What's a number?

[Birdjaguar]

Math is God, math alone is.

Amen, brother.

 

[Perfection]

Of course that means math isn't just in your mind then!

 

[Greizer85]

@Birdjaguar: quit calling things by names and then try to make that flat grid that Souron talked about. It's still impossible. There are objective truths in the world even if we all turn into Zen buddhists overnight.

 

[Formaldehyde]

Mathematical patterns in nature is an interesting topic, and I gave it some thought. For simplicity's sake I choose the nautilus as a pretty clear example

That is a classic example of the Fibonacci sequence and the Golden Ratio.


Link to video.


Link to video.

Amen, brother.

Praise the Mathematical Lord.

 

[Birdjaguar]

Of course that means math isn't just in your mind then!

well that depends....

@Birdjaguar: quit calling things by names and then try to make that flat grid that Souron talked about. It's still impossible. There are objective truths in the world even if we all turn into Zen buddhists overnight.

The question: do mathematical concepts (right word?) exist independently of our minds and the system of math that we have created around them?

How does explaining why there are no pentagon shaped honeycombs mean that that math is something universal and exists beyond our human brains?

If we can find Fibonacci sequences and gold ratios in nature, does that mean that these universal concepts are influencing evolution? How would they do so?

If these concepts exist outside of people, where do they exist and what is their relationship to the physical universe?

Now one could say that we see things in nature and from them we develop the concepts. Interesting, but I would bet it is not supported by facts.

 

[Perfection]

well that depends....

On what?

 

[warpus]

The question: do mathematical concepts (right word?) exist independently of our minds and the system of math that we have created around them?

How does explaining why there are no pentagon shaped honeycombs mean that that math is something universal and exists beyond our human brains?

If we can find Fibonacci sequences and gold ratios in nature, does that mean that these universal concepts are influencing evolution? How would they do so?

If these concepts exist outside of people, where do they exist and what is their relationship to the physical universe?

Mathematics is a representation of an abstract system used for computation and analysis. As such, it is a framework within which everything is perfect and follows certain rules. This makes it easy to perform calculations and to figure out answers to questions that the framework was designed to examine. (by perfect I mean that every object has a defined and certain purpose and that each time you add 2 and 2 together you'll get the same result, assuming you are working in the same vector space)

It so happens that the sorts of questions we can analyze with this system can be applied to almost everything in life that's related to counting, measuring, figuring out certain problems, etc. That's why math is so powerful as a tool! With it you can figure out that once you eat an apple, it will be gone, or that if you mix 4 apples with 2 apples you will have 6 apples, which can last you 6 days if you eat an apple a day. That can be a life saving conclusion to reach, and math makes it happen.

But either way, neither the system, nor the objects in it, such as numbers, exist in any tangible way. It is just an abstract system that's useful for answering certain questions.

The examples of fibonacci occurring in nature might make you think: "but wait! maybe there's more to it, or something!" But nope, not really, it's just a sequence that happens to be frequently naturally occurring. I mean, keep adding up the previous 2 numbers, I can see why certain natural processes might take on that form, it's a very simple formula.

Now one could say that we see things in nature and from them we develop the concepts. Interesting, but I would bet it is not supported by facts.

the discovery/innovation of certain mathematical concepts were for sure inspired by nature. be it a guy sitting under a tree, swimming with a bunch of cows, looking at a snail, or eating a salad, the relationship between nature and math is apparent. It is that connection between the two that makes it easy for you to look at nature and say: "Wow.. math!".. or to look at math and say: "Wow.. nature!"

But that is not where math comes from. Math is just an abstract system that we have built ourselves. The parallels between math and nature are there because math is built using some very basic and naturally occurring axioms. It makes sense that we can see a hint of math in naturally formed objects, such as galaxies, snails, and planets.

 

[Birdjaguar]

Math is God, math alone is.

Of course that means math isn't just in your mind then!

On what?

I was wrong, it doesn't depend. It would just be.

 

[Souron]

Several definitions of numbers have been presented in this thread, and it is not apparent that they are consistent.

Here's a list:
1) Pete Atoms observed that numbers represent the concept "how much more", effectively of ratios. This seems to be a physical property.
2) Birdjaguar seems to claim that numbers are a convention, much like language. Most other posters appear to disagree.
3) suiraclaw and ParadigmShifter seem to be saying that numbers are things that are constructed from a small set of axioms and operations. They are therefore purely abstract mathematical concepts.
4) Duo claims numbers are the physical property associated with quantity.

 

[Birdjaguar]

And what does Souron say?

 

[Souron]

I wouldn't have started this thread if I knew the answer.

 

[stormbind]

At a high level they are a "convention", an agreed upon way of organizing things. In the beginning they were for counting and measuring.

Have you thought that it might have been the other way around?

There is a lot of evidence to show that disjointed human civilisations derived unique topologies to understand their environment. I hypothesise that counting could be an extention of topological thinking.

 

[Birdjaguar]

Have you thought that it might have been the other way around?

There is a lot of evidence to show that disjointed human civilisations derived unique topologies to understand their environment. I hypothesise that counting could be an extention of topological thinking.

In the Mideast numbers and writing developed over many, many years from the increasing need for complex accounting.

