Implications for mathematics

Copyright 2017 Graham Berrisford. One of more than 300 papers at http://avancier.website. Last updated 12/01/2021 12:06

 

This article is a supplement to this description theory

It reviews other philosophical triangles and revises them to match our epistemological triangle.

This is not to say the existing triangles are “wrong”; all of them are mental models.

It is to suggest that, revised as suggested here, the other triangles are simpler and clearer.

Contents

Our epistemological triangle RECAP. 1

Where do numbers come from?. 1

From family resemblances to quantification and numbers. 1

What does it mean say a number “exists”?. 1

Mathematical entities as instruments. 1

The collection of things that instantiate a type. 1

Do aliens create and use the same mathematics?. 1

Conclusions and remarks. 1

 

Our epistemological triangle RECAP

 

A map is an abstract model or representation of a physical territory.

The triangle below relates maps to the territories they represent.

 

Cartography

Maps

<create and use>          <represent>

Mappers  <observe and envisage> Territories

 

Epistemology is about what we know of reality, through observation, testing, reasoning and learning from others.

This article uses this triangle to relate epistemological concepts.

 

Epistemology

Descriptions

<create and use>        <represent>

Describers <observe and envisage> Phenomena

 

The triangle is only simple graphical device, telling a small part of the story

The semantics of the triangle are defined below.

 

Describers are actors (natural or artificial) that can encode and decode descriptive models of phenomena.

Descriptions embrace all forms of mental, documented, digital and physical models.

Phenomena are entities, events and processes that can be observed or envisaged in time and space.

 

The relationship between each pair of concepts is many-to-many.

One describer can create several descriptions of the same thing.

Those descriptions may be compatible or in conflict (is light waves or particles?).

Also, several describers can contribute to creating one description of the same thing.

It may well be that none of those describers (e.g. system architects) can hold the whole description in mind.

 

Both describers and descriptions can be observed as phenomena               .

Describers are physical actors (natural or artificial), which may be described.

Descriptions are physical matter/energy structures, which can be described.

 

To describe a thing is to classify it (after A J Ayer).

A description represents, specifies or idealises a thing that embodies or instantiates the description.

A class or type represents, specifies or idealises a thing that embodies or instantiates the type.

A type is a description; a description is a type.

 

“Intensional definition” is the process of creating a type or description.

A description expresses a type in the symbols of a particular language.

What gives the description meaning is the action of an actor in creating or using it.

Encoding is the process of creating the symbols.

Decoding is the process of reading and using the symbols.

 

(The encoding and decoding of information is a theme of cybernetics, after Ashby.

See article/chapter 4 for that and others ideas drawn from Ashby’s cybernetics.

 

Many don’t at first grasp the radical nature of this psycho-biological and cybernetic view of description and reality.

Note especially

·       Descriptions in the mind are at the top (not the left)

·       Descriptions are often recoded into other descriptions

·       Descriptions are physical phenomena

On the nature of description

This article starts by saying “holism is not wholeism” and “the map is the territory we understand”.

All written here about systems is based on the idea that systems are patterns we abstract from physical phenomena.

 

This reflects the outcome of a famous debate between two mathematicians about the meaning of descriptions.

Another mathematician (to help me) has distilled the argument thus.

 

Frege posited that descriptions (axioms) are imperfect representations of thoughts.

And that mathematics is carried out at the level of thoughts rather than descriptions.

The presumption is that we know what geometric entities, such as points and lines, actually are.

 

Hilbert said that, even if we did know, this is irrelevant to understanding of geometry.

Since geometry merely defines some relations between some entities.

He argues mathematics is carried out at the level of descriptions or models.

In geometry, a description is a holistic model - it asserts that particular relationships exist between basic, unanalysed, entities

Those entities can be anything (large or small) that follow the relationships stipulated in the model.

 

Hilbert is now regarded as the winner of the debate, according to the Stanford Encyclopedia of Philosophy,

https://plato.stanford.edu/entries/frege-hilbert/

 

I am told the debate is whether you regard what is described in geometry as

·       a concrete entity, of which every detail is potentially relevant to answering questions about it

·       an abstract set of relationships between unanalysed entities.

 

In other words, does geometry addresses the whole of a thing (Frege), or only selected features of it that are describable by geometry (Hilbert).

As related articles show, the cybernetic answer to this question is firmly in the Hilbert camp.

Where do numbers come from?

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.

 

Abstraction of quantities from sets

Quantities

<create and use>      <represent totals of>

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.

·       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.

 

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 (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.

From family resemblances to quantification and numbers

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"

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 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 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.

What does it mean say a number “exists”?

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.

Mathematical entities as instruments

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?

 

There were always things that an intelligent observer will regard as similar.

In the history of the cosmos, this was first true at the level of atomic particles, then stars and planets.

So, there were always numerous similar things – which we can now regard instances of a type.

But numbers only existed the form of types when people started to create, remember and communicate types.

The collection of things that instantiate a type

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.

Is a collection universal and eternal, or limited in space and time?

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.

Can a collection be infinite?

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.

Do aliens create and use the same mathematics?

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.

Conclusions and remarks

Did mathematical concepts exist before life in an ethereal/metaphysical sense?

Or do they exist only in a real/physical sense, in the models we create?

 

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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