A contribution to disambiguation of the concepts ‘number’, ‘time’ and ‘frequency’


The antonym for the term 'knowledge' is not  the 'ignorance'. 'Ignorance' is a lack of education, unfamiliarity with otherwise existing knowledge. The opposite of ‘knowledge’ is an infinitely large, unexplored, undiscovered, unanticipated variety of relations, processes and something else within and beyond us. Stock of our existing knowledge records is growing faster than the substance of knowledge itself, and it is increasingly difficult to browse and share this voluminous structure of information. To state that this present problems in use and creation of knowledge would be rather paradoxical.

It seems rather obvious that more attention should be devoted to the modes of knowledge presentation and sharing. Even a limited and somewhat arbitrary selection of publications [1-19] allows for observing homonymous, synonymous and otherwse ambiguous definitions of important concepts such as 'number', 'time' and 'frequency' . 

Engineering and scientific publications do not frequently comprise terms  such as “respect”  and “sharing” or “ambiguity” and “misinterpretation”.  However,  the mind provoking frequency of omittances of the former two modes of communication, and the abundance of the later pair of consequences of the lack of communication, both call for a closer look of the causes for this state of the art.  The concepts 'number', 'time' and ‘frequency’ are chosen because of their fundamental importance in science, engineering and education, and because their use exemplifies the problems listed above. These concepts are affiliated with differing interpretations, henceforth associated with  inevitable ambiguity when otherwise verified knowledge and theories are shared between differing disciplines. Moreover, when differences in  these key concepts are ignored in publishing scientific advances, one is prompted to sense certain disrespect to others.  Finally, the recipients and users of knowledge, such as the students and engineers can fall into traps of misinterpretations. There is a growing evidence which indicates that there can be a serious disruption to a learner’s progress when confronted by apparent misalignments of meaning. Reasoning modality is influenced by perceptions of similarity, which can be blunted by conceptual inconsistencies and terminological contradictions [1]. 

Knowledge broadening, dissemination and use are in the core of our progress. The way how we share theories affects efficiency of  knowledge applications,  i.e. the rate and direction of advances in our civilisation. Sharing is important for verification of knowledge. Missions of the academe include sustaining the knowledge shareability and applicability. Any successful application of knowledge provides  additional supporting evidence to its validaty. 

The differences in interpretations of concepts, which are criticized in this treatise, must be distinguished from the category of differences that result from the rivaling hypotheses, each attempting to explain the same phenomena, or even the same evidence. It is important to note that these competing hypotheses do not simply ignore the differences. On the contrary, all care is taken to explain these differences, and to present the preferred arguments in most clear fashion.  Such transparent approach to differences has enabled advances such as the discovery of penicillin and quantum physics.

This treatise focuses on the cases where the differences are ignored. These differences can vary from the simple cases of homonymy to quite substantial conceptual misalignments.  None of them should be tolerated.  Defining knowledge in terms of its application and sharing is a precondition for sustainability of our civilisation. 

Numerous sources point at the damaging consequences of ambiguities associated with  the concepts of ‘number’, ‘time’ and ‘frequency’. [2, 3] 

We propose hereby the modyfied definitions that are intended to contribute to the disambiguation in the usage of these three concepts.

Number, numeral, cipher, digit and count

Concept of the ‘numbers’ can be defined by departing from axioms outlined in [2] and [4]. A proposed definition is then derived as follows:

A ‘figure’ is an arrangement of points made within two-dimensional space to present a visual static impression (a perception) of something (e.g., a figure printed on a book page, showing the front view of a home).

‘Metric’ is a relation used to enable a comparison of (the groups of) phenomena.

‘Comparison’ is a definition indicating whether (or to what degree) one phenomenon differs from other phenomena. When comparison indicates that some phenomena are sufficiently similar, such phenomena can be counted using numbers. When some group (set) of phenomena can be classified within the same category, their attributes can be compared using numbers (by quantifying the relevant levels at a corresponding numerical scale).

‘Number’ is a most general metric, systematically ordered within mathematics; a quantity comparison derived from zero and units; every number has a unique position in the sequence, which starts with numerals. 

‘Numeral’ is any one of the elements that can be combined to form numbers in a number system (e.g., decimal system, binary system, hexadecimal system, etc.). 

