Quantum Structure in Cognition: Human Language as a Boson Gas of Entangled Words
Abstract
We model a piece of text of human language telling a story by means of the quantum structure describing a Bose gas in a state close to a Bose–Einstein condensate near absolute zero temperature. For this we introduce energy levels for the words (concepts) used in the story and we also introduce the new notion of ‘cogniton’ as the quantum of human thought. Words (concepts) are then cognitons in different energy states as it is the case for photons in different energy states, or states of different radiative frequency, when the considered boson gas is that of the quanta of the electromagnetic field. We show that Bose–Einstein statistics delivers a very good model for these pieces of texts telling stories, both for short stories and for long stories of the size of novels. We analyze an unexpected connection with Zipf’s law in human language, the Zipf ranking relating to the energy levels of the words, and the Bose–Einstein graph coinciding with the Zipf graph. We investigate the issue of ‘identity and indistinguishability’ from this new perspective and conjecture that the way one can easily understand how two of ‘the same concepts’ are ‘absolutely identical and indistinguishable’ in human language is also the way in which quantum particles are absolutely identical and indistinguishable in physical reality, providing in this way new evidence for our conceptuality interpretation of quantum theory.
Keywords
Human language Bose–Einstein statistics Zipf’s law Identity Indistinguishability Bose gas1 Introduction
Human language is a substance consisting of combinations of concepts giving rise to meaning. We will show that a good model for this substance is the one of a gas of entangled bosonic quantum particles such as they appear in physics in the situation close to a Bose–Einstein condensate. In this respect we also introduce the new notion of ‘cogniton’ as the entity playing the same role within human language of the ‘bosonic quantum particle’ for the ‘quantum gas’. There is a gas of bosonic quantum particles that we all know very well, and that is the electromagnetic field, which we will also briefly call ‘light’, which is a substance of photons. Often we will use ‘light’ as an example and inspiration of how we will talk and reason about human language where ‘concepts’ (words), as ‘states of the cogniton’, are then like ‘photons of different energies (frequencies, wave lengths)’. With the new findings we present here, we also make an essential and new step forward in the elaboration of our ‘conceptuality interpretation of quantum theory’, where quantum particles are the concepts of a proto-language, in a similar way that human concepts (words), are the quantum particles (cognitons) of human language (Aerts 2009a, 2010a, b, 2013, 2014; Aerts et al. 2018d, 2019c).
There are several new results and insights that we will put forward in the coming sections. We summarize them here, referring also to earlier work on which they are built, guaranteeing however that the article is self-contained, so that it is not necessary to have studied these earlier works for understanding its content. The reason we can present here a self-contained theory of human language is because most of our earlier results take a simple and transparent form in the model of a boson gas that we elaborate here for human language. Since we also introduce the basics of the physics of a boson gas, our presentation will remain self-contained also from a physics’ perspective. In the article, we will use the terms ‘words’ and ‘concepts’ interchangeably because their difference does not play a role in the aspects of language we study.
We will see that the state of the gas of bosonic quantum particles which we identify explicitly to also be the state of a piece of text such as that of a story is one of very low temperature, i.e. a temperature in the neighborhood of where also the fifth state of matter appears, namely the Bose–Einstein condensate. This means that the interactions between ‘words’, which are the boson particles of language in our description, is mainly one of ‘quantum superposition’ and ‘quantum entanglement’, or more precisely one of ‘overlapping de Broglie wave functions’. This corresponds well with some of our earlier findings, when studying the combinations of concepts in human language, namely that superposition and entanglement are abundant, and the type of entanglement is deep, namely it also violates additionally to Bell’s inequality the marginal laws (Aerts 2009b; Aerts et al. 2011; Aerts and Sozzo 2011, 2014; Aerts et al. 2012, 2015a, 2016, 2018a, b, c, 2019a, b; Aerts Arguëlles 2018; Beltran and Geriente 2019).
When we present our model in the next sections, we will see that it contains several new explanations of aspects of human language which we brought up in earlier work. For example, we elaborated an axiomatic quantum model for human concepts, which we called SCoP (state context property system), and in which different exemplars of a specific concept are considered as different states of this concept (Gabora and Aerts 2002; Aerts and Gabora 2005a, b; Aerts 2009b; Aerts et al. 2013a, b). In the theory of the boson gas for human language that we develop here, we will not only introduce these states explicitly, but also introduce them as eigenstates for specific values of the energy and a detailed energy scale for all the words appearing in a considered piece of text will be introduced. If we compare this with the quantum description of light, it means that the cognitons of our piece of text of human language will radiate their meaning with different frequencies to the human mind, engaging in the meaning of this piece of text.
For each of the energy levels \(E_i\), \(N(E_i)E_i\) is the amount of energy ‘radiated’ by the story ‘In Which Piglet Meets a Heffalump’ with the ‘frequency or wave length’ connected to this energy level. For example, the energy level \(E_{54} = 54\) is populated by the concept Thought and the word Thought appears \(N(E_{54})=10\) times in the story ‘In Which Piglet Meets a Heffalump’. Each of the 10 appearances of Thought radiates with energy value 54, which means that the total radiation with the wave length connected to Thought of the story ‘In Which Piglet Meets a Heffalump’ equals \(N_{54}E_{54} = 10 \cdot 54 = 540\).
We started this investigation with the idea that ‘concepts within human language behave like bosonic entities’, an idea we expressed earlier as one of the basic pieces of evidence for the ‘conceptuality interpretation’ (Aerts 2009a). The origin of the idea is the simple direct understanding that if one considers, for example, the concept combination Eleven Animals, then, on the level of the ‘conceptual realm’ each one of the eleven animals is completely ‘identical with’ and ‘indistinguishable from’ each other of the eleven animals. It is also a simple direct understanding that in the case of ‘eleven physical animals’, there will always be differences between each one of the eleven animals, because as ‘objects’ present in the physical world, they have an individuality, and as individuals, with spatially localized physical bodies, none of them will be really identical with the other ones, which means that each one of them will also always be able to be distinguished from the others. Even if all the animals are horses, simply because they are ‘objects’ and not ‘concepts’, they will not be completely identical and hence they will be distinguishable. The idea is that it is ‘this not being completely identical and hence being distinguishable’ which makes the Maxwell–Boltzmann statistics being applicable to them. However, when we consider ‘eleven animals’ as concepts, such that their ontological nature is conceptual, they are all ‘completely identical and hence intrinsically indistinguishable’. Within the conceptuality interpretation of quantum theory, where we put forward the hypothesis that quantum entities are ‘conceptual’ and hence are not ‘objects’, their ‘being completely identical and hence intrinsically indistinguishable’, would also be due to their being conceptual instead of objectual entities.
In earlier work we already investigated this idea by looking at simple combinations of concepts with numerals, such as indeed Eleven Animals and then considering two states of Animal, namely Cat and Dog. We then checked whether the twelve different exemplars of them that form in these two states, namely Eleven Dogs, One Cat And Ten Dogs, Two Cats And Nine Dogs,..., Ten Cats And One Dog, Eleven Cats, in their appearance in texts follow a Maxwell–Boltzmann or rather a Bose–Einstein statistical pattern. In a less convincing way because of a collection of limited data (Aerts 2009a; Aerts et al. 2015b), but with an abundance of data and very convincingly (Beltran 2019), it was shown that indeed the Bose–Einstein statistics delivers a better model for the data as compared to the Maxwell–Boltzmann statistics.
The result that we put forward in the present article, namely that the Bose–Einstein statistics as explained above models entire texts of any size, is a much stronger one, although it expresses the same idea. Consider any text, and then consider two instances of the word Cat appearing in the text, if then one of the concepts Cat is exchanged with the other concept Cat, absolutely nothing changes in the text. Hence, a text contains a perfect symmetry for the exchange of cognitons (concepts, words) in the same state. This is not true for physical reality and its physical objects. Suppose one considers a physical landscape where two cats are within the landscape, exchanging the two cats will always change the landscape, because the cats are not identical and are distinguishable as physical objects. If we introduce a quantum description of the text, the wave function must be invariant for the exchange of the two cats, which would again be not the case if the wave function would describe the physical landscape containing two cats as objects. This is the result we will present in Sect. 2.
Section 3 is devoted to a self-contained presentation of the phenomenon of Bose–Einstein condensation in physics. We illustrate the different aspects of the Bose–Einstein condensation valuable for our discussion, by means of two examples of Bose gases, the rubidium 87 atom gas and the sodium atom gas, that also originally where the first ones to be used to realize a Bose–Einstein condensate (Anderson et al. 1995; Davis et al. 1995). We compare the Bose–Einstein condensates of the gases and how their energy level distribution is modeled by the Bose–Einstein distribution function with our Bose–Einstein modeling of pieces of texts of stories and point out the points of correspondence.
Another finding that we will put forward, in Sect. 4, was completely unexpected. The method of attributing an energy level to a word depending on the number of appearances of the word in a text, introduces the typical ranking considered in the well-known Zipf’s law analysis of this text (Zipf 1935, 1949). When we look at the \(\log /\log\) graph of ranking in function of the number of appearances, we indeed see the linear function, or a slight deviation of it, which represents the most common version of Zipf’s law. Zipf’s law is an experimental law, which has not yet been given any theoretical foundation, hence perhaps our finding, of its unexpected connection with Bose–Einstein statistics, might provide such a foundation. We also show, in Sect. 4, how the connection with Zipf’s law allows us to develop more in depth the Bose–Einstein model of texts of different sizes, short stories and long stories of the size of novels.