"The invention of zero and place notation has been heralded as a major accomplishment of the civilized world, but the literature does not treat the advent of abstract numerals because of the common but erroneous assumption that abstract numbers are intuitive to humans. The token system is one piece of artifactural evidence proving that counting, like anything else, is not spontaneous. Instead, counting is cultural and has to be learned." ~Denise Schmandt-Besserat, "Two Precursors of Writing: Plain and Complex Tokens," in Wayne M. Senner, editor, The Origins of Writing, (Lincoln: University of Nebraska Press, 1989), p. 38

Brief summary of Denise Schmandt-Besserat's work on the origins of counting:

Spoiler :

Around 8000 BCE the hunter gatherers of the near East began to domesticate plants and animals with sufficient skill to change their lifestyle from one of migration to fixed villages. Wheat and barley were the main crops, cattle and sheep the important animals. Increases in food led to larger populations and changes in the way people organized their lives. Egalitarian hunter gatherer groups gave way to rank oriented villages where the headman collected and distributed the surplus resources. Simple clay tokens appear in the archaeological record and give evidence of the increasing complexity of economic life and the expansion of their way of counting to include concrete counting. “Accounting” was added to the repertoire of skills used by “educated” citizens. Simple tokens for the everyday products of an agricultural way of life were used to track payments and accounts.

As society become more complex new tokens were added to represent more types of goods and services. Villages became towns and small cities. Around 4000 BCE the cities of southern Mesopotamia reorganized themselves into city states built around temple bureaucracies and economic control of the surplus goods produced by the growing non agricultural workforce. Around 3700 BCE we find complex tokens strung together on cords or sealed in clay balls. These security measures were attempts to produce tamper free records. The clay envelopes were sometimes marked on the outside with the impressions of the tokens found on the inside. That way you didn’t have to break the envelope to know what was inside. Personal seals often authenticated the contents.

By 3500 BCE, the clay envelopes were being replaced by simple tablets with impressions of tokens. At this stage the rules of concrete counting prevailed; tablets accounting for “three jars of oil” were written as “one jar of oil” and “one jar of oil” and “one jar of oil.” The increased demand placed on temple and bureaucratic accounting taxed the ability of the token system to keep up. Then around 3100 BCE in the city of Uruk in Sumeria, someone invented numerals. They separated the “how many” from the “of what.” They invented abstract counting. Initially, an impressed mark counted the “how many” and an incised pictograph indicated the “of what.” These beginning number forms were based on the old impressions of tokens for measures of grain and number of animals. The Sumerians developed and used both base 6 and base 10 counting systems.

Once numbers and objects were separated, pictographs could be used to express far more than accounting transactions and objects. Pictographs took a phonetic path and soon names were added, quickly followed by the whole spectrum of ideas. The full transition to abstract counting and writing took several centuries, but like all good ideas it spread quickly and changed as it spread. From abstract counting we got mathematics. The Sumerians gave us Cuneiform, the art of writing on soft clay with a reed stylus, which became the dominant writing in the Middle East for the next 2000 years.

Spoiler :

Egalitarian Society – Individuals receive a share of common resources equal to status in group.
Rank Society – Collection and distribution of resources by head man
State – Collection and redistribution of resources by central authority. Dependent upon industry, large surpluses of basic goods like grain and cattle, taxes, bureaucracy, coercion of workforce. Uses accounting, tribute, seals of approval, control by scribes (knowledge, education). Rise of monumental temple architecture.
Hunter Gatherer – Egalitarian Paleolithic society
Agricultural – Sedentary society based on crops and domesticated animals.
Tallies – Simple counting (the addition of one more unit of something) that shows up in Paleolithic finds as notches in a stick or bone.
Concrete counting – One to one correspondence. A token that represents “one jar of oil” or “one measure of grain” illustrates this concept. In modern society “twins” and “quartet” are examples of concrete counting.
Abstract counting – Numbers separated from objects. “Twoness” is independent of any particular object and may be associated with many items and used mathematically. 2+4 = 6, but twins cannot be added to quartets.
Computation – using numbers to count objects
Accounting – tracking economic transactions
Simple tokens – unadorned clay objects in simple shapes
Complex tokens – complex shaped clay objects or those incised with lines or designs
Impressed marking – when an object is pushed into clay to replicate its shape.
Incised marking – when an object is drawn over a clay surface to make a mark.

 

[ParadigmShifter]

The abacus predates the invention of zero so place notation came first, but in a non-written form.

 

[warpus]

Numbers were around a long time before formal mathematical systems were developed. As such the definition of what a number is has changed over the years.

I am assuming we are discussing numbers, as they exist in a modern mathematical framework. As such, they are abstract mathematical formulations.

 

[ParadigmShifter]

Integers and rational numbers (fractions) are natural constructs though. (Negative numbers represent debt, fractions represent ratios of lengths or division of goods).

It's just real numbers that had to be "discovered". Irrational numbers were discovered pretty early on by a student of Pythagoras (apparently he was drowned for discovering them). Transcendental real numbers were discovered much later on.

Complex numbers were invented as well to form an algebraically closed extension field to the real numbers.

 

[Souron]

Numbers were around a long time before formal mathematical systems were developed. As such the definition of what a number is has changed over the years.

I am assuming we are discussing numbers, as they exist in a modern mathematical framework. As such, they are abstract mathematical formulations.

It's true that the construction, and axioms that define particular number sets have changed over the years. But I do not think that effects a layman definition of a number. And I think a layman is qualified to give a reasonable definition of a number -- numbers are not advanced mathematics.

 

[PeteAtoms]

It's true that the construction, and axioms that define particular number sets have changed over the years. But I do not think that effects a layman definition of a number. And I think a layman is qualified to give a reasonable definition of a number -- numbers are not advