Number

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Number

Description

A number N is a count N_X [x] of elementary entity X divided by the unit-entity U_X [x]. X must represent the same entity in both occurences. The counting-unit [x] cancels by division, such that numbers (for example, numbers 8 or 24) are abstracted from the counted entity (we write 8 and 24, although 8 x·x^-1 and 24 x·x^-1 would be equally correct; distinguished from a count of 8 x or 24 x if we count an entity-type X=apple). It is difficult to separate the concept of 'number' from the realization of number words or number symbols. The number symbols are called numerals; a numeral is the figure of a number, with different notation types used as a figure (VIII and 8 for Roman and Arabic numerals; 八 and 捌 for practical and financial Chinese). Consider the symbol 9 written into MitoPedia as elementary entity X=9. Then counting "9 9 9 9 9 9 9 9" yields a count N₉ = 8 x, and the count N₉ [x] divided by the unit-entity U₉ [x] yields the number N = 8, using the figure eight as the numberal in Arabic notation type. The human number concept has not only quantitative cardinal meaning related to the count (8 or 24 elementary entities), but is applied in expressing the ordinal rank of objects or events arranged in a sequence (in the Fibonacci-sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, .. the 8^th number is 13, whereas in an older representation of the Fibonacci-sequence 1, 1, 2, 3, 5, 8, 13, 21, .. the 5th number is 5; the 24^th day of a month), and in nominal labelling (drawing lot #24; serial number #8.007; DOI number doi10.26124bec2020-0001). Counting numbers (1, 2, 3, 4, 5, 6, 7, 8, ..) are unified multiplicities required for cardinal counting or ordinal nomination of the endpoint in a sequence. It is debatable, if one can have a zero count; a no-object, or an object that is not there to be counted. If this possibility is not denied, then counting numbers are equivalent to natural or whole numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, ..). Numbers are represented by numerals as words, iconic symbols, or entirely abstract symbols. The word 'snake', the numeral 'eight', the symbol '8' written in ink on a piece of white paper are as different from the real "object snake", as they differ from the "concept ////////", or "concept §§§§§§§§", or "concept 88888888", or "concept ∞∞∞∞∞∞∞∞", or "concept 'number eight'". We are so deeply used to these symbols, that we easily take the iconic or abstract symbol 8 — that represents the number eight — as the number eight itself, without a need to give the symbol 8 an interpretation and ask for its meaning.The numeral has to be distinguished from it's interpretation as the number that the numeral represents.

Abbreviation: N

Communicated by Gnaiger Erich 2020-06-30

Formats and meanings of numbers

It is instructive to compare some terms related to count and number, to units and first numbers in sequence in different languages, illustrating the depth of interpreration and meaning residing in etymology. Terms in English are mainly based on Latin, compared with German which includes more terms with their roots in Northern languages (Table 1). If you are multilingual, try this with other languages. Observe how predominant are the roots count and Zahl, unit and eins. These terms are highly differentiated. This makes it even more surprising or dubious, why the formalized language of the International System of Units (SI) lacks a term for the 'unit-entity' U_X.

Table 1. Count, unit, number, and number one: knowledge and meaning within words of different languages

Term	English	German	Comment
count NX	count (account)	Anzahl (Konto)
generating a count	counting (paying)	zählen (zahlen)
narrating a count	telling, recounting	erzählen	Telling a tale and Zahl have a common Proto-Indo-European root (tala-zahla).
unit	unit	Einheit	Single undivided whole; a member of a group; a unit quantity as a standard of measure.
unity	unity, uniqueness, oneness	Einheit, Einigkeit, Einzigartigkeit	Being or appearing as one (Latin unus, one).
union	union	Vereinigung	Being one; international union .., IUPAC (Latin unus, one).
unified	united	vereinigt	Assembled in one (Latin unus, one).
unitary	unitary	einheitlich
number N	number	Zahl	antonym: letter / Buchstabe
numeral X	numeral, cipher, digit, notation, form, entity-type	Ziffer	(Arabic sifr, from India for zero), chiffre
interpreting a numeral	decipher	entziffern, dechiffrieren	(from French)
number series	one, 1; two, 2; three, 3	eins, 1; zwei, 2; drei, 3	(Proto-Indo-European origin)
serial numbering	first (primary), 1st; second (secondary), 2nd; third, 3rd; fourth, 4th	erste, zweite, dritte, vierte	One / eins is not used for the 'first' / 'erste' (antonym last / letzte); two is not used for the 'second' / but zwei is used for 'zweite'; further up, three / drei is used for "third" / "dritte"; etc.
the feeling of being alone	loneliness	Einsamkeit	antonym: Gemeinsamkeit
single	single, separate	einzeln, einzig, einfach	antonym: double /doppelt (Latin singulus, one, consisting of one unit)
singular	singular	Einzahl	antonym: plural / Mehrzahl (Latin singulus, one, consisting of one unit)
single component	detail	Einzelheit
uniqueness	singularity, uniqueness	Einmaligkeit

Figure 1. Formats and meanings of numbers.