In Sect. 5, we reflect about the issue of ‘identity and indistinguishability’ from the perspective we developed in the foregoing sections, taking into account the conundrum this issue actually still is in quantum theory with respect to quantum particles (Dieks and Lubberdink 2019). Confronting the theoretical view where bosons and fermions are considered to be identical and indistinguishable even if they are in different states, we note that experimentalists take another stance in this respect considering, for example, photons of different frequencies as distinguishable. A recent experiment shows that if this experimentally accepted possibility to distinguish them is erased by means of a quantum eraser, these different frequency photons behave as indistinguishable (Zhao et al. 2014). This makes us put forward the proposal that ‘the way in which we clearly see and understand the identity and indistinguishability of concepts (words, cognitons) in human language’ is also ‘the way in which identity and indistinguishability for quantum particles can be understood’. More specifically, it shows that ‘identity and indistinguishability’ are contextual notions for a quantum particle, depending on the way a measuring apparatus or a heat bath interacts with the quantum particle, similarly to how ‘identity and indistinguishability’ are contextual notions for a human concept, depending on how a mind interacts with the concept. We elaborate with examples this new way of interpreting ‘identity and indistinguishability’ and show how it is a strong confirmation of our conceptuality interpretation of quantum theory.
2 Human Language as a Bose Gas
Let us consider again the Winnie the Pooh story ‘In Which Piglet Meets a Heffalump’ as published in Milne (1926). In Table 1, we have presented the list of all words that appear in the story (in the column ‘Words concepts cognitions’), with their ‘number of appearances’ (in the column ‘Appearance numbers \(N(E_i)\)’), ordered from lowest energy level to highest energy level (in the column ‘Energy levels \(E_i\)’), where the energy levels are attributed according to these numbers of appearances, lower energy levels to higher number of appearances, and their values are given as proposed in (1).
The word And is the most often appearing word, namely 133 times, hence the cognitons in this state populate the ground state energy level \(E_0\), which as per (1) we put equal to zero. The word He is the second most often appearing word, namely 111 times, hence the cognitons in this state populate the first energy level \(E_1\), which following (1) we put equal to 1. Hence, the ‘words’, their ‘energy levels’ and their ‘numbers of appearances’ are in the first three columns of Table 1.
The question can be asked ‘what is the unity of energy in this model that we put forward?’, is the number ‘1’ that we choose for energy level \(E_1\) a quantity expressed in joules, or in electronvolts, or still in another unity? This question gives us the opportunity to reveal already one of the very new aspects of our approach. Energy will not be expressed in ‘\({\mathrm{kg\,m}}^2/{\mathrm{s}}^2\)’ like it is the case in physics. Why not? Well, a human language is not situated somewhere in space, like we believe it to be the case with a physical boson gas of atoms, or a photon gas of light. Hence, ‘energy’ is here in our approach a basic quantity, and if we manage to introduce—this is one of our aims in further work—what the ‘human language equivalent’ of ‘physical space’ is, then it will be oppositely, namely this ‘equivalent of space’ will be expressed in unities where ‘energy appears as a fundamental unit’. Hence, the ‘1’ indicating that ‘He radiates with energy 1’, or ‘the cogniton in state He carries energy 1’, stands with a basic measure of energy, just like ‘distance (length)’ is a basic measure in ‘the physics of space and objects inside space’, not to be expressed as a combination of other physical quantities. We used the expressions ‘He radiates with energy 1’, and ‘the cogniton in state He carries energy 1’, and we will use this way of speaking about ‘human language within the view of a boson gas of entangled cognitons that we develop here’, in similarity with how we speak in physics about light and photons.
The words The, It, A and To, are the four next most often appearing words of the Winnie the Pooh story, and hence the energy levels \(E_2\), \(E_3\), \(E_4\) and \(E_5\) are populated by cognitons respectively in the states The, It, A and To carrying respectively 2, 3, 4 and 5 basic energy units. Hence, the first three columns in Table 1 describe the experimental data that we extracted from the Winnie the Pooh story ‘In Which Piglet Meets a Heffalump’. As we said, the story contains in total 2655 words, which give rise to 542 energy levels, where energy levels are connected with words, hence different words radiate with different energies, and the size of the energies are determined by ‘the number of appearances of the words in the story’, the most often appearing words being states of lowest energy of the cogniton and the least often appearing words being states of highest energy of the cogniton. In Table 1, we have not presented all 542 energy levels, because that would lead to a too long table, but we have presented the most important part of the energy spectrum, with respect to the further aspects we will point out.
More concretely, we have represented the range from energy level \(E_0\), the ground state of the cogniton, which is the cogniton in state And, to energy level \(E_{78}\), which is the cogniton in state Put. Then we have represented the energy level from \(E_{538}\), which is the cogniton in state Whishing, to the highest energy level \(E_{542}\) of the Winnie the Pooh story, which is the cogniton in state You’ve.
These last five highest energy levels, from \(E_{538}\) to \(E_{542}\), corresponding respectively to the cogniton in states Whishing, Word, Worse, Year and You’ve, all have a number of appearance of ‘one time’ in the story. They do however radiate with different energies, but the story is not giving us enough information to determine whether Whishing is radiating with lower energy as compared to Year or vice versa. Since this does not play a role in our actual analysis, we have ordered them alphabetically. So, different words which radiate with different energies that appear an equal number of times in this specific Winnie the Pooh story will be classified from lower to higher energy level alphabetically.
An energy scale representation of the words of the Winnie the Pooh story ‘In Which Piglet Meets a Heffalump’ by A. A. Milne as published in Milne (1926)
Words concepts cognitions | Energy levels \(E_i\) | Appearance numbers \(N(E_i)\) | Bose–Einstein modeling | Maxwell–Boltzmann modeling | Energies from data \(E(E_i)\) | Energies Bose–Einstein | Energies Maxwell–Boltzmann |
---|---|---|---|---|---|---|---|
And | 0 | 133 | 129.05 | 28.29 | 0 | 0 | 0 |
He | 1 | 111 | 105.84 | 28.00 | 111 | 105.84 | 28.00 |
The | 2 | 91 | 89.68 | 27.69 | 182 | 179.36 | 55.38 |
It | 3 | 85 | 77.79 | 27.40 | 255 | 233.36 | 82.19 |
A | 4 | 70 | 68.66 | 27.11 | 280 | 274.65 | 108.43 |
To | 5 | 69 | 61.45 | 26.82 | 345 | 307.23 | 234.09 |
Said | 6 | 61 | 55.59 | 26.53 | 366 | 333.55 | 159.20 |
Was | 7 | 59 | 50.75 | 26.25 | 413 | 355.24 | 183.76 |
Piglet | 8 | 47 | 46.68 | 25.97 | 376 | 373.40 | 207.78 |
I | 9 | 46 | 43.20 | 25.70 | 414 | 388.82 | 231.27 |
That | 10 | 41 | 40.21 | 25.42 | 410 | 402.05 | 254.24 |
Pooh | 11 | 40 | 37.59 | 25.15 | 440 | 413.52 | 276.69 |
Of | 12 | 39 | 35.30 | 24.89 | 468 | 423.55 | 298.64 |
Had | 13 | 28 | 33.26 | 24.62 | 364 | 432.38 | 320.09 |
Would | 14 | 26 | 31.44 | 24.36 | 364 | 440.21 | 341.05 |
As | 15 | 25 | 29.81 | 24.10 | 375 | 447.19 | 361.53 |
In | 16 | 25 | 28.34 | 23.86 | 400 | 453.44 | 381.53 |
But | 17 | 23 | 27.00 | 23.59 | 391 | 459.07 | 401.07 |
Heffalump | 18 | 23 | 25.79 | 23.34 | 414 | 464.15 | 420.15 |
His | 19 | 23 | 24.67 | 23.09 | 437 | 468.77 | 438.78 |
Very | 20 | 23 | 23.65 | 22.85 | 460 | 472.96 | 456.97 |
You | 21 | 23 | 22.70 | 22.61 | 483 | 476.79 | 474.72 |
Then | 22 | 21 | 21.83 | 22.37 | 462 | 480.30 | 492.05 |
Honey | 23 | 20 | 21.02 | 22.13 | 460 | 483.51 | 508.95 |
So | 24 | 20 | 20.27 | 21.89 | 480 | 486.47 | 525.43 |
Up | 25 | 20 | 19.57 | 21.66 | 500 | 489.19 | 541.51 |
They | 26 | 19 | 18.91 | 21.43 | 494 | 491.71 | 557.19 |
If | 27 | 18 | 18.30 | 21.20 | 486 | 494.03 | 572.47 |
Jar | 28 | 18 | 17.72 | 20.98 | 504 | 496.18 | 587.37 |
There | 29 | 18 | 17.18 | 20.75 | 522 | 498.18 | 601.89 |
At | 30 | 17 | 16.67 | 20.53 | 510 | 500.03 | 616.03 |
Be | 31 | 15 | 16.19 | 20.32 | 465 | 501.75 | 629.80 |
Got | 32 | 15 | 15.73 | 20.10 | 480 | 503.34 | 643.21 |
Just | 33 | 15 | 15.30 | 19.89 | 495 | 504.83 | 656.26 |
What | 34 | 15 | 14.89 | 19.68 | 510 | 506.22 | 668.97 |
Christopher | 35 | 14 | 14.50 | 19.47 | 490 | 507.51 | 681.33 |
This | 36 | 14 | 14.13 | 19.26 | 504 | 508.71 | 693.35 |
Trap | 37 | 14 | 13.78 | 19.06 | 518 | 509.83 | 705.03 |
About | 38 | 13 | 13.44 | 18.85 | 494 | 510.88 | 716.40 |
All | 39 | 13 | 13.12 | 18.65 | 507 | 511.86 | 727.44 |
Should | 40 | 13 | 12.82 | 18.45 | 520 | 512.77 | 738.17 |
For | 41 | 12 | 12.53 | 18.26 | 492 | 513.62 | 748.59 |
Like | 42 | 12 | 12.25 | 18.06 | 504 | 514.41 | 758.70 |
Robin | 43 | 12 | 11.98 | 17.87 | 516 | 515.15 | 768.51 |
See | 44 | 12 | 11.72 | 17.68 | 528 | 515.84 | 778.03 |
When | 45 | 12 | 11.48 | 17.49 | 540 | 516.48 | 778.26 |
Down | 46 | 11 | 11.24 | 17.31 | 506 | 517.08 | 796.20 |
Heffalumps | 47 | 11 | 11.01 | 17.12 | 517 | 517.64 | 804.87 |
With | 48 | 11 | 10.79 | 16.94 | 528 | 518.15 | 813.26 |
Do | 49 | 10 | 10.58 | 16.76 | 490 | 518.63 | 821.39 |
Go | 50 | 10 | 10.38 | 16.58 | 500 | 519.08 | 829.25 |
Off | 51 | 10 | 10.19 | 16.41 | 510 | 519.49 | 836.85 |
On | 52 | 10 | 10.00 | 16.23 | 520 | 519.87 | 844.19 |
Think | 53 | 10 | 9.82 | 16.06 | 530 | 520.22 | 851.29 |
Thought | 54 | 10 | 9.64 | 15.89 | 540 | 520.54 | 858.13 |
More | 55 | 9 | 9.47 | 15.72 | 495 | 520.83 | 864.74 |
No | 56 | 9 | 9.31 | 15.56 | 504 | 521.10 | 871.11 |
Out | 57 | 9 | 9.15 | 15.39 | 513 | 521.35 | 877.25 |
Pit | 58 | 9 | 8.99 | 15.23 | 522 | 521.57 | 883.15 |
Went | 59 | 9 | 8.84 | 15.07 | 531 | 521.77 | 888.84 |
Don’t | 60 | 8 | 8.70 | 14.91 | 480 | 521.95 | 894.30 |
Good | 61 | 8 | 8.56 | 14.75 | 488 | 522.11 | 899.55 |
Head | 62 | 8 | 8.43 | 14.59 | 496 | 522.25 | 904.58 |
Know | 63 | 8 | 8.29 | 14.44 | 504 | 522.37 | 909.41 |
Oh | 64 | 8 | 8.16 | 14.28 | 512 | 522.48 | 914.03 |
Right | 65 | 8 | 8.04 | 14.13 | 520 | 522.57 | 918.45 |
Well | 66 | 8 | 7.92 | 13.98 | 528 | 522.64 | 922.67 |
Bed | 67 | 7 | 7.80 | 13.83 | 469 | 522.70 | 926.70 |
Could | 68 | 7 | 7.69 | 13.68 | 476 | 522.74 | 930.54 |
Deep | 69 | 7 | 7.58 | 13.54 | 483 | 522.77 | 934.20 |
Did | 70 | 7 | 7.47 | 13.40 | 490 | 522.78 | 937.67 |
First | 71 | 7 | 7.36 | 13.25 | 497 | 522.79 | 940.96 |
Have | 72 | 7 | 7.26 | 13.11 | 504 | 522.78 | 944.08 |
Help | 73 | 7 | 7.16 | 12.97 | 511 | 522.76 | 947.02 |
Himself | 74 | 7 | 7.06 | 12.84 | 518 | 522.72 | 949.79 |
How | 75 | 7 | 6.97 | 12.70 | 525 | 522.68 | 952.40 |
Looked | 76 | 7 | 6.88 | 12.56 | 532 | 522.63 | 954.85 |
Now | 77 | 7 | 6.79 | 12.43 | 539 | 522.56 | 957.13 |
Put | 78 | 7 | 6.70 | 12.30 | 546 | 522.49 | 959.27 |
... | ... | ... | ... | ... | ... | ... | ... |
... | ... | ... | ... | ... | ... | ... | ... |
Wishing | 538 | 1 | 0.67 | 0.09 | 538 | 359.92 | 48.65 |
Word | 539 | 1 | 0.67 | 0.09 | 539 | 359.58 | 48.22 |
Worse | 540 | 1 | 0.67 | 0.09 | 540 | 359.24 | 47.80 |
Year | 541 | 1 | 0.66 | 0.09 | 541 | 358.90 | 47.38 |
You’ve | 542 | 1 | 0.66 | 0.09 | 542 | 358.55 | 46.96 |
2655 | 2655.00 | 2654.96 | 242,891 | 242,891.01 | 242,889.76 |
The result is expressed in the graphs of Fig. 1a, where the blue graph represents the data, hence the numbers in column ‘Energies from data \(E(E_i)\)’ of Table 1, the red graph represents the quantities obtained by the Bose–Einstein model, hence the quantities in column ‘Bose–Einstein modeling’ of Table 1, and the green graph represents the quantities obtained by the Maxwell–Boltzmann model, hence the quantities of column ‘Energies Maxwell–Boltzmann’ of Table 1. We can easily see in Fig. 1a how the blue and red graphs almost coincide, while the green graph deviates abundantly from the two other graphs which shows how Bose–Einstein statistics is a very good model for the data we collected from the Winnie the Pooh story, while Maxwell–Boltzmann statistics completely fails to model these data.When we determine the two constants A and B, respectively C and D, in the Bose–Einstein distribution function (9) and Maxwell–Boltzmann distribution function (12), by putting the total number of particles of the model equal to the total number of words of the considered piece of text, (10) and (13), and by putting the total energy of the model to the total energy of the considered piece of text, (11) and (14), we find a remarkable good fit of the Bose–Einstein modeling function with the data of the piece of text, and a big deviation of the Maxwell–Boltzmann modeling function with respect to the data of the piece of text.
To construct the two models, we also considered the energies, and expressed as a second constraint the conditions (11), (14), that the total energy of the Bose–Einstein model and the total energy of the Maxwell–Boltzmann model are both equal the total energy of the data of the Winnie the Pooh story. The result of both constraints, (10), (13) and (11), (14) on the energy functions that express the amount of energy per energy level—or, to use the language customarily used for light, the frequency spectrum of light—can be seen in Fig. 2. We see again that the red graph, which represent the Bose–Einstein radiation spectrum, is a much better model for the blue graph, which represents the experimental radiation spectrum, as compared to the green graph, which represents the Maxwell–Boltzmann radiation spectrum.
Due to their shape, the graphs in Fig. 1a are not easily comparable, and although quite obviously the blue and red graphs are almost overlapping, while the blue and green graphs are very different, which shows that the data are well modeled by Bose–Einstein statistics and not well modeled by Maxwell–Boltzmann statistics, it is interesting to consider a transformation where we apply the \(\log\) function to both the x-values, i.e. the domain values, and the y-values, i.e. the image values, of the functions underlying the graphs. This is a well-known technique to render functions giving rise to this type of graphs more easily comparable.
In Fig. 1b, the graphs can be seen where we have taken the \(\log\) of the x-coordinates and also the \(\log\) of the y-coordinates of the graph representing the data, which is again the blue graph in Fig. 1b, of the graph representing the Bose–Einstein distribution model of these data, which is the red graph in Fig. 1b, and of the graph representing the Maxwell–Boltzmann distribution model of the data, which is the green graph in Fig. 1b. For readers acquainted with Zipf’s law as it appears in human language, they will recognize Zipf’s graph in the blue graph of Fig. 1b. It is indeed the \(\log /\log\) graph of ‘ranking’ versus ’numbers of appearances’ of the text of the Winnie the Pooh story ‘In Which Piglet Meets a Heffalump’, which is the ‘definition’ of Zipf’s graph. As to be expected, we see Zipf’s law being satisfied, the blue graph is well approximated by a straight line with negative gradient close to -1. We see that the Bose–Einstein graph still models very well this Zipf’s graph, and what is more, it also models the (small) deviation from Zipf’s graph of the straight line. Zipf’s law and the corresponding straight line when a \(\log /\log\) graph is drawn is an empirical law. Intrigued by the modeling of the Bose–Einstein statistics by the Zipf graph, we have analyzed this correspondence in detail in Sect. 4.
In the next section, however, we want to describe what a Bose gas is in physics, when it is brought nearby its state of Bose–Einstein condensate, with the aim of identifying the physical equivalent to the Winnie the Pooh story ‘In Which Piglet Meets a Heffalump’ and other pieces of texts which we will also consider.
