin and the air column of the flute.

But do the overtones and resonances play a role in the musical scales nonetheless? – They do! You can read about this in the next post, which explains how a fifth comes about.

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Theory of the Three Worlds

Real Constraints for Musical Scales

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This is a post about the theory of the three worlds and continues the post about the perception of the octave.

Does the coincidence of the three worlds only work for the octave?

The octave demonstrates how mathematics (Platonic world) enters the physical world, and how this convergence of mathematics (integral numbers) and physics (vibrating matters) gives rise to a very special phenomenon, namely resonance. This resonance, in turn, is perceived by us human subjectively (Penrose: mental world) in quite a special way: We subjectively recognise two tones at an octave’s distance from each other as identical tones. Each of us – regardless of our cultural background – perceives a tone with twice the frequency as the “same” tone (Happy Birthday experiment).

If the frequency ratio (mathematics) deviates even only very slightly, the resonance will disappear (physics) and the tones will (mentally) strike us as different, and their simultaneous sound as a dissonance.

The octave as the first overtone thus combines the three worlds. Can we extend this success and use the further overtones following the octave for our musical scale too? The answer to this question is not a simple yes, for the mathematical series of integers has to fit in with the constraints of the physical and mental worlds.

What constraints are these? And what is a musical scale in the first place?

Constraints in the physical and mental worlds

Sounds are used to communicate, and mammals and human beings communicate acoustically. They are able to generate sounds and to hear them. These physical/mental circumstances must be taken into account if we consider how musical scales came into being.

We are unable to produce pitches of any frequency with our voices. And if two tones are far apart from their frequency, we find it difficult to measure their mutual distance (mental, subjective world). Therefore the notes of a musical scale must not be too far apart. This is the first constraint of the physical and mental world in the formation of musical scales.

This physical/mental constraint can be further substantiated and specified: since we perceive a second note an octave higher as the “same” tone (Happy Birthday experiment), a musical scale must not exceed the range of an octave. If it did, the musical scale would overlap with itself because tones outside the octave would immediately find a “same” tone within the octave. For this reason, a musical scale is always limited to the range of an octave, as is the case in all musical cultures.

On the other hand, the notes must not be too close together, for if they were, we would not be able to differentiate between them any longer. A musical scale must therefore not have any number of notes – even though this would be perfectly conceivable in mathematical terms. Yet not everything that is mathematically possible makes sense in reality.

The consequences of these conditions for musical scales can be summarised in two points:

  1. The notes of a musical scale must be inside the range of an octave.
  2. A musical scale must not consist of too many notes.

This is the physical/mental constraint for musical scales.

How can convincing musical scales come into being under these constraints? Can they still be scales with simple mathematical ratios?  →  next post

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Theory of the Three Worlds

The Perception of the Octave in the Mental World

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This is a post about the theory of the three worlds and continues the post about the resonance of the octave.

The subjective side

The mathematical world (Pythagoras) with its simple ratios and the physical world with its resonance phenomena provide us with an understanding of the octave but still fail to explain why this interval is the basis of all musical scales in all cultures. To understand this, we will also have to look at the mental world, i.e. the world of our subjective perception.

This world is accessible to everyone, but it will always remain your own and subjective perception. I can’t read your mind. Although imaging techniques such as MRI or PET are capable of observing which areas of the brain are active at what time, what they thus make perceptible is the flow of blood in a specific place, not the thought of how you experience it.

Happy Birthday

The mental world is your very personal world, but it makes quite a contribution to the primacy of the octave. Again, I propose a little experiment, not an objective one as in the preceding post, but still one that is verifiable. It has the advantage that in all likelihood, you will already have conducted it several times.

It can also be a Christmas carol in a family setting. Several people sing together, and if we are lucky, we sing with one voice. At any rate, this is usually our intention. It works better if all the singers have roughly the same register. But what if women and men and children sing together? We still recognise it if they all sing in unison. Although we do not sing in the same frequencies, but with frequencies that are an octave distant from each other, we practically don’t notice it. We perceive the distance of an octave as the same tone. If I, as a bass singing beside an alto, fail to hit the lower octave, I’m out of tune; if I do hit it, I sing in tune. This is the subjective effect of the octave: it is the same note.

The resonance in the physical world facilitates this subjective concurrence of the tones that are an octave distant from each other, and the resonance ratios on the basilar membrane of the inner ear support us in subjectively bringing the two frequencies together in our mental world, too.

