Posts Tagged ‘numbers’

Numbers and mathematics are possible only because time flows

August 13, 2022

It is probably just a consequence of ageing that I am increasingly captivated (obsessed?) by the origin of things. And of these things, I find the origins of counting, numbers and mathematics (in that order) particularly fascinating. In that order because I am convinced that these developed within human cognition – and could only develop – in that order.  First counting, then numbers and then mathematics. The entire field of what is called number theory, which studies the patterns and relationships between numbers, exists because numbers are what they are. All the patterns and relationships discovered in the last c. 10,000 years all existed – were already there – as soon as the concept of numbers crystallised. Whereas counting and numbers were invented, all the wonders of the patterns and relationships that make up number theory were – and are still being – discovered. And what I find even more astonishing is that the entire edifice of numbers is built upon just one little foundation stone- the concept of identity which gives the concept of oneness.

Croutons in the soup of existence

The essence of identity lies in oneness. There can only be one of any thing once that thing has identity. Once a thing is a thing there is only one of it. Half that thing is no longer that thing. There can be many of such things but every other such thing is still something else.

Numbers are abstract and do not exist in the physical world. They are objects (“words”) within the invented language of mathematics to help us describe the physical world. They enable counting and measuring. The logical one or the philosophical one or the mathematical one all emerge from existence and identity. Neither logic nor philosophy nor mathematics can explain what one is, except that it is. Every explanation or definition attempted ends up being circular. It is what it is.

Given one (1), all other numbers follow.

Where numbers come from

Numbers start with one (1), and without a one (1) there can be no numbers. …… . Given the abstract concepts of identity (oneness, 1) and arithmetical addition (+), all natural numbers inevitably follow. With a 1 and with a +, and the concept of a set and a sum, all the natural numbers can be generated.

1 + 1 + 1 + 1 ……

…. Numbers, ultimately, rest on the concept of identity (oneness).

Equally fascinating are the questions that existence, time and causality are answers to. I am coming to the conclusion that the flow of time (whatever time is) does not emerge from existence but, in fact, enables existence.

Revising Genesis

…. What time is remains a mystery but the first act of creation is to set it flowing. Note that the flow of time does not need existence. To be, however, requires that time be flowing. Time itself, whatever it is, is a prerequisite for the flow of time and the flow of time is prerequisite for existence. ………. For even the concept of existence to be imaginable, it needs that the flow of time be ongoing. It needs to be present as a permanent moving backdrop. The potential for some particular kind of existence then appears, or is created, only when some particular rules of existence are defined and implemented. These rules of existence must therefore also be in place before the concept of things, whether abstract or material or otherwise, can be conjured up.

It is inevitable that my views have evolved and they may well evolve further but my current conclusion is that for mathematics to exist time needs to be flowing.

The bottom line:

  1. All branches of mathematics, though abstract, are existentially dependent upon the concept of numbers.
  2. Numbers depend on the concept of counting.
  3. Counting derives from the concept of oneness (1).
  4. Oneness depends upon the concept of a unique identity.
  5. The existence of a unique identity requires a begin-time.
  6. Beginnings require time to be flowing.
  7. Existence is enabled by the flow of time

Therefore

Numbers and mathematics are possible only because time flows

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Numbers emerge from the concept of identity

December 18, 2020

Numbers are abstract. They do not have any physical existence. That much, at least, is fairly obvious and uncontroversial.

Are numbers even real? The concept of numbers is real but reason flounders when considering the reality of any particular number. All “rational” numbers (positive or negative) are considered “real numbers”. But in this usage, “real” is a label not an adjective. “Rational” and “irrational” are also labels when attached to the word number and are not adjectives describing the abstractions involved. The phrase “imaginary numbers” is not a comment about reality. “Imaginary” is again a label for a particular class of the concept that is numbers. Linguistically we use the words for numbers both as nouns and as adjectives. When used as a noun, meaning is imparted to the word only because of an attached context – implied or explicit. “A ten” has no meaning unless the context tells us it is a “ten of something” or as a “count of some things” or as a “measurement in some units” or a “position on some scale”. As nouns, numbers are not very pliable nouns; they cannot be modified by adjectives. There is a mathematical abstraction for “three” but there is no conceptual, mathematical difference between a “fat three” and a “hungry three”. They are not very good as adjectives either. “Three apples” says nothing about the apple. “60” minutes or “3,600” seconds do not describe the minutes or the seconds.

The number of apples on a tree or the number of atoms in the universe are not dependent upon the observer. But number is dependent upon a brain in which the concept of number has some meaning. All of number theory, and therefore all of mathematics, builds on the concept and the definition of one.  And one depends, existentially, on the concept of identity.