3 The Bose–Einstein Condensate in Physics
We will explain in this section different aspects related to the experimental realization of a Bose gas near to it being a Bose–Einstein condensate where most of the bosons are in the lowest energy state. The awareness of the existence of this special state of a Bose gas came about as a consequence of a peculiar exchange between the Indian physicist Satyendra Nath Bose and Albert Einstein (Bose 1924; Einstein 1924, 1925). Bose actually devised a new way to derive Planck’s radiation law for light—which has the form of a Bose–Einstein statistics, hence, like we now know, being a consequence of the indistinguishability of the photon as a boson, but that was not known in these pre-quantum theory times—and sent the draft of his calculation to Einstein. Although what Bose did was far from being fully understood in that time, the new method of calculation must have caught right away the full attention of Einstein, because he translated the article from English to German and supported its publication in one of the most important scientific journals of that time (Bose 1924). Einstein himself then, inspired by Bose’s method, worked out a new model and calculation for an atomic gas consisting of bosons, and predicted the existence of what we now call a Bose–Einstein condensate, an amazing accomplishment, taken into account that the difference between bosons and fermions and the Pauli exclusion principle were not yet known (Einstein 1924, 1925). Because of the intense study of Bose–Einstein condensates that took off after their first experimental realizations (Anderson et al. 1995; Bradley et al. 1995; Davis et al. 1995), a lot of new knowledge, experimental, but also theoretical, has been obtained, material on which we built upon for some of the details of the present article (Ketterle and van Druten 1996; Parkins and Walls 1998; Dalfovo et al. 1999; Ketterle et al. 1999; Görlitz et al. 2001; Henn et al. 2008).
The principle idea is still the one foreseen by Einstein, namely to take a dilute gas of boson particles and then stepwise lower its temperature and as a consequence its total energy such that at a certain moment there is so little energy in the gas that all boson particles are forced to transition to the lowest energy state. At that moment, all boson particles are in the same state, namely this lowest energy state, and the gas behaves then in a way for which there is no classical equivalent—we will see that given our conceptuality interpretation of quantum theory and the boson gas model we built here for human language, we will be able to put forward a new way to view the indistinguishability that lies at the heart of a Bose–Einstein condensate (see Sect. 5).
The Bose–Einstein condensates that have been realized so far all consist mainly of massive boson particles, hence generally atoms with integer spins, which makes them bosons. Indeed, the situation of the bosons of light, i.e. of photons, is more complicated, because photons interact so abundantly with matter that their number is never constant, which makes it difficult to realize a thermal equilibrium in this case, albeit not impossible (Klaers et al. 2010a, b, 2011; Klaers and Weitz 2013). We do want to keep using our analogy of language with light, although of course the pieces of texts that we will study contain a fixed number of words, but a dynamic use of human language will also give rise to a continuous coming into existence of new words, which means that for such a dynamic situation the example of light is probably even more representative than gases with a fixed number of atoms. In this stage of our analysis, also because they are the more easy to realize Bose–Einstein condensates, we however focus on massive bosons, hence atoms with integer spins.
Energy and length scales of the sodium Bose–Einstein condensate
Energy scale E | \(\approx h^2/2{\mathrm{ml}}^2\) | Length scale l | \(\approx h / \sqrt{2mE}\) | |
---|---|---|---|---|
limiting temperature for s-wave scattering | \(\approx 1\, {{\mathrm{mK}}}\) | Scattering length | \(a \approx l / 2\pi\) | \(\approx 3\, {\mathrm{nm}}\) |
Bose–Einstein condensate transition temperature | \(\approx 2\, \upmu {\mathrm{K}}\) | Separation between atoms | \(n^{-{1 \over 3}} \approx l / \sqrt{\pi }(2.612)^{1 \over 3}\) | \(\approx 200\, {\mathrm{nm}}\) |
Temperature T | \(\approx 1\, \upmu {\mathrm{K}}\) | Thermal de broglie wave length | \(\lambda _{th} = l / \sqrt{\pi }\) | \(\approx 300\, {\mathrm{nm}}\) |
harmonic oscillator level spacing \(h\nu\) | \(\approx 0.5\, {{\mathrm{nK}}}\) | Oscillator length \(\nu = 10\, {\mathrm{Hz}}\) | \(a_{HO}= l / \sqrt{2}\pi\) | \(\approx 6.5\, \upmu {{\mathrm{m}}}\) |
A temperature of around \(1\, \upmu {\mathrm{K}}\) gives rise to a thermal de Broglie wavelength of around \(300\, {\mathrm{nm}}\).
The largest length scale is related to the confinement characterized by the size of the box potential or by the oscillator length \(a_{HO} = {1 \over 2\pi }\sqrt{h / m \nu }\), which is the typical size of the ground state wave function in a harmonic oscillator potential of frequency \(\nu\) (see “Appendix 2”). With \(\nu = 10\, {\mathrm{Hz}}\), we get a value for \(a_{HO}\) of about \(6.5\, \upmu {{\mathrm{m}}}\). The energy scale related to the confinement is characterized by the harmonic oscillator energy level spacing, given by \(h \nu\). Again, for \(\nu = 10\, {\mathrm{Hz}}\) we get an energy value for the spacing of about \(0.5\, {{\mathrm{nK}}}\).
Energy and length scales of rubidium Bose–Einstein condensate
Energy scale E | \(\approx h^2 / 2ml^2\) | Length scale l | \(\approx h / \sqrt{2mE}\) | |
---|---|---|---|---|
limiting temperature for s-wave scattering | \(\approx 0.1\, {{\mathrm{mK}}}\) | Scattering length | \(a=l / 2\pi\) | \(\approx 5\, {\mathrm{nm}}\) |
Bose–Einstein condensate transition temperature | \(\approx 170\, {{\mathrm{nK}}}\) | Separation between atoms | \(n^{-{1 \over 3}} \approx l / \sqrt{\pi }(2.612)^{1 \over 3}\) | \(\approx 300\, \mathrm{nm}\) |
Temperature T | \(\approx 50\, {\mathrm{nK}}\) | Thermal de broglie wave length | \(\lambda _{th} = l / \sqrt{\pi }\) | \(\approx 800\, \mathrm{nm}\) |
harmonic oscillator level spacing \(h\nu\) | \(\approx 1\, {\mathrm{nK}}\) | Oscillator length \(\nu \approx 10\, \mathrm{Hz}\) | \(a_{HO}= l / \sqrt{2}\pi\) | \(\approx 4\, \upmu {\mathrm{m}}\) |
We want to show now that our Bose–Einstein distribution model of the Winnie the Pooh story ‘In Which Piglet Meets a Heffalump’ is well modeled by a Bose gas close to the Bose–Einstein condensate of this gas, and will take the rubidium and sodium gases that we described in as inspiration. What is important to notice is the difference in order of magnitude between the energy level spacings of the harmonic trap oscillator, they are of the order of \(1\, {\mathrm{nK}}\), and the energies involved with the gas itself, of the order of \(1\, \upmu \mathrm{K}\). The Winnie the Pooh story ‘In Which Piglet Meets a Heffalump’ is not in a Bose–Einstein condensate state, because then all the words of the story should be the word And, populating the zero energy level. So, it is in a state which is close to a Bose–Einstein condensate.
The rubidium condensate is a better example for the Winnie the Pooh story, as also the number of atoms, 2000, is of the same order of magnitude as the number of words, 2655, of the Winnie the Pooh story. The energy levels of the trap for the rubidium condensate are of the order of \(1\, {\mathrm{nK}}\), while the temperature of the gas is \(170\, {\mathrm{nK}}\) (Table 3), which is 170 times bigger. We see for the Winnie the Pooh story that if we take 1 unit of energy for the energy level spacings, we have \(B = kT = 593\), following (15), and hence \({1 \over 2} kT\), being a good estimate for the average energy per atom of a one-dimensional gas, gives for the latter 271, which means that we are in this respect also in the same order of magnitude for the Winnie the Pooh story and the rubidium condensate. Hence, we can say that the Winnie the Pooh story can be looked at as behaving similarly to a Bose gas of rubidium 87 atoms in one-dimension at a temperature of \(170\, {\mathrm{nK}}\). We will see in Section 4, where we consider the text of the novel ‘Gulliver’s Travels’ of Jonathan Swift (Swift 1726), that the sodium condensate is a better example for this text.
Let us introduce a second piece of text in Table 4, namely a story entitled ‘The magic shop’ written by Herbert George Wells (Wells 1903), with which we want to illustrate an aspect of our ‘Bose gas representation of human language’ that we have not yet touched upon. For the Winnie the Pooh story, If we look at Fig. 2 and Table 1, we can see that the ‘energy spectrum’ does not cover the whole range of possible energy values. Indeed, the red graph of Fig. 2 on the right hand side of the graph has still a substantial value, and is not at all close to zero. Hence one can wonder what happens further on for the higher energy spectrum with this graph?
On the low energy spectrum, the amount of radiation increases starting from zero radiation for energy level \(E_0\), hence for the words that are captured in the zero energy level of the Bose–Einstein condensate, there is no radiation emerging from them following the considered choice of zero in the energy scale—for the case of the Winnie the Pooh story, the zero level energy state puts the cogniton in state And—and then the amount of radiation increases steeply—we have already a radiation of 111 energy units (and 105.84 in the Bose–Einstein model) for \(E_1\) for the Winnie the Pooh story and the cogniton in state He. The energy radiation keeps increasing steeply—182 for \(E_2\) (179.36 for the Bose–Einstein model) for the cogniton in state The, 255 for \(E_3\) (233.36 for the Bose–Einstein model) for the cogniton in state It, 280 for \(E_4\) (274.65 for the Bose–Einstein model) for the cogniton in state A, 345 for \(E_5\) (307.23 for the Bose–Einstein model) for the cogniton in state To, etc.—to reach a maximum at \(E_{71}\) with a radiation level of 522.79 energy units for the cogniton in state First. Then the radiation starts to decrease slowly. But, remark that at energy level \(E_{542}\), with the cogniton in state You’ve, which is the highest energy level of Table 1, we still have a radiation of 385.55 energy units, which is more than half of the maximum radiation reached at energy level \(E_{71}\) for the cogniton in state First.
