Description
A number N is a count NX [x] of elementary entity X divided by the unit-entity UX [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 xΒ·x-1 and 24 xΒ·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; ε « and ζ 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 8th 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 24th 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 ββββββββ", or "concept 'number eight'". We are so deeply used to these symbols, that we easily take the iconic or abstract symbol 8 β that represents the number eight β 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 2020-06-29
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' UX.
Term English German Comment count NX count (account) Anzahl (Konto) generating a count counting (paying) zΓ€hlen (zahlen) narrating a count telling, recounting erzΓ€hlen 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 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) 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, 1st; second, 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'. the feeling of being alone loneliness Einsamkeit antonym: Gemeinsamkeit single component detail Einzelheit uniqueness singularity, uniqueness Einmaligkeit
- Figure 1: Formats and meanings of numbers.
- 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 β 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.
- 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.
- 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.
- 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.
- 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?
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 β 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 looking glass is substituted by the magic kaleidoscope.
- We learn early about odd and even numbers: Numerical parity becomes an integral part of number representation from about the 4th 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-Stacking 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 Stacking), 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 stacking, 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 stacking ears and eye brows and hair get more complex with differentiation, and with taking a face as a CASE of a person (Figure 8).
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 stacked further into a new H=person: H(1F[1M,1A{2E}]). The Counting-Assembling-Stacking Experience (CASE) goes on.
Figure 4. Two faces: one eye β 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 β H(1M,1N,1A[2E]).
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 β where the denominator must not be zero β, 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; β2 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 -β to +β. 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 numberβword 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 |
Brown 2018 Metrologia | Brown RJC (2018) The evolution of chemical metrology: distinguishing between amount of substance and counting quantities, now and in the future. Metrologia 55:L25. https://doi.org/10.1088/1681-7575/aaace8 | 2018 |
Brown 2021 Metrologia | Brown RJC (2021) A metrological approach to quantities that are counted and the unit one. Metrologia 58:035014. https://doi.org/10.1088/1681-7575/abf7a4 | 2021 |
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 |
Cooper 2012 Synthese | Cooper G, Humphry SM (2012) The ontological distinction between units and entities. Synthese 187:393β401. https://doi.org/10.1007/s11229-010-9832-1 | 2012 |
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 |
Giaquinto 2015 Oxford Univ Press | Giaquinto M (2015) Philosophy of number. In Kadosh RC, Dowker A (ed) The Oxford handbook of numerical cognition. Oxford Univ Press:17-32. https://doi.org/10.1093/oxfordhb/9780199642342.013.039 | 2015 |
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 β canonical reviewer's comments on: Bureau International des Poids et Mesures (2019) The International System of Units (SI) 9th ed. MitoFit Preprints 2020.04.v2. https://doi.org/10.26124/mitofit:200004.v2 | 2021 |
Gnaiger 2024 MitoFit | Gnaiger E (2024) Addressing the ambiguity crisis in bioenergetics and thermodynamics. MitoFit Preprints 2024.3. https://doi.org/10.26124/mitofit:2024-0003 | 2024 |
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 5:38β54. | 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 |
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