Copyright 2017 Graham Berrisford. One of more than 300 papers at http://avancier.website. Last updated 23/02/2021 11:58
Where do numbers
come from?
Contents
Descriptions
and described things in mathematics
What does it
mean say a number “exists”?
Do aliens
create and use the same mathematics?
Mathematicians
define a type (like “point”, “line” and “triangle”) to describe the
properties of a thing that instantiates that type. A type is a pattern for one member of a set. An instance is an exhibition or embodiment of the type in
one set member.
Mathematics |
Types <create and
use> <are instantiated
in> Mathematicians
<observe and envisage> Set
members |
Two
mathematicians had a famous debate about the meaning of descriptive types or
axiomatic assertions. The debate might be distilled as follows.
Frege
believed mathematics is carried out at the level of thoughts about real things,
rather than descriptions of them. He believed descriptions are imperfect
representations of our thoughts. We “know” what a geometric entity (like “point”,
“line” and “triangle”) is without reference to a description of it, or how it
relates to other entities. And every detail of it may be relevant to
answering questions about it.
Hilbert said that, even if we do know
other details of an entity (say, its color), they are entirely irrelevant to
understanding geometry, which merely defines relations between entities.
Geometric descriptions are holistic in that they define how geometric entities
are related, rather than what those entities are or contain; they can be
anything (large or small) that is relatable as described. Mathematics is
carried out at the level of descriptions, regardless of what the entities are
in reality.
According
to the Stanford Encyclopedia
of Philosophy, Hilbert is now regarded as the winner of the
debate. Geometry does not address the whole of a thing (Frege); rather, it
addresses only those features of it that are describable by geometry (Hilbert).
If an animal can remember a general pattern or type, then a startling new idea emerges. It becomes possible to count things of that type, and communicate the total number to others.
Counting things of a type |
Quantities <create and
use> <are instantiated
in> Counters <observe and envisage> Things of a type |
The argument here is that logic and mathematics cannot exist until intelligences have crystallized family resemblances into types. You can’t say a statement is true or false, until you know the types referred to in the statement. You can’t say “This man is my brother” is true or false, until you know the properties of the brother type. You can’t count your siblings until you know the properties of the sibling type. You can’t count any things until you know the type to which they must conform, or the set they belong to.
Primitive animals surely don’t classify things into rigid “types”. But they can recognize “family resemblances” between similar things (e.g. food items, friends and cliff edges). And learn to respond to similar things in appropriate ways.
Scientists have studied how far honey bees, dolphins and babies can understand quantities. We know many animals can recognize when a smallish family of similar things gains or loses a member. Experiments show dolphins can recognize which of two boards has, say, five dots rather than six. And babies (before they have words) can recognize when a small group of things gains or loses a member.
Astonishingly, experiments suggest honey bees can count up to four and communicate that amount to other bees.
Honeybees are clever little creatures. They can form
abstract concepts, such as symmetry versus asymmetry. And they use symbolic
language — the celebrated waggle dance — to direct their hivemates
to flower patches. New
reports suggest that they can also communicate across species, and can count —
up to a point.”
http://www.livescience.com/2909-bees-count.html
A number is a description made by a describer who can do three things.
1. observe or envisage several discrete things
2. either aggregate them into a set (e.g. items in basket) or match them to a given type
3. count the things in the aggregate or matching the type
Types enable counting; numbers imply types. And types can be organised in type hierarchies.
Clearly, sentient animals evolved to recognize family
resemblances. Humans go further; they formalise the description of a family
member into a “type”. The proposal here is that all types and mathematical
concepts emerged out of 1) the animal brain's ability to recognize
"family members" and 2) the particularly human ability to more
formally describe/symbolise a family member using words.
Numbers emerge from enumerating things – the members of a
family - that resemble each other. As soon as we have a family in mind, we can
count the members of that family. As soon as we can count the members of a
family, we find some families have something in common. That is, they share the
number that enumerates how many members belong to the family.
Thus, a number acquires the status of a type (quality or
concept) that can be instantiated many times. Numbers are types that represent
what families of the same size have in common:
· “oneness” is the property shared by
all families with one member
· “twoness” is the property shared
by any one-thing family to which we have added one.
· “empty (zeroness)”
is the property of any empty nest or family (observed or envisaged) that
currently has no members.
It appears the Sumerians were the first people to develop a
counting system. And the number zero was invented later,
perhaps independently by the Babylonians, Mayans and Indians. But surely
the concept of an empty nest or empty family was understood eons before that.
Quantifiable variables, such as “speed” or “height” can be regarded as types
Type name |
Type qualities or attributes |
Height |
The distance from
bottom to top of a standing object |
One thing that instantiates the type “height” is me. My instantiation of the type is measurable as 1.84 metres, or 6 feet and 0.5 inches.
A type does not exist in a thing that instantiates it. So where is it? To answer that question, we must decide what “exist” means.
Some mathematicians and take the view that a type is eternal, it exists outside of space and time. They think of a type like “even number” as a “universal” or “Platonic ideal” that has existed since the cosmos began. The alternative view here is that a type is a tool created to describe a thing. It only exists when encoded in some mind (mysteriously) or record (using a known symbology).