How can we understand this, because we have in Table 1 exhausted all the words of the Winnie the Pooh story and hence seemingly represented all possible energy levels. But is this true? To see clear in this, we have to reflect about the difference of the numbers in the third and the fourth column of Table 1, respectively the ‘numbers of appearances’ of the specific words in the Winnie the Pooh story and the ‘values of the Bose–Einstein distribution that we used to model these numbers of appearances’. The values in the fourth column are of a probabilistic nature and express averages of stories ‘similar’ to the one of Winnie the Pooh with respect to the numbers of appearances of the specific words, while the values in the third column express real counts for one specific story. More concretely, by ‘similar’ we actually mean ‘containing the same total number of words, and containing the same total amount of energy’. Remember indeed that the Bose–Einstein distribution function only contains two parameters, which hence will be determined by the total number of words and the total amount of energy. Or to put it even more concretely, suppose we would collect a vast number of pieces of ‘meaningful’ texts all containing the same total number of words N and the same amount of total energy E, the Bose–Einstein distribution function (9) is then supposed to model a specific type of average that can be obtained for all these texts, and the more numerous these texts the better this average will correspond with the Bose–Einstein distribution function. The reason is that this function is the consequence of the limit process in statistical mechanics of a micro-canonical ensemble of states of particles with the same N and E (Bose 1924; Einstein 1924, 1925; Huang 1987).
The above reasoning indicates that we can consider to introduce a ‘place for words that do no appear in the considered text but could have appeared’. Remark that these new words do not add to the sum N of all words, since they have ‘number of appearance zero’, which means that this operation of ‘adding new words’ leaves N unchanged. In the ranking of energy levels, they have to be classified by ‘additional energy levels higher than the highest one we now identified with respect to the last alphabetically classified word that appears one time in the text’. Remark that also E remains unchanged by this adding of words that could have appeared. Indeed, although these new added words carry high energies, since all of them have appearance number zero, they do not add to the total amount of energy because the product of the energy of an even very high energy level with the zero of its number of appearances equals zero. Since N and E are left unchanged by the adding of these new words that could have appeared also the micro-canonical ensemble and its thermodynamical equilibrium remain unchanged. However the adding of the new words does alter substantially the Bose–Einstein distribution function and the Maxwell–Boltzmann distribution function calculated to model the data, because they both do not have appearance values equal to zero for these words, which means that there will be contributions to the total number of words and the total energy of their modeling. Hence, this operation of adding words such that the energy spectrum completes itself over the whole range is a necessary operation in the modeling with Bose–Einstein or Maxwell–Boltzmann.
Again more concretely, let us consider the words that appear one time in the Winnie the Pooh story, and look for synonyms of these words, then a word that appears now one time could not have appeared and instead its synonym could then have appeared. So, the synonyms can be listed in a new set of words to add with zero appearance, as ‘could have appeared’, and indeed, the Bose–Einstein distribution function will not be zero for them, which expresses exactly this ‘they could have appeared’.
An energy scale representation of the words of the story ‘The magic shop’ by H. G. Wells as published in Wells (1903)
Words concepts cognitions | Energy levels \(E_i\) | Appearance numbers \(N(E_i)\) | Bose–Einstein modeling | Maxwell–Boltzmann modeling | Energies from data \(E(E_i)\) | Energies Bose–Einstein | Energies Maxwell–Boltzmann |
---|---|---|---|---|---|---|---|
The | 0 | 202 | 201.4 | 18.84 | 0 | 0 | 0 |
And | 1 | 176 | 157.28 | 18.75 | 176 | 157.28 | 18.75 |
A | 2 | 125 | 128.99 | 18.66 | 250 | 257.97 | 37.33 |
I | 3 | 113 | 109.3 | 18.57 | 339 | 327.89 | 55.72 |
Of | 4 | 95 | 94.81 | 18.48 | 380 | 379.22 | 73.94 |
Was | 5 | 72 | 83.69 | 18.4 | 360 | 418.46 | 91.98 |
To | 6 | 71 | 74.9 | 18.31 | 426 | 449.41 | 109.85 |
He | 7 | 67 | 67.77 | 18.22 | 469 | 474.41 | 127.54 |
In | 8 | 67 | 61.87 | 18.13 | 536 | 495.00 | 145.06 |
It | 9 | 63 | 56.92 | 18.05 | 567 | 512.24 | 162.41 |
Said | 10 | 59 | 52.69 | 17.96 | 590 | 526.86 | 179.59 |
That | 11 | 51 | 49.04 | 17.87 | 561 | 539.42 | 196.61 |
Gip | 12 | 48 | 45.86 | 17.79 | 576 | 550.29 | 213.45 |
With | 13 | 45 | 43.06 | 17.7 | 585 | 559.80 | 230.13 |
His | 14 | 43 | 40.58 | 17.62 | 602 | 568.16 | 246.65 |
My | 15 | 36 | 38.37 | 17.53 | 540 | 575.58 | 263.00 |
You | 16 | 33 | 36.39 | 17.45 | 528 | 582.18 | 279.19 |
Had | 17 | 31 | 34.59 | 17.37 | 527 | 588.10 | 295.22 |
Shopman | 18 | 27 | 32.97 | 17.28 | 486 | 593.42 | 311.09 |
There | 19 | 27 | 31.49 | 17.2 | 513 | 598.22 | 326.80 |
As | 20 | 25 | 30.13 | 17.12 | 500 | 602.58 | 342.35 |
At | 21 | 25 | 28.88 | 17.04 | 525 | 606.54 | 357.74 |
Magic | 22 | 25 | 27.73 | 16.95 | 550 | 610.16 | 372.98 |
But | 23 | 24 | 26.67 | 16.87 | 552 | 613.46 | 388.07 |
Little | 24 | 23 | 25.69 | 16.79 | 552 | 616.49 | 403.00 |
One | 25 | 22 | 24.77 | 16.71 | 550 | 619.27 | 417.78 |
... | ... | ... | ... | ... | ... | ... | ... |
... | ... | ... | ... | ... | ... | ... | ... |
What | 65 | 9 | 10.04 | 13.79 | 585 | 652.41 | 896.44 |
Which | 66 | 9 | 9.89 | 13.73 | 594 | 652.47 | 905.87 |
Behind | 67 | 8 | 9.74 | 13.66 | 536 | 652.51 | 915.19 |
Boy | 68 | 8 | 9.6 | 13.59 | 544 | 652.54 | 924.40 |
Do | 69 | 8 | 9.46 | 13.53 | 552 | 652.55201 | 933.50 |
Door | 70 | 8 | 9.32 | 13.46 | 560 | 652.55204 | 942.50 |
Genuine | 71 | 8 | 9.19 | 13.4 | 568 | 652.54 | 951.38 |
Glass | 72 | 8 | 9.06 | 13.34 | 576 | 652.51 | 960.16 |
Hat | 73 | 8 | 8.94 | 13.27 | 584 | 652.48 | 968.83 |
Moment | 74 | 8 | 8.82 | 13.21 | 592 | 652.43 | 977.40 |
More | 75 | 8 | 8.7 | 13.14 | 600 | 652.37 | 985.87 |
... | ... | ... | ... | ... | ... | ... | ... |
... | ... | ... | ... | ... | ... | ... | ... |
Yard | 1149 | 1 | 0.25 | 0.08 | 1149 | 292.03 | 87.03 |
Yes | 1150 | 1 | 0.25 | 0.08 | 1150 | 291.78 | 86.68 |
You’d | 1151 | 1 | 0.25 | 0.08 | 1151 | 291.53 | 86.34 |
You’re | 1152 | 1 | 0.25 | 0.07 | 1152 | 291.28 | 86.01 |
Youngster | 1153 | 1 | 0.25 | 0.07 | 1153 | 291.02 | 85.67 |
Garden | 1154 | 0 | 0.25 | 0.07 | 0 | 290.77 | 85.33 |
Okay | 1155 | 0 | 0.25 | 0.07 | 0 | 290.52 | 85.00 |
Store | 1156 | 0 | 0.25 | 0.07 | 0 | 290.27 | 84.66 |
Meter | 1157 | 0 | 0.25 | 0.07 | 0 | 290.02 | 84.33 |
Junior | 1158 | 0 | 0.25 | 0.07 | 0 | 289.76 | 84.00 |
... | ... | ... | ... | ... | ... | ... | ... |
... | ... | ... | ... | ... | ... | ... | ... |
Continued | 3494 | 0 | 0.01 | 0 | 0 | ||
Adding | 3495 | 0 | 0.01 | 0 | 0 | 27.71 | 0.003 |
Mention | 3496 | 0 | 0.01 | 0 | 0 | 27.68 | 0.003 |
Similar | 3497 | 0 | 0.01 | 0 | 0 | 27.65 | 0.003 |
Criterion | 3498 | 0 | 0.01 | 0 | 0 | 27.61 | 0.003 |
Obviously | 3499 | 0 | 0.01 | 0 | 0 | 27.58 | 0.003 |
Appearing | 3500 | 0 | 0.01 | 0 | 0 | 27.55 | 0.003 |
Totalities | 3934 | 3934.00 | 3934.00 | 817415 | 817415.00 | 817414.18 |
The only criterion is that ‘they appear in a meaningful story with the same total number of words and the same total energy’. Hence, adding synonyms is a simple way to ensure that the whole story remains meaningful, but also a completely new meaningful part to the story can be added with words that are no synonyms’.