From Croutons in the soup of existence

The properties of one are prescribed by the assumptions (the “grammar”) of the language. One (1,unity), by this “grammar” of mathematics is the first non-zero natural number. It is the integer which follows zero. It precedes the number two by the same “mathematical distance” by which it follows zero. It is the “purest” number. Any number multiplied by one or divided by one remains that number. It is its own factorial. It is its own square or square root; cube or cube root; ad infinitum. One is enabled by existence and identity but thereafter its properties are defined, not discovered. 

The question of identity is a philosophical and a metaphysical quicksand. Identity is the relation everything has to itself and nothing else. But what does that mean? Identity confers uniqueness. (Identical implies sameness but identity requires uniqueness). The concept of one of anything requires that the concept of identity already be in place and emerges from it. It is the uniqueness of identity which enables the concept of a one.

Things exist. A class of similar things can be called apples. Every apple though is unique and has its own identity within that class of things. Now, and only now, can you count the apples. First comes existence, then comes identity along with uniqueness and from that emerges the concept of one. Only then can the concept of numbers appear; where a two is the distance of one away from one, and a three is a distance of one away from two. It is also only then that a negative can be defined as distance away in the other direction. Zero cannot exist without one being first defined. It only appears as a movement of one away from one in the opposite direction to that needed to reach two. Negative numbers were once thought to be unreal. But the concept of negative numbers is just as real as the concept for numbers themselves. The negative sign is merely a commentary about relative direction. Borrowing (+) and lending (-) are just a commentary about direction. 

But identity comes first and numbers are a concept which emerges from identity.

image from personality-psychology.com/

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Decimals are too simple

April 22, 2018

Of course all attempts to create a 10 hour day with 100 minutes to each hour and 100 seconds to each minute have failed. Similarly all attempts to divide the circle into 100 parts have not caught on.

The use of 60 is almost as basic as the use of 10.

The origins of base 60

All the non-decimal systems I learnt were embedded in memory before I was 20. I don’t expect many will remember these.

As a child I learned the use of 12 and 60 from my grandmother. The use of 12 was automatic with 4 and 3. Three pice to a pie. 4 pies to an anna and 16 annas to the rupee. When driving around India with my father, miles and furlongs and yards and feet came naturally. Bushels and pecks and gallons and quarts and pints came later as an apprentice in England.

Decimals are simple. But they are also simplistic.

Perhaps too simple.

Rupee, anna, pice, pies

Pounds, shillings, pence, farthings

Ton, hundredweights, pounds, ounces

Mile, furlongs, yards, feet

Bushel, pecks, gallons, quarts, pints

 

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Counting was an invention

March 19, 2017

A new book is just out and it seems to be one I have to get. I am waiting to get hold of an electronic version.

Number concepts are a human invention―a tool, much like the wheel, developed and refined over millennia. Numbers allow us to grasp quantities precisely, but they are not innate. Recent research confirms that most specific quantities are not perceived in the absence of a number system. In fact, without the use of numbers, we cannot precisely grasp quantities greater than three; our minds can only estimate beyond this surprisingly minuscule limit.

Numbers fascinate me and especially how they came to be.

The earliest evidence we have of humans having counting ability are ancient tally sticks made of bone and dating up to 50,000 years ago. An ability to tally at least up to 55 is evident. One of the tally sticks may have been a form of lunar calendar. By this time apparently they had a well developed concept of time. And concepts of time lead immediately and inevitably to the identification of recurring time periods. By 50,000 years ago our ancestors counted days and months and probably years. Counting numbers of people would have been child’s play. They had clearly developed some sophistication not only in “numbering” by this time but had also progressed from sounds and gestures into speech.  They were well into the beginnings of language.

Marks on a tally stick tell us a great deal. The practice must have been developed in response to a need. Vocalisations – words – must have existed to describe the tally marks. These marks were inherently symbolic of something else. They are evidence of the ability to symbolise and to think in abstract terms. Perhaps they represented numbers of days or a count of cattle or of items of food or of number of people in the tribe. But their very existence suggests that the concept of ownership of property – by the individual or by the tribe – was already in place. Quite probably a system of trading with other tribes and protocols for such trade were also in place. At 50,000 years ago our ancestors were clearly on the threshold of using symbols not just on tally sticks or in cave paintings but in a general way and that would have been the start of developing a written language. …….