1. Counting and notation types: (1.1) dyce, (1.2) Roman numerals, (1.3) Mandarin-Chinese signs, (1.4) Arabic numerals, (1.5) English words. The dyce format requires hardly any interpretation, since the signal for counting is given in a series of linear expansion; this works well up to :::, but does not work for 66 or 666. Similarly, Roman and Mandarin symbols from I to III do not need interpretation due to the signal for counting, but IV to VI is more complex in Roman and Mandarin notations by compression required for extension towards higher numbers. Interpretation of Arabic numerals and English words needs learning from beginning with 1 and one, since the formal relation to counting is abandoned in favor of reduction; these investments pay off in the long run — once Arabic numerals have been learned, these symbols can be recognized and distinguished most rapidly, be written most economically, and be extended to high numbers by combination in the decimal number system. English words are much less economical in writing, but they connect isomorphically the image of the written number-word with the acoustic form of the spoken number-word.
2. Cardinal counting and ordinal ranking of dice: There are 15 dice in the figure. Dice 1 to 5 are in row 1; dice 6 to 10 are in row 2; dice 11 to 15 are in row 3.
3. Nominal labelling: Dice of tye (1) are with single notation and positioned on the marging of the figure; (2) dice with single notation and positioned in the center of the figure; (3) dice with multiple notations and positioned on the margin of the figure; (4) dice with multiple notations and positioned in the center of the figure.
4. Number magnitude and space: Dice with different notation types have an increasing magnitude from left to right. This spacial association is less pronounce for Mandarin notation type.
5. Sex of numbers and numerical parity: Even numbers such as 6 are associated with female sex, and are likened more than odd numbers such as 3 which connotate masculinity (Wilkie 2015 Front Psychol). Even odd and even numbers are gendered. Isn't it odd to be the odd man?

Dice have figures 1 to 6 on their six sides (Figure 1). These figures can be expanded backwards (downwards?) from 6 to 1 to 0. One way to get a zero with a die is to (1) push it off the table. Other ways are to (2) scrape off any dot on any side of the die, (3) cover any dot on any side of the die with an untransparent square, (4) put the untransparent square on the table and ignore the die (Table 2, 1. row).

Emptiness and zero are a non-matter of Zen, and that's where the concept of the number 0 comes from. We cannot count a dot that is not there. Therefore, zero cannot be a count. In addition, zero does not count: N_X + 0 = N_X. But zero devours a count in multiplication: N_X·0 = 0. And zero undetermines a count in division: N_X/0 = ∞.

This presentation given above in practical form needs further consideration in canonical form.

1. "Zero cannot be a count" has to be generalized: A number is not a count. A number cannot be added to a count, as much as a number cannot be added to a mass, or a mass cannot be added to a volume. If two things are to be added, the two occurrences have to be expressed in the same quantity and units. Addition requires an identity of quantity and units. Back to the X-mass party: m_X + 4 is as meaningless as m_X + 0. Zero is neither a mass nor a count, it is a number. A mass of 4 kg of peares can be added to a defined mass of apples, both occurrences expressed in an identical unit [kg]. But neither 4 nor 0 can be added to a defined mass of apples, or to a defined number of apples. The numbers 4 and 0 are unitless. The mass of X is expressed in the SI unit [kg]. The "number of X" is a count expressed in the canonical unit [x]. A number of 4 apples, N_a = 4 x, can be added to a number of 5 pears, N_p = 5 x; then the total number of pieces of fruit in the Assembly A is N_A = 4 x + 5 x = 9 x. If you

Thus N_X + 0 is as meaningless as N_X + 4.

Table 2. Symbols for numbers — numerals — notations.

The header of the table shows underlined numerals indicating the ordinal rank of each column. The index column on the left shows italic numerals as nominal labels as an index for the format of each row. Each column within the table represents a number in different notations. Counting all symbols within the table yields a cardinal number of 42 number-symbols, or 56 number-symbols when the header and index column are included. Each number is represented by "::: VI 六 6 six sechs" symbols, which illustrates that a symbol cannot be a number. "::: VI 六 6 six sechs" can be seen as a single counting-symbol (it is composed of six symbol-components) representing the number 6, or as six different symbols, each representing the number 6 in a different format:

1. Symbols for 1 to 6 on dice showing figures of dots as cardinal counting-numerals. The numeral 0 does not exist on a numerical system of dice, it is a no-dot or even a no-die, since no die has seven sides. The dots yield iconic symbols or figures as counting-numerals, such that the count of dots yields the explicit code for interpretation of each symbol. It is suggestive to deduce, that dice deliver an intuitive understanding of the numbers 1 to 6, but this is wrong, since the dots on dice yield the experience of counts from 1 dot to 6 dots. How do we get from a number of dots to a number?
2. Roman numerals have the same problem with the number 0 as a die. Lines are shown for counting-numerals as iconic symbols I, II, and III, comparable to the dots on a die. IV shows a line as a negative integer on the left of V. This is part of the code to decipher Roman numerals, but the code is more complex. For instance, X can be seen as two times V (V + Λ).
3. The Chinese Mandarin includes zero as a number, and shows counting-numerals as lines comparable to the Roman iconic counting-numerals I, II, and III. Higher numbers are represented as abstract symbols comparable to Arabic numerals, or as symbols that may be seen as words.
4. The Arabic number system adopted the zero from Asian Indian mathematicians. Cipher (German Ziffer) is rooted in the Arabic sifr, which stems from India for zero. The Arabic numeral 1 can be seen as an iconic symbol or counting-numeral comparable to the Roman I and Chinese —.
5. English words.
6. German words.

Nominal numbers 1 to 6 in circels label six categories of cells in the table:

(0) The symbol for number 0 is outside of this system and does not count.

(1) The symbol for number 1 that is positioned in rank 1 and the 1. line.

(2) The symbol for number 2 that is positioned in rank 1 and the 1. line.

Does a number make sense?

Something heavy makes sense. We feel the weight on the basis of interpreting sensory signals. Mass does not make sense. Even if we are familiar with the concept of mass, we tend to say 'weight' for some kg of potatoes, and tend to forget what we learned about weight as a force in contrast to the base quantity mass. Even if we tend to forget similarly what we learned about irrational or imaginary numbers, "numerousness, like shape, size and color, is a basic property of our perceptual world" (Agrillo 2020 PLOS ONE). Numerousness is so basic, that we perhaps never had to learn to think deeply about the sense of numbers — not in the sense of numerology, but in the sense of a sensory and neuronal processing system for numerosity. The predecessors of numerical cognition in animals and human infants rely on finite and iconic representations that are limited to cardinality and do not support a unified concept of number. Extraction of numerical information from optical images and acoustic frequencies may not depend initially on actual counting, but represents the pillars in the evolutionary framework of counting. Thus numbers are like counts in the looking glass or events in the echo. It makes sense to think that the object or event is the cause for the mirror image or echo. But if the mirror image looks at me, how do I resist identifying me with the image? If a number reflects a count, how do I avoid confusion between count and number? If a looking glass may cause confusion, then numbers and counts are entangled on multiple angels. The SI gives the count a unit of a number: The unit of a count is devoured by the number one, the number reflects the count, the entity is number one - the looking glass is substituted by the magic NUCE-kaleidoscope.

We learn early about odd and even numbers: Numerical parity becomes an integral part of number representation from about the 4^th grade onward (Wilkie 2015 Front Psychol). But structure encoding of faces occurs between 4 and 6 months of age (Farzin 2012 J Vision). A first step is to discern an odd and even number of objects nested in a single face. Visual object categorization and semantic identification require Counting-Associating-Sorting Experience (CASE). In Figure 3 there are four objects (Counting). But CASE makes us seeing immediately two eyes as a single pair (Counting and Associating) and identify one mouth and the pair of eyes as components of a face (Associating and Sorting), very close to an automatic reflex (requiring little Experience). CASE makes us look into a single unit 'face', without ambiguitiy of seeing two eyes as two faces (Figure 4). This would be different, if our species is born and lives with a single eye patch mask (Figure 5). Assembling in CASE of two nose holes occurs in parallel with the pair of eyes (Figure 6), and a deeper level of sorting, when the nose is recognized not only as a pair of holes but as an assembled object (Figure 7). As in Figure 3, the nose with two nose holes is recognized as a single assembly rather than seeing three objects within the face (Figure 3). Counting, assembling and sorting ears and eye brows and hair get more complex with differentiation, and with taking a face as a CASE of a person (Figure 8).

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  Figure 3. H(1M,1A[2E]): Unit face as the Highest-level System H. The four objects are not seen as 4 (1 vertical oval + 1 mouth + 1 eye + 1 eye), but are a CASE: accounted for as 1M + 1A(2E) (Counting and Assembling: 2 eyes = 1 Assembled pair of eyes), and associated with a single face {Stacking: H(1M,1A[2E])}. A new face is sorted further as part of a new H=person: H(1F[1M,1A{2E}]). The Counting-Assembling-Stacking Experience (CASE) goes on.

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  Figure 4. Two faces: one eye — one face. The three objects seen in the face are not interpreted as 1 + 2. Based on CASE, the part of the face seen is recognized as a single Highest-level System H with two objects, Mouth and Nose, and one assembly A(2E) that is completed by the invisible second eye — H(1M,1N,1A[2E]).