So, we added many more energy levels, namely till the cogniton being in energy level \(E_{3500}\). We have only shown the seven last ones of these words in Table 4, namely Continued, Adding, Mention, Similar, Criterion, Obviously and Appearing, having zero number of appearances in the H. G. Wells story, but their Bose–Einstein value in the Bose–Einstein model, as well as their Maxwell–Boltzmann value in the Maxwell–Boltzmann model, being not zero.
In Fig. 3a, b, we have represented, respectively, the numbers of the appearing and not appearing words with respect to the energy levels, a graph very steeply going down, and the \(\log /\log\) graphs of these numbers of appearances, where we take the logarithm of both y and x. In Fig. 4, we have represented the amounts of radiated energy with respect to the energy levels, and we see that this time the red graph representing the Bose–Einstein model of the data, after steeply going up and reaching a maximum, goes slowly down to touch closely the zero level of amount of energy radiated for high energy level cognitons. We see again, like in Fig. 1, that the Bose–Einstein distribution function, the red graph, gives an almost complete fit with the data, the blue graph, and gives definitely a much better fit than the Maxwell–Boltzmann distribution function, the green graph, does. Let us look more carefully to the amounts of energy graphs in Fig. 4. Also here we see that the red graph, which is the Bose–Einstein distribution, is a much better fit for the blue graph of the data, than the green graph, which is the Maxwell–Boltzmann distribution. We see that the maximum amount of radiation is reached at energy level \(E_{70}\) in the state of the cogniton characterized by Door and the amount is 652.55204 energy units. So the frequency of Door would be the dominant color with which the story ‘The magic shop’ shines.
4 Zipf’s Law and the Bose Gas of Human Language
Consulting Table 1, we can see that the biggest difference is at the zero point of the graph, where on the x-axis \(E_0 = 0\) and \(R_0 = 1\), hence between the product \(R_0 \times N_0\), which equals \((E_0+1) \times N_0\), that is between \(1 \times 133 = 133\) and \(E_0 \times N_0 = 0 \times 133 = 0\). This can not easily be seen as a difference between the graphs of Fig. 5 and the graphs of Fig. 2, since 133 is still little compared to the values the functions take at \(R_1\) and \(E_1\). Again consulting Table 1, we indeed see that \(R_1 \times N_1 = (E_1+1) \times N_1 = 2 \times 111 = 222\), while \(E_1 \times N_1 = 1 \times 111 = 111\). This means that both the ‘product graph’ of Fig. 5 and the ‘energy distribution graph’ of Fig. 2 go quickly up between \(R_0\) and \(R_1\) and between \(E_0\) and \(E_1\), the first from value 113 to value 222, and the second from value 0 to value 111, which is almost with the same steepness. Both graphs will then remain increasing quite quickly and then slowly flatten till they reach their maxima at Zipf rank \(R_{70}\) and energy level \(E_{71}\). Then, from this maximum on, both the Zipf product and the energy distribution slowly decrease from their maxima to a lower value. More specifically, the maximum value is 522.79 in both cases, and for the last considered Zipf rank \(R_{542}\) and energy level \(E_{542}\) we find values 359.22 and 358.55 respectively. This shows that there is a decreasing for the Zipf products and not constancy like Zipf’s law predicts.
In the foregoing reasoning on Zipf’s law, we have always considered the two graphs, the blue and the red one, in both Figs. 5 and 2. Of course, Zipf did not know of the Bose–Einstein distribution that is represented by the red graph in both figures, and which we used to model the data, represented by the blue graph in both figures. Hence Zipf only had the blue graph in Fig. 5 available to come up with the hypothesis that the product of rank and number of appearances is a constant. If one considers the blue graph in Fig. 5, one could indeed imagine it to vary around a constant function, certainly in the middle part of the graph. The beginning part can then be considered as a deviation, which is also what Zipf did when noting that in the first ranks the law did not hold up well. It was also known to Zipf that the end part of the graph, as a consequence of how ranks and numbers of appearances behave there, making the product go up and down heavily, did not behave very well with respect to his law either, and the slight downward slope all at the end was identified by Zipf as well. We see it explicitly pictured by the red graph, representing the Bose–Einstein distribution modeling of the data.
There is however another aspect of the situation which was overlooked by Zipf. It is self-evident that ‘if Zipf’s law is a law, it has to be a probabilistic law’. Let us specify what we mean by this. Suppose we had a large number of texts available with exactly the same number of different words in it, such that a Zipf analysis would lead to the same total number of ranks for each of the texts. Zipf’s graphs, including the ‘product graph’, i.e. the blue graph in Fig. 5, will then show a statistical pattern for the set of texts where it is tested on. Suppose we make averages for the numbers of appearances pertaining to the same rank over the available texts, then the function representing these averages of the numbers of appearances for the different texts will be a distribution function with a steep upward slope in the first ranks going towards a maximum and then a slow downwards slope in the ranks after this maximum. It will be a function similar to the Bose–Einstein distribution we have used to model texts as Bose gases, i.e. the red graph. This will be even more so when we add the two constraints that in our case follow naturally from our modeling, namely that the different texts need to count the same total number of words, and the sum of the products, which in our interpretation of the Bose gas model is the total energy, needs to be the same for each one of the texts. What is however more important still is that ‘if Zipf’s law is a probabilistic law, we should also introduce rankings that represent words with a zero number of appearances’, exactly like what we have done for the H. G. Wells story ‘The magic shop’, for which we have represented the data and the Bose–Einstein model in Table 4, and the graphs representing these data in Figs. 3a, in b and in 4.
The foregoing analysis is meant to provide evidence to the Bose–Einstein distribution being a better model for the Zipf data than a constant, or also still than later more complex versions of Zipf’s law along the lines of still believing that the product graph is in good approximation a constant, and the \(\log /\log\) version in good approximation a straight line. There is however another aspect of Zipf’s finding that we want to put forward here, since it will be important for our model of a Bose gas for human language.
In this we will also be inspired by the global foundational work we have done in our Brussels group (Aerts 1986, 1990, 1999, 2009b; Aerts et al. 2010, 2012, 2013a, 2018a, 2019a, 2011; Aerts and Gabora 2005a, b; Aerts et al. 2013b; Aerts and de Bianchi 2014, 2017; Aerts et al. 2016; Aerts and Sozzo 2011, 2014; Aerts et al. 2015a, 2016; Sassoli de Bianchi 2011, 2013, 2014, 2019; Sozzo 2014, 2015, 2017, 2019; Veloz et al. 2014; Veloz and Desjardins 2015), and by the more specific work on the ‘conceptuality interpretation’ (Aerts 2009a, 2010a, b, 2013, 2014; Aerts et al. 2018d, 2019c). To mention a concrete aspect in need of a more foundational approach, there is yet no well identified spatial domain for human language, which means that we will have to build a ‘quantum cognition’ without reference to space (Aerts 1999; Sassoli de Bianchi 2019).
The eleven lowest energy levels of the novel Gulliver’s Travels by Jonathan Swift (Swift 1726). The values of the Bose–Einstein model are compared with the data, i.e. the numbers of appearances of the words in the text in (a) without the introduction of a power coefficient and in (b) with the introduction of a power coefficient. The comparison for all energy levels can be seen for (a) in Fig. 7a and for (b) in Fig. 7b
Cogniton state | Energy level | Appearance number | Bose–Einstein value |
---|---|---|---|
(a) Gulliver’s Travels without power coefficient | |||
The | \(E_0 = 0\) | 5838 | 16,454.07 |
Of | \(E_1 = 1\) | 3791 | 6297.00 |
And | \(E_2 = 2\) | 3633 | 3893.39 |
To | \(E_3 = 3\) | 3400 | 2817.73 |
I | \(E_4 = 4\) | 2852 | 2207.73 |
A | \(E_5 = 5\) | 2442 | 1814.80 |
In | \(E_6 = 6\) | 1976 | 1540.59 |
My | \(E_7 = 7\) | 1593 | 1338.35 |
That | \(E_8 = 8\) | 1280 | 1183.03 |
Was | \(E_9 = 9\) | 1263 | 1060.00 |
Me | \(E_{10}=10\) | 991 | 960.14 |
(b) Gulliver’s Travels with power coefficient | |||
The | \(E_0 = 0\) | 5838 | 5305.75 |
Of | \(E_1 = 1\) | 3791 | 4164.08 |
And | \(E_2 = 2.11\) | 3633 | 3358.88 |
To | \(E_3 = 3.28\) | 3400 | 2795.26 |
I | \(E_4 = 4.47\) | 2852 | 2384.16 |
A | \(E_5 = 5.69\) | 2442 | 2073.04 |
In | \(E_6 = 6.92\) | 1976 | 1830.30 |
My | \(E_7 = 8.18\) | 1593 | 1636.12 |
That | \(E_8 = 9.45\) | 1280 | 1477.55 |
Was | \(E_9 = 10.73\) | 1263 | 1345.80 |
Me | \(E_{10}=12.02\) | 991 | 1234.70 |
For the lowest energy level, with cognitons in state The, we find the Bose–Einstein distribution to have a value of 16454.07 while The appears only 5838 times in the Gulliver’s Travels text. This is indeed a big difference, the Bose–Einstein is more than three times the experimental value of the number of appearances. We find a similar too high value for the Bose–Einstein distribution for the two next states of the cognitons, the state Of has a Bose–Einstein distribution value of 6297.00, while Of appears only 3791 in the text, the state And has a Bose–Einstein distribution value of 3893.39, while And appears only 3633 times in the text. For the next states of the cognitons the Bose–Einstein model, however, gives values too low with respect to the experimental data. For To the Bose–Einstein distribution value is 2817.73 while To appears 3400 times in the text, for I the Bose–Einstein distribution value is 2207.73 while I appears 2852 times, for A the Bose–Einstein distribution value is 1814.80 while it appears 2442 times, for In the Bose–Einstein distribution value is 1540.59 while it appears 1976 times, for My the Bose–Einstein distribution value is 1338.35 while it appears 1593 times, for That the Bose–Einstein distribution value is 1183.03 and it appears 1280 times, for Was the Bose–Einstein distribution value is 1060.00 and it appears 1263 times, and for Me the Bose–Einstein distribution value is 960.14 while Me appears 991 times in the text of the Gulliver’s Travels story.