First and second overtone

The octave is the first mathematically and physically possible overtone and in this respect it differs from the second overtone, which in the musical scale coincides with a fifth. To illustrate this mathematical correlation, I will again show the vibration ratios of the fundamental tone and the first overtones:

Fig. 1: Octave and fifth as overtones

Why, though, is the octave the characteristic of monophony rather than the fifth, although both of them are most closely related to the fundamental tone in mathematical and physical terms? Although mathematically the fifth is slightly farther distant from the fundamental tone than the octave, the double octave is still farther away, and yet we mentally perceive the double octave as the “same” tone as the fundamental tone, precisely like the octave.

In the mental world, i.e. in what we experience, there is a clear difference between the octave and the fifth. In this world, the octave (and all the multiple octaves) are the “same” tone, whereas the fifth is another tone. This applies all over the world, in all cultures. Since a tone which is an octave higher is perceived as the same tone, the musical scales repeat themselves an octave higher, but not a fifth.

An experiment to distinguish the octave from the fifth in the mental world

The Happy Birthday experiment described above can be extended to demonstrate the difference between the fifth and the octave, as well as the special role of the fifth. At the next birthday party, singers may try, for example, to sing the song not an octave, but a fifth lower (or higher). This is likely to be very difficult for you, precisely because you don’t sing the “same” notes as the others. And if you succeed, the others will look at you in amazement, precisely because you sing the fifth and “not the same tone”. The octave is the same tone, the fifth isn’t.

About access to the mental world

As is well known, the mental world is difficult to prove since it is completely subjective. Although all of us permanently live in this world with our thoughts and feelings, it is only indirectly accessible to objective scientific exploration. You can communicate the contents of your mental experience to other people, but you can never be quite sure that others will experience them in the same way. You can only hope that others will be able to understand your experience. Yet precisely this subjective experience and its comprehension make music so interesting: we share our subjectivity in a very specific way.

Conclusion

We can see how the mathematical, physical and mental worlds precisely meet in the octave. The uniform significance of the octave in all the world’s musical cultures can only be understood when we include all three worlds.

The next post will be about the other notes of the musical scales. Can they also be explained as simply as the octave?

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Theory of the Three Worlds

The Platonic world

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Why “Platonic”?

Penrose calls one of the three worlds in the theory of three worlds as Platonic. Why?

Plato

The rich Athenian citizen Plato was a follower of the philosopher Socrates. He set up a school of philosophy in the 4th century B.C., which was fundamental for European philosophy and has crucially shaped philosophical discussions until the present day. If Roger Penrose thus calls one of the three worlds “Platonic”, he refers to Plato and specifically to one particular question and the discourse about it, which is still of great significance today. This question is, “Are ideas real?”

Plato’s realism of ideas

Subsequent philosophers often presented the issue as a conflict between Plato and his disciple Aristotle. Plato is ascribed the attitude that ideas were not only real but even constitute actual reality, while what we describe as reality were mere shadows of the original ideas. Penrose calls the abstract world of mathematics “Platonic”, thus referring to Plato’s thought that abstract ideas had the quality of reality.

The world of ideas as one of the three worlds

Of course the Platonic reality of ideas was also disputed, and Europe’s history of philosophy is full of pros and cons about it, which under the headings of realism, nominalism and the problem of universals have shaped the philosophers’ discourse for many centuries and exert an influence in the background even now. Like Plato, Penrose’s theory attributes a reality to the abstract Platonic world, but not an exclusive reality, as would an uncompromising Platonic realism, but as one of the three real worlds, which interact with each other. Thus the theory of the three worlds is not about which world is the real and true one – which was discussed at length as the problem of universals – but about how the interaction takes place between them.

But let’s return to the Platonic world. What distinguishes it from the other two worlds?

Characteristics of the Platonic world

 Nonlocality

Where is the number “3”? Can you point your finger at it somewhere in your environment?

Of course you can point at three apples, at three pencilled lines or at three coffee cups, but this is not the number three; rather, they are apples, pencilled lines and coffee cups. The number three remains abstract. No one can point at it.

Naturally, you can also point at the word “three” or at the “3” in this text, but these are only symbols of the number and not the number itself. The number itself remains abstract; it simultaneously exists everywhere and nowhere.