My time-line then becomes:

  • 8 million YBP           Human Chimpanzee divergence
  • 6 million YBP           Rudimentary counting among Archaic humans (1, 2, 3 many)
  • 2 million YBP           Stone tools
  • 600,000 YBP          Archaic Human – Neanderthal divergence
  • 400,000 YBP          Physiological and genetic capability for speech?
  • 150,000 YBP           Speech and counting develop together
  • 50,000   YBP           Verbal language, counting, trading, calendars in place (tally sticks)
  • 30,000   YBP           Beginnings of written language?
Clearly our counting is dominated by the base of 10 and our penchant for 12-based systems. The joints on the fingers of one hand allows us to count to 12 and that together with the five fingers of the other clearly led to our many 60-based counting systems.
The origins of base 60

I like 60. Equilaterals. Hexagons. Easy to divide by almost anything. Simple integers for halves, quarters, thirds, fifths, sixths, tenths, 12ths, 15ths and 30ths. 3600. 60Hz. Proportions pleasing to the eye. Recurring patterns. Harmonic. Harmony.

The origins of the use of base 60 are lost in the ancient past. By the time the Sumerians used it about 2,500 years ago it was already well established and continued through the Babylonians. But the origin lies much earlier. ……

Why then would the base 60 even come into being?

image sweet science

The answer, I think, still lies in one hand. Hunter-gatherers when required to count would prefer to use only one hand and they must – quite early on and quite often – have had the need for counting to numbers greater than five. And of course using the thumb as pointer one gets to 12 by reckoning up the 3 bones on each of the other 4 fingers. 

My great-grandmother used to count this way when checking the numbers of vegetables (onions, bananas, aubergines) bought by her maid at market. Counting up to 12 usually sufficed for this. When I was a little older, I remember my grandmother using both hands to check off bags of rice brought in from the fields – and of course with two hands she could get to 144. The counting of 12s most likely developed in parallel with counting in base 10 (5,10, 50, 100). The advantageous properties of 12 as a number were fortuitous rather than by intention. But certainly the advantages helped in the persistence of using 12 as a base. And so we still have a dozen (12) and a gross (12×12) and even a great gross (12x12x12) being used today. Possibly different groups of ancient man used one or other of the systems predominantly. But as groups met and mixed and warred or traded with each other the systems coalesced.

If we had 4 bones on each finger we would be using 5 x 16 = 80 rather than 60.

 

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Baboons can tell “more” from “less” – but that is still a long way from counting

May 4, 2013

Being able to distinguish between “more” and “less” is – most likely – a capability that is a pre-requisite for the evolutionary development of the ability to count which itself must lead to the invention of numbers. Recent experiments with baboons demonstrates that they have a clear ability to make quite complex more/less distinctions.

Allison M. Barnard, Kelly D. Hughes, Regina R. Gerhardt, Louis DiVincenti, Jenna M. Bovee and Jessica F. Cantlon.Inherently Analog Quantity Representations in Olive Baboons (Papio anubis). Frontiers in Comparative Psychology, 2013 DOI: 10.3389/fpsyg.2013.00253

From the University of Rochester press release:

… Now a new study with a troop of zoo baboons and lots of peanuts shows that a less obvious trait—the ability to understand numbers—also is shared by man and his primate cousins.

“The human capacity for complex symbolic math is clearly unique to our species,” says co-author Jessica Cantlon, assistant professor of brain and cognitive sciences at the University of Rochester. “But where did this numeric prowess come from? In this study we’ve shown that non-human primates also possess basic quantitative abilities. In fact, non-human primates can be as accurate at discriminating between different quantities as a human child.”

“This tells us that non-human primates have in common with humans a fundamental ability to make approximate quantity judgments,” says Cantlon. “Humans build on this talent by learning number words and developing a linguistic system of numbers, but in the absence of language and counting, complex math abilities do still exist.” ……

……… The baboons’ choices, conclude the authors, clearly relied on this latter “more than” or “less than” cognitive approach, known as the analog system. The baboons were able to consistently discriminate pairs with numbers larger than three as long as the relative difference between the peanuts in each cup was large. Research has shown that children who have not yet learned to count also depend on such comparisons to discriminate between number groups, as do human adults when they are required to quickly estimate quantity. 
Studies with other animals, including birds, lemurs, chimpanzees, and even fish, have also revealed a similar ability to estimate relative quantity, but scientists have been wary of the findings because much of this research is limited to animals trained extensively in experimental procedures. The concern is that the results could reflect more about the experimenters than about the innate ability of the animals. ……..

……… To rule out such influence, the study relied on zoo baboons with no prior exposure to experimental procedures. Additionally, a control condition tested for human bias by using two experimenters—each blind to the contents of the other cup—and found that the choice patterns remained unchanged.

A final experiment tested two baboons over 130 more trials. The monkeys showed little improvement in their choice rate, indicating that learning did not play a significant role in understanding quantity.

“What’s surprising is that without any prior training, these animals have the ability to solve numerical problems,” says Cantlon. The results indicate that baboons not only use comparisons to understand numbers, but that these abilities occur naturally and in the wild, the authors conclude. …….

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