‘Digit’ is a figure representing a numeral; examples: "0", "1", "A".

‘Cypher’  is a figure representing a number; examples: "10001001", "10,001.001".

The usage of the concept of  ‘number’ must be distinguished with regard to purpose of that ussage. For example, the cases where the number represents a ‘count’ from cases where it represent a quantitative level of some attrbute: e.g.  12 kg represent a mass of some solid object. A statement that the “frequency is the number of observed occurences” [5] is ambiguous. More clear formulation is to state that frequency is the count of observed occurrences.

Time, duration and datum

Concept of  ‘time’ is discussed widely in many disciplines, including the historic views [6-9].

One class of definitions indicates that time is a scalar quantity that can be correlated with a number line stretching from - ¥ to +¥. Advances in physics required introducing hypotheses where time is defined along the fourth axis of the spacetime manifold. Relativity theory requires that the distance between the points on that axis depend on the relative speed of the observers of the events. Special relativity postulates that time cannot be understood except as part of space-time, a combination of space and time. General relativity has further changed the notion of time by introducing the idea of curved space-time. [10-12] 

Philosophical considerations [6] add further questions to this concept of ‘time’. Although, nearly a century  after its publication, general relativity remains a highly successful model, there are indications that the theory is incomplete, and the question of the reality of spacetime singularities remain open. [9, 13, 14]

It is proposed hereby to sustain this concept, termed ‘time’, in line with the advances in science. Its measure, ‘second’, defined by ISO standards [15], page 112, and [16] allows for both theoretical and practical applications and advances in relevant knowledge with a sufficient and continuously increasing precision and accuracy.

However, it is recommended hereby to further distingush the above concept of ‘time’from its special case termed ‘duration’ (which is also mesured in seconds). ‘Duration’ is a time measurement that allows for comparisons such as to which of the two observed phenomena takes less time. For example, if we denote time by t, then                                              

                                  duration = t2 – t1,    seconds

where ti = i-th point on the time number line. Point ti+1 occurs at any time after the point ti.

Moreover, the concept of ‘time’ is often used to refer to the hour of the day reckoned by the position of a celestial reference point relative to a corresponding celestial meridian. Time may be designated solar, lunar, or sidereal, as the reference is the sun, moon, or vernal equinox, respectively. Solar time may be further classified as mean or astronomical (if the mean sun is the reference), or as apparent (if the apparent sun is the reference).

Time may also be designated according to the reference meridian, either the local or Greenwich meridian or, additionally, in the case of mean solar time, a designated zone meridian. Standard and daylightsaving time are variations of zone time. [9,  17]

Sources [18, 19] present further examples of usage of this term with the above meanings.

 It is recommended herewith to associate the above meanings with the terms ‘datum’ and ‘instant’ defined in accordance to Coordinated Universal Time (UTC) unless otherwise stated. An example of recommended usage is: “The plenary session started at 2 pm on 24th August 2009 . At the same instant, in the adjacent room started a worksession devoted to coordination of multi-language translating services.”

Frequency and frequence

Application, dissemination and broadening of knowledge requires observations of phenomena (relations, events, systems etc). Some phenomena can be classified in a category that allows for counting their occurences.

‘Frequency’ is the count of observations of such classifiable phenomena within the analysed scope (within some boundaries defined by the purpose of observations). 

For example, if we observe a a large container filled in with a mix of thousands of yellow, green and red apples, we can count how many of red apples have been found in small boxes filled each time with ten apples chosen at random from the above large container. If all colours have been mixed well in the large container, then the frequency of red apples in each box can vary from 0 to 10.

At the same time, many authors us the term ‘frequency’ with another special meaning, particularly related to periodic phenomena and the concept of duration.  This homonymy itself can be easily eliminated, for example by introducing a term ‘frequence’.  

It is proposed therefore to consider defining this concept, ‘frequence’, as follows: ‘frequence’ (f) is the ratio of the count of the observed cycles, e.g. wavelengths (nλ1), and  a multiple of the count of wavelengths of  some convenient, measurable, stable and standardized radiation (Nls) that occurs simultaneously.