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  Figure 5. One face: When meeting with a familiar face that lost the sense for the CASE, just ware a single eye patch mask to avoid confusing your partner and justify unambiguously that you are one CASE — just one.

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  Figure 6. H(1M,2A[2E,2H]): Unit face H with one object M, and two assemblies A — A pair of eyes and A pair of nose holes 2A(2E,2H), not seen as two noses.

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  Figure 7. H(1M,2A[2E,1N{2H}]): Unit face H with one object M, and two assemblies A — A pair of eyes A(2E), and A nose with a sorted pair of nose holes N(2H).

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  Figure 8. Unit face with many components: the counting of components and nesting in assemblies increases in complexity as a function of differentiation.

A taxonomy of numbers

Class	Example	Comment
counting number, natural number	1, 2, 3, 4, ..	Positive integers. All natural numbers are whole numbers.
whole number	0, 1, 2, 3, 4, ..	Natural numbers including zero. All whole numbers are integers.
negative integer	-1, -2, -3, -4, ..	Negative integers are integers excluding whole numbers.
integer	.., -4, -3, -2, -1, 0, 1, 2, 3, 4, ..	Negative integers and whole numbers. Orders: (1) even numbers, (2) odd numbers, (3) prime numbers. All integers are rational numbers.
rational number	-2/1, -0.5, 0.0, 0.3, 1/2, 7.8	Any number that can be written as a ratio of two integers — where the denominator must not be zero —, or as the resulting fraction written with decimal digits after the decimal dot. All rational numbers are real numbers, and are not imaginary numbers.
irrational number	square root of 2; √2	Irrational numbers cannot be written as a ratio of two integers; the have an infinite number of decimal places.
real number		Real numbers include all types of numbers listed above and are defined as points on a line from -∞ to +∞.
immaginary number	square root of -1	An imaginary number squared yields a negative real number.
complex number		Complex numbers are combinations of real and imaginary numbers.

References

Bioblast link	Reference	Year
Agrillo 2020 PLOS ONE	Agrillo Christian, Piffer Laura, Bisazza Angelo, Butterworth Brian (2020) Evidence for two numerical systems that are similar in humans and guppies. PLOS ONE 7:e31923.	2020
Baroody 1983 J Research in Mathematics Education	Baroody AJ, Price J (1983) The development of number–word sequence in the counting of three-year-olds. J Research in Mathematics Education 14:361-8.	1983
Bell 1999 Springer	Bell John L (1999) The art of the intelligible. An elementary survey of mathematics in its conceptual development. Springer Science+Business Media Dordrecht:249 pp.	1999
Bureau International des Poids et Mesures 2019 The International System of Units (SI)	Bureau International des Poids et Mesures (2019) The International System of Units (SI). 9th edition:117-216. ISBN 978-92-822-2272-0	2019
Farzin 2012 J Vision	Farzin Faraz, Hou Chuan, Norcia Anthony M (2012) Piecing it together: Infants' neural responses to face and object structure. J Vision 12.6.	2012
Gnaiger 2020 BEC MitoPathways	Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5th ed. Bioenerg Commun 2020.2. https://doi.org/10.26124/bec:2020-0002	2020
Gnaiger 2020 MitoFit x	Gnaiger E (2021) The elementary unit — canonical reviewer's comments on: Bureau International des Poids et Mesures (2019) The International System of Units (SI) 9th ed. https://doi.org/10.26124/mitofit:200004.v2	2021
Gong 2019 J Numerical Cognition	Gong Tianwei, Li Baichen, Teng Limei, Zhou Zijun, Gao Xuefei, Jiang Ting (2019) The association between number magnitude and space is dependent on notation: evidence from an adaptive perceptual orientation task. J Numerical Cognition 5:38–54.	2019
Grosholz 2007 Oxford Univ Press	Grosholz Emily R (2007) Representation and productive ambiguity in mathematics and the sciences. Oxford Univ Press 312 pp.	2007
Kadosh 2015 Oxford Univ Press	Kadosh Roi Cohen, Dowker Ann, ed (2015) The Oxford handbook of numerical cognition. Oxford Univ Press:1185 pp.	2015
Singh 1997 Fourth Estate	Singh Simon (1997) Fermat's last theorem. Fourth Estate, London 340 pp.	1997
Spiegelhalter 2015 Profile Books	Spiegelhalter David (2015) Sex by numbers: What statistics can tell us about sexual behaviour. Profile Books, London 368 pp.	2015
Wilkie 2015 Front Psychol	Wilkie James EB, Bodenhausen Galen V (2015) The numerology of gender: gendered perceptions of even and odd numbers. Front Psychol 6:810.	2015

MitoPedia concepts: Ergodynamics