For example, the current “largest known prime” is a number that exists in current records. The next one exists only as a concept or type in current minds, as an envisaged possibility. The next number (N) is not yet known, does not yet exist as instantiating the “largest known prime” type.
Suppose an instance of N already appears in a list of “odd” numbers, where its primeness goes unrecognized. A mathematician may say that this N already exists in the three eternal, ethereal sets of “odds”, “primes”, and “largest known primes”. But that is to use the word “exist” in a different way.
By this logic, everything that exists already instantiates infinite as yet undefined types, and is a member of infinite possible sets. So, do all types and sets (all possibly-conceivable ones) exist eternally and ethereally? This view of what it means to “exist” is an untestable and useless assertion, surely better removed using Occam’s razor.
Now suppose the “largest known prime” algorithm runs further and generates a second copy of N. The new actor (the second N) plays the role labelled “largest known prime”. The old actor (the first N) still plays the role called “odd number”, where its primeness goes unrecognized
But at any moment you can read any number (a discrete thing) and prove by testing it instantiates any number of type(s). Because the type does not exist in the thing itself; the match of a thing to a type is an encoding or decoding process.
Many mathematicians are reluctant to believe that there were no numbers before life. But surely, you cannot have numbers until you have types, of which instances can be counted? And you cannot typify things until there is some kind of intelligence?
A type describes one thing, of which there is a collection. That collection may contain any number of things (zero, one or many). That number is the cardinality or quantity or extension of the collection. But counting the members of a collection only became possible when animals were able to recognise the existence of a collection.
We could speak of all things that instantiate a type across the cosmos. But our practical interest is more usually in things within our sphere of influence on planet earth.
We could speak of all things that instantiate a type over all time (past, now and future). But our practical interest is more usually in things that instantiate a type right now. Or things that instantiate the type for a period of time we are interested in.
Mathematicians
usually think of a type as defining instances across all space and time. However, you can limit the
collection by extensional definition, or by including space and time
constraints on the intensional definition of a set member. Or by making set membership a
decision, signified by assigning an identifier to a thing.
Type name |
Type qualities or attributes |
Registered vehicle |
A vehicle with a
registration number (regardless of its other attributes). |
E.g.
the designers of a vehicle registration system create types like “vehicle
owner” and “vehicle”. They
don’t mean to an express an interest in every instantiation of those types in
the lifetime of the cosmos. They
mean to classify only things observed or envisaged in the system of interest.
It is perfectly legitimate to discuss infinite abstract mathematical objects and sets. But the practical interest here is in modelling “real machines” in societies and businesses that are finite in reality.
Things have life times; and (because types are things we create and use) types have life times too. We model things that exist in time and space (for a while), using descriptive types that also exist in time and space (for while).
E.g. consider modelling some regular behavior of a “real machine” as a “system”. Having described a system, we can imagine it being instantiated by infinite real machines. But in finite time and space, the number of real machines that can instantiate one system description is finite.
E.g. consider modelling the size of a collection of things using a number. Given a number (say 8), we can imagine infinite collections that contain 8 things. But in finite time and space, the number of real-world collections is finite.
E.g. consider types of number. Given a type of number (such as “even number”) we can imagine an infinitely extendable collection of such numbers. But in finite time and space, the number of numbers that can exist in minds, records and computers is finite.
For sure, we can posit the infinite
extension of the set of prime numbers. But
our main concern is with things and types that demonstrably exist, or can be
made to exist, in time and space. And in the processes by which new
instances of a type can be generated when needed.
The largest known prime number exists in material reality. The prime number beyond the highest calculated so far does not exist yet. However, we can describe the process for generating the next largest known number.
An alien species may live on another planet, but still lives in the same universe. Intelligent aliens will surely recognize the same patterns we do.
Aliens will evolve and learn to |
leading them to articulate the concepts of |
detect the same “family resemblances” between similar things |
types (“star”, “planet”, “plant”, “parent” and “river”). |
judge whether a thing is an instance of type or not |
truth and falsehood |
count the instances of a type, then add to and subtract from that total |
arithmetic |
recognize when subtraction exhausts the instances of a type (leaving zero instances) |
further mathematics |
typify when an instance of one type leads inevitably to an instance of another type |
the rules of logic |
The existence of numbers and logic in the discourse of alien species will naturally follow from their typifying things in the universe we share. So, there is no reason to think that their mathematics or logic would evolve along significantly different lines.
Surely,
there were no concepts before conceivers, no descriptions before describers. All
descriptions and types, including mathematical concepts are real rather than
ethereal. There exist only in descriptions encoded in physical forms by
intelligent entities. And when all descriptions have gone from the cosmos,
types too will disappear.
People plausibly argue the laws of logic must have existed before any intelligent entities thought of them, recorded them or used them. But it is not necessary or useful for them to have existed before life. It could never be proved, and it requires you to posit the existence of ethereal things. You can use Occam’s razor to eliminate them from your philosophy. And suffer no loss to the credibility or usefulness of mathematics. That seems a consistent philosophical position. And by requiring no recourse to ethereal metaphysics, it is more economical.
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