We have tested the Bose–Einstein model on a large number of stories, short stories and long stories of the size of novels, and when we allow the energy spacings between different energy levels to vary according to a power law, we have been able to construct a perfectly matching Bose–Einstein model for the data for all of the considered stories. The power that was each time needed was situated between 0.75 and 1.25.
We want to emphasize that it is remarkable how the application of the power 1.08 to the linear version of the text of the novel of Gulliver’s Travels makes the Bose–Einstein model fit so well the data, and we observed the same effect of the introduction of a power on an original linear version of the model for many of the other example texts that we investigated. We mentioned already how those who studied Zipf’s law came to add a power to take into account that the gradient of the best fitting straight line in the \(\log /\log\) version of the graphs was not equal to \(-1\). However, also the concave slightly curbed nature of the lowest energy level ranks was noticed and tried to be remedied by making the law more general still, however in purely ad hoc ways with the only aim to fit the data (Mandelbroth 1953; Mandelbrot 1954; Edmundson 1972). That this slight concave curb appears in the Bose–Einstein distribution as a consequence of adding a power to the spacings between energy levels in exactly a way to make it fit with the data is in this sense remarkable, and since we saw it happening in many of the other examples for different values of the power, it is a strong indication of the Bose–Einstein model touching onto a fundamental property of human language.
5 Identity and Indistinguishability
We want to reflect now on what can the obtained results teach us about the notions of ‘identity and indistinguishability’ with respect to how they are used in human language and in quantum theory. We also want to reflect on the way in which these results support the ‘conceptuality interpretation of quantum theory’ (Aerts 2009a, 2010a, 2013, 2014; Aerts et al. 2018d, 2019c). Before we start our analysis, we repeat that all the words appearing in the stories that we considered are ‘states’ of the ‘cogniton’, which is the entity that for human language is what a ‘photon’ is for light, or what a ‘rubidium 87 atom’ is for the rubidium gas used to fabricate the Bose–Einstein condensate (Anderson et al. 1995).
Let us first analyze how the issue of ‘identity and indistinguishability’ appears in quantum theory. It is structurally speaking a consequence of the generally adopted mathematical rule that wave functions should be symmetrized or anti-symmetrized, depending of whether the quantum particles in question are bosons or fermions. This entails that a multi-particle wave functions is always a superposition of products of the single particle building blocks of the multi-particle wave function, such that the different product pieces are chosen in a way that the total wave function is symmetric or anti-symmetric, depending on whether the composed quantum entity is a boson or a fermion. Let us make concrete what this means when we apply a quantum model to the text of the Winnie the Pooh story. The set of energy levels \(\{E_0, \ldots , E_{542}\}\) shown in Table 1 are in principle the energy levels for a one particle situation in quantum theory, and the many particle situation of a text is then described in a Hilbert space which is the tensor product of, in the case of the Winnie the Pooh story, 2655 Hilbert spaces of which each one describes a one particle situation. The symmetrization is obtained by a superposition of all possible permutations of the original products and a renormalization to make the wave function a unit vector.
Such a symmetrization for bosons and anti-symmetrization for fermions, following quantum theory, exists for all bosons and all fermions, which literally means that all identical quantum particles are entangled in this strong way, giving rise to non-local correlations of the EPR type. This state of affairs is still nowadays a serious unsolved and not understood conundrum for theoretical physics and philosophy of physics (Black 1952; Van Fraassen 1984; French and Redhead 1988; Saunders 2003, 2006; Muller and Seevinck 2009; Krause 2010; Dieks and Lubberdink 2011, 2019), and this stands in great contrast with how experimentalists go along with it, for example, photons pertaining to different energy levels, hence carrying different frequencies, are treated by them as distinguishable (Hong et al. 1987; Knill et al. 2001; Zhao et al. 2014). The way in which experimentalists look at the ‘indistinguishability’ of photons was expressed clearly in more recent times, because of the actual importance of the creation of entangled photons for different reasons, e.g. for the fabrication of optically based quantum computers, and hence the focus in quantum optics on how to achieve this. Spontaneous parametric down conversion, which is a nonlinear optical process that converts one photon of higher energy into a pair of photons of lower energy has been historically the process for the generation of entangled photon pairs for the well-known Bell’s inequality tests (Aspect et al. 1982; Weihs et al. 1998). Parametric down conversion is however an inefficient process because it has a low probability and hence physicists looked for other ways to produce entangled photons. Hence, when a scheme for using linear optics in function of the needs of the production of qubits was presented (Knill et al. 2001), this made arise an abundance of new research. Most of the applications of this new research rely on the two-photon interference effect with two ‘indistinguishable photons’ entering from different sides of a beam splitter and leaving in the same direction after undergoing the so called Hong-Ou-Mandel interference effect (Hong et al. 1987). The crucial aspect of Hong-Ou-Mandel interference is the ‘indistinguishability of the two photons in the spectral, temporal and polarization degrees of freedom’.
Let us formulate the reason why it makes sense to state our view as just expressed above given the conceptuality interpretation of quantum theory. The main hypothesis of the latter is that ‘the role played by the human mind in relation with language is the same as the role played by a measuring apparatus (but also a heat bath and also a context that is perhaps not willingly used by a human being to make a measurement) in relation with a collection of quantum entities’. The statement above in italics follows directly from this hypothesis.The way in which we understand in a straightforward way ‘what identity and indistinguishability are with respect to human language and human mind’ teaches us ‘what identity and indistinguishability are in quantum theory’.
Let us become more concrete and consider the text of the Winnie the Pooh story of which the words can be found in Table 1. We see that—and the reasoning we develop now can be made for any other of the considered words—the word Piglet corresponds to the cogniton being with energy \(E_8\), and it appears 47 times in the text of the story. In the quantum wave function that represents the story, which is a multipartite wave function formed by 2655 parts (the total number of words), Piglet is the state associated with 47 of its parts, or components. It is straightforward that each of the Piglet in each of the components can be interchanged with each other of the Piglet in each other of the components without the story being changed even in the slightest way. This means, in physics jargon, that the wave function is symmetric (or anti-symmetric) with respect to the interchange of all these Piglet components. And, the symmetry (or anti-symmetry) is a consequence of their ‘absolute indistinguishability’. It is also easy to understand that this ‘absolute indistinguishability’ is due to Piglet being a concept, and not an object. Indeed, let us imagine for a moment, just to make the above more clear still, that the scenery of the story would be pictured in some physical theatrical form with real piglets on the places where now the concept Piglet appears in the text. If we interchanged these real piglets, of course this would influence the physical scenery of the story. It is indeed not possible to ‘interchange a real physical piglet with another real physical piglet without changing the whole of the physical scenery’. That is why real piglets when put in baskets will follow a Maxwell–Boltzmann statistics and not a Bose–Einstein statistics as conceptual piglets do. The ‘interchanging of concepts in a piece of text’, hence in the components of the wave function representing this piece of text, is an intrinsically different operation than the ‘interchange of objects in space’, and the basic hypothesis of the conceptuality interpretation of quantum theory consists in believing that quantum particles are like concepts, and that the reason why we find their behavior not understandable is because we think of them as objects. One of the crucial difficulties when thinking of quantum particles as objects comes to the surface exactly in their behavior as indistinguishable entities, as for objects this is something impossible to understand, while for concepts it is something straightforward and natural.