For the convenience, rather than counting Nl each time we need to measure the frequence of some cycles, a metrological device termed a ‘clock’ is developed. Such a clock enables measuring Nls to be expressed in ‘seconds’ (s). [15, 16]

Frequence is measured in units termed hertz  (Hz):

                      1 Hz = 1 s-1

Bearing in mind that Nlis subject to some level of  uncertainty, value of the derived frequence f  is always just an approximation characterized by some estimate of an average value and variance. The current state of the art in sciences allows for reducing the relative uncertainty to the order of 10-15  ; [16].

In the above definition, the use of the concepts of ‘duration’ (and especially that of the ‘time’) was deliberatelly avoided, until it became necessary to introduce some observable metrics – namely, the second. It is instructive to emphasise that a ‘second’ is defined in terms of the count Nls.


Differentiation of disciplines is useful as long as it does not prevent knowledge shareability.

Knowledge can be thought to be composed of definitions, and appropriate and transparent definitions are infinitelly shareable.

Impedances to knowledge shareability range from relatively easy resolvable cases of homonymy, to substantial ambiguities that are usually consequence of isolationism.

Homonymy can be resolved by attributing differing terms to differing concepts. More complex ambiguities will be extenuated by exposing relevant definitions and theories to the scrutiny of other disciplines.


[1] K. Abhary, H. K. Adriansen, F. Begovac, Z. Kovacic, C. N. Shpigelman, C. Stevens, S. Spuzic, F. Uzunovic and K. Xing "A Contribution to Transparency of Scientific and Engineering Concepts" The International Journal of Knowledge, Culture and Change Management, (2009) Volume 9, Issue 5, pp.93-106. http://ijt.cgpublisher.com

[2] S. Spuzic, K. Xing and K. Abhary "Some Examples of Ambiguities in Cross-disciplinary Terminology", The International Journal of Technology, Knowledge and Society (2008), Volume 4, Issue 2, pp.19-28

[3] S. Spuzic and F. Nouwens "A Contribution to Defining the Term ‘Definition’", Issues in Informing Science and Information Technology Education, Volume 1 (2004) pp 645 - 662, http://proceedings.informingscience.org/InSITE2004/090spuzi.pdf

[4] Axioms postulated within the project De-CITE   http://figures.yolasite.com/axioms.php     at     http://figures.yolasite.com/de-cite.php   (mirror site   http://www.zlatko.info/moodle/course/view.php?id=11)   accessed 10th Sept 2010

[5] I. Diamond and J. Jefferies ”Beginning statistics: an introduction for social scientists” Sage Publications Ltd,  2000

[6] Ned Markosian  “Time” Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/time/  accessed 10 Sept 2010.

[7] R A Serway and J. W. Jewett “Physics for Scientists and Engineers with Modern Physics”,  International Edition 8e,  BrooksCole, 2010 

[8] S  Hawking “A Brief History of Time”  Bantam 1998

[9]  “Time”  http://en.wikipedia.org/wiki/Time    Wikipedia, Wikimedia Foundation, http://en.wikipedia.org   (accessed 11th September 2010)

[10]  J. Barbour "The End of Time: The Next Revolution in Physics" Oxford University Press,1999

[11] P. Davies "About Time: Einstein’s Unfinished Revolution"  Simon & Schuster, 1996

[12] R. Feynman "The Character of Physical Law" Cambridge (Mass): The MIT Press, 108-126, 1994

[13] J Maddox “What Remains To Be Discovered”, Macmillan, 1998

[14]  R Penrose  “The Road to Reality: A Complete Guide to the Laws of the Universe”  Vintage (2007)

[15] “The International System of  Units”, Organisation Intergouvernementale de la Convention du Mètre,  2006 http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf    (accessed 10th September 2010)

[16] “Second”    Wikipedia, Wikimedia Foundation, http://en.wikipedia.org   & http://en.wikipedia.org/wiki/Second#Modern_measurements   (accessed 10th September 2010)

[17] R Marks  "The New Dictionary & Handbook of Aerospace"  Praeger, 1969

[18] The UNESCO Thesaurus, http://databases.unesco.org/thesaurus/ (accessed on 13 October 2005)

[19] “UN glossaries and UN interpreters’ resource page…”, 
http://un-interpreters.org/glossaries.html     &  
http://databases.unesco.org/thesaurus/other.html (accessed on 13 October 2005)

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