Symbols are always in a certain place, they are thus localised. The number itself, however, is non-local, i.e. there is no place in the universe wh  ere the number is found; rather, it can be found everywhere. It exists on earth, on the moon and equally in Andromeda. This nonlocality is a very elementary property of the objects of the Platonic world; in particular, it distinguishes them from the objects in the physical world, in which objects are locally defined, i.e. localised.

Timelessness

 The Platonic world’s relationship with time is analogous to locality:

1 plus 2 is 3 – this is true now, was true yesterday and will be true tomorrow and for ever thereafter. In this sense, we can describe the Platonic world as a place of eternal truths, in stark contrast to the physical world, which is subject to constant change. If it rains today, the sun may shine tomorrow; 1 plus 2 is 3 every day and any day. This timelessness applies to all mathematical statements, but also to their objects, again in contrast to the objects of the physical world: the number 3 is timeless, whereas 3 apples are not.

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Logic, Theory of the Three Worlds

The theory of the three worlds (Penrose)

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The theory of the three worlds

There are practical questions which concern our specific lives, and there are theoretical questions which seemingly don’t. However, there are also theoretical considerations which definitely concern our practical everyday lives. One of these is the three worlds theory, which deals with questions as to which worlds we specifically live in.

On what foundation is our everyday existence based? The theory of the three worlds points to the fact that we simultaneously live in three completely different worlds. Practically, this does not constitute a problem for us; theoretically, however, the question arises as to how three worlds which are so different from each other are able to meet in reality at all.

Roger Penrose has named the three worlds as follows:
A) the Platonic world,
B) the physical world,
C) the mental world.

This is Roger Penrose’s original graph:

 

 

 

 

 

 

 

 

Platonic world: The world of ideas. Mathematics, for example, is completely located in the Platonic world.

Physical world: The real, physical world with things that are in a specific place at a specific time.

 Mental world: My subjective perceptions without which I would not be able to recognise the other worlds, but also my thoughts and ideas as I experience them.

The circular relationship between the three worlds

The arrows between the spheres indicate the circular relationship that these worlds engage in together:

Platonic → physical: Behind physics, there is mathematics. Physics is inconceivable without higher mathematics. Evidently, the physical world complies with mathematical laws with a staggeringly accurate precision. Is the real world therefore determined by mathematics?

 Physical → mental: My brain is part of the physical world. According to common understanding, the neurons of the brain tissue determine my brain performance with their electric switches.

 Mental → Platonic: Great thinkers are capable of formulating the laws of mathematics in their thoughts (mental world); these laws “come into being” in their heads.

This, then, is the circular process: The Platonic world (mathematics) determines the physical one, which is the basis of human thought. In human thought, in turn, mathematics (and other ideas) are located. These mathematical laws … and here we come full circle.

The scope of the three worlds

What is also interesting are the opening funnels in Penrose’s sketch, which together with the arrows point from one world to the next. Penrose uses them to indicate the fact that the world that follows in the circular process merely requires part of the world from which it emerges during the generation process.

Platonic → physical: Only a small part of mathematical findings can be used in physics. Seen in this light, the physical laws only need (are?) an excerpt from mathematics.

Physical → mental: My brain is a very small part of the physical world.

Mental → Platonic: My brain deals with many things; mathematics and abstract ideas are only a part of it.

The Platonic world is then the origin of the physical world again. However, the proportions do not appear to work out properly. This resembles the famous impossible staircase:

figure: the impossible staircase

The impossible staircase

As an aside:
The impossible staircase was discovered by Roger Penrose’s father, Lionel Penrose, and is also called the Penrose steps – or the Escher-Penrose steps after the Dutch graphic artist who, inter alia, inspired Douglas Hofstadter to write his book Gödel, Escher, Bach. The endlessness with which the steps ascend can seemingly be graphically represented without any problems, but from a logical point of view it is eminently intricate (self-referential taboo).

For Penrose, there is a mystery in the three worlds. He writes that undoubtedly there are not three separate worlds in reality but only one, and at present we are not even able to divine the true nature of this world. This is therefore about three worlds in one – and thus about their differences and the form of their interlinkage.

Not an abstract theory

The three worlds are not an abstract theory but can be recognised in our own world of private experiences. They play an important part in music, for instance. The example of music also enables us to see how the three worlds interact. More about this on this website.

Translation: Tony Häfliger and Vivien Blandford