Let us show now how we can also easily understand the difference we indicated above between theoretical physicists who are struggling with the issue that, following quantum theory, all photons should be identical, in contrast with experimental physicists who pragmatically consider photons of different frequency as distinguishable and hence not identical. Consider again the Winnie the Pooh story, although we all understand right away that all concepts in the Piglet state are ‘absolutely indistinguishable’, we also are convinced that two different energy states of the cogniton are distinguishable. For example, energy state \(E_{43}\), which is the concept Robin, appearing 12 times in the text, is distinguishable from, Piglet. It is even very important for the meaning carried by the story that these two states are distinguishable. In a very similar way, for any measuring apparatus that is sensitive to the frequency of light, it is very important that a red photon is distinguishable from a blue photon, e.g. for our eyes, but also, we suppose, for plants practicing photosynthesis. It is even the ‘essence of the measuring apparatus’ to ‘distinguish these two states’. However, when a special purpose apparatus is fabricated that, when we would read the Winnie the Pooh story, the points where Piglet appears are made not distinguishable any longer with the points where Robin appears—and there is a multitude of ways we can imagine this to be done—the two cognitons that are still read by us, will be indistinguishable. Again, such an operation consisting of completely erasing the Piglet nature and Robin nature of both concepts, can only work ‘because both are concepts and not objects’. Underneath all of the words of the Winnie the Pooh text is indeed the more abstract notion of Concept, and hence we can bring all words into this abstract state of just being an unspecified concept in the text, which would make all of them indistinguishable. There are different ways of ‘erasing’, some ways more close to the ontology of the concepts, other more close to the measuring itself, and that is also why the quantum eraser effect can be understood very well within the conceptuality interpretation (see Aerts 2009a Section 4.4).
In Fig. 9 we have represented two particles, the balls, in two states, the boxes, and three different configurations of this situation. The first configuration consists of the two particles in the first state, the second configuration of the two particles in the second state, and the third configuration consists of one particle in one state and the other particle in the other state. If the two particles are indistinguishable in the way that customarily is looked upon quantum indistinguishability, which is also the reason that this example is often displayed, the probabilities that are attached within a Bose–Einstein statistics model are 1/3, 1/3 and 1/3 for each of the configurations. However, if the the two particles are indistinguishable classically, the probabilities that are attached within a Maxwell–Boltzmann statistics are 1/4, 1/4 and 1/2. The reason is that the last configuration of one particle in one state and the other particle in the other state is realized in two ways classically, one way, and its permuted way are different realities. Within the ‘quantum indistinguishability’ these two are not different realities, and given our conceptuality interpretation this would be explained by them indeed not being different realities if they are concepts. What however in case we consider the three configurations of Fig. 9 for distinguishable states of the cogniton, hence for distinguishable concepts? To make things more concrete, suppose we consider the concepts Cat and Dog and the configurations Two Cats, Two Dogs and A Cat And A Dog. Let us remark that this is exactly the situation we have studied already in great detail showing Bose–Einstein statistics to be a better representation as compared to Maxwell–Boltzmann statistics (Aerts 2009a; Aerts et al. 2015b; Beltran 2019). How can we understand that even for distinguishable concepts Bose–Einstein is a better statistics than Maxwell–Boltzmann? The reason is the presence of ‘entanglement’ and ‘superposition’ also for distinguishable concepts like Cat and Dog. Indeed, the probabilities 1/3, 1/3, 1/3 with Bose–Einstein, versus 1/4, 1/4, 1/2 with Maxwell–Boltzmann, actually mean that for Maxwell–Boltzmann there are much more microstates in the third configuration than there are in the first two configurations, actually the double amount. When there is no entanglement and no superposition, and hence Cat and Dog are ‘separated’, we can understand this. This ‘is’ what happens when Cat and Dog are objects, hence a real cat and a real dog. Let us make this concrete, suppose we visit a farm with a lot of cats and dogs living at the farm, equal in number, and we receive as a present two of them randomly chosen for us by the farmer, then we will have the double chance that the gift will be a cat and a dog as compared to the gift being two cats or two dogs. What however if we ask a child to which it is promised that he or she can have two pets and he or she can choose for each pet whether it is a cat or a dog. The microstates that come into play in this case exist in the conceptual realm of the child’s conceptual world, and there is no reason that within this conceptual world there will be a double amount of microstates for the choice of a cat and a dog as compared to the choices for two cats or two dogs. If there are two children that each apart choose one pet and do this independently of each other Maxwell–Boltzmann statistics will be the better one again, because the amount of microstates of the combination of the two choices will be the double of the amount of microstates playing a role for each child apart. This situation was investigated by us in many different and more complex configurations of this type with the result of Bose–Einstein being a better statistics than Maxwell–Boltzmann to model the situation (Aerts 2009a; Aerts et al. 2015b; Beltran 2019). Actually, we noticed already in our study of quantum entanglement with concept combinations that the violation of Bell’s inequalities comes about due to the combined exemplars (microstates) being exemplars of the combined concept directly (giving rise to the Bose–Einstein situation) and not being exemplars of the concepts apart that then afterwards are combined (giving rise to the Maxwell–Boltzmann situation) (Aerts and Sozzo 2011, 2014; Aerts et al. 2019b, a). In our investigation of the quantum superposition with concept combinations the situation is even more Bose–Einstein, because the exemplars of the combined concepts that play a role (microstates) are no longer combinations of exemplars of the single concepts, which means that their amount in average will be equal to the amount of exemplars of the single concepts, the situation hence fulfilling the basic requirement to be modeled by Bose–Einstein statistics (Aerts and Gabora 2005b; Aerts et al. 2010, 2012; Aerts 2011; Sozzo 2014; Aerts et al. 2015a; Sozzo 2015; Aerts et al. 2017). The insight that also combined distinguishable concepts tend to give rise to Bose–Einstein rather than Maxwell–Boltzmann statistics explains why it is so important for the thermal de Broglie wave-lengths to be large with respect to the distance between the quantum particles, the equivalent for human language always being fulfilled, for the Bose–Einstein statistics to be applicable and why the original Rayleigh Jeans radiation law for light, which is the Maxwell–Boltzmann version of the Planck radiation law, is satisfied for low frequencies.
We have not yet reflected about ‘identity’ in itself. With respect to ‘the identity’ of a quantum particle, it can be proven that when the wave function of two identical quantum particles is considered, there does not exist a self-adjoint operator in the Hilbert space of their states that can represent a measurement that would identify one of the quantum particles (French and Redhead 1988; Butterfield 1993). Can a concept be said to have an identity? Not in the way we understand identity for an object. What can be attributed to a concept is a ‘number’ indicating ‘the number of times it is’, and that, one could say, is what can be seen as substituting what identity is for an object. The fact that also a ‘number of times it is’ can be attributed to a quantum particle is again a support for the hypothesis of our conceptuality interpretation.
Taking into account our above analysis, what we can understand about the nature of reality goes further than what we have formulated till now, in case we interpret quantum theory following the conceptuality interpretation. Like we mentioned already, we showed in earlier work that ‘combinations of concepts’ give rise to quantum superposition (Aerts et al. 2015a). Every sentence in a text is a combination of concepts. Also every paragraph in a text is a combination of concepts, since sentences, as combinations of concepts, combine amongst each others to form paragraphs. Depending on the nature of the text, this process, of increasingly larger pieces of the text being essentially ‘combinations of concepts’, keeps going on, certainly up to the level of stories, where the overall meaning content of a story glues all its concepts together in specific combinations. This implies that superpositions will also form for large subsets of combined concepts, and we believe that this is exactly the mechanism which we call ‘understanding’ when the human mind is engaging in these pieces of text. More concretely, suppose the human mind reads a piece of text. When reading, there is no direct focus on single words as a collection, on the contrary, when the words are read, a ‘new state is being formed’, which integrates ‘the meaning carried by the combination of all the concerned concepts’. This new state carrying the meaning of the piece of text formed by the combination of these words is exactly the superposition state which we identified already in earlier work (Aerts et al. 2015a), and it are these superposition states that form again and again by combining concepts of sentences or paragraphs that again superpose in the course of the reading of the whole text, and lead to the understanding of the whole piece of text. A similar process takes place when talking, thinking or writing, albeit in general in a more discontinuous and complex way than when reading. We believe that what happens with a physical Bose gas close to its Bose–Einstein condensate state can be understood similarly. The role played by the human mind with respect to the text is now played by the heat bath and the measuring apparatuses applied with respect to the Bose gas. When the temperature is low enough and the diluteness of the gas is such that the phase space density (25) satisfies (26), hence the thermal de Broglie wave length (17) is larger than the distance between the atoms, this process of superposition formation starts to happen. Indeed, the de Broglie waves of the different atoms will overlap heavily and give rise to these superpositions, which means that the process which we call ‘understanding’ when the human mind and text are involved takes place in the Bose gas with the heat bath. These superpositions are new emergent states that do not pertain to one of the atoms any longer, but represent several atoms joining in a new entity, just like the several combined concepts represent an emergent meaning. The more the temperature is lowered and the density of the gas is kept such that the de Broglie waves overlap on larger and larger regions of the gas, the more new states are formed containing a synthetic material reality different from single atoms. The Bose–Einstein condensate is an ultimate state where all the atoms have been gathered in the lowest energy state so that for the whole gas a single new state has emerged. The stories that we have studied are in states close to this Bose–Einstein condensate state, where synthetic parts of combined concepts emerge in superposition states and the sizes of these parts are determined by the state of understanding of the human mind of the stories.
Footnotes
- 1.
We are happy, although it is of course a coincidence, that it is also the ‘first’ story we analyzed and also use in this article.
Notes
Acknowledgements
This work was supported by QUARTZ (Quantum Information Access and Retrieval Theory), the Marie Sklodowska-Curie Innovative Training Network 721321 of the European Union’s Horizon 2020 research and innovation program. We thanks Massimiliano Sassoli de Bianchi, Sandro Sozzo and Tomas Veloz for their comments to a first version of the present article which helped improve and fine tune the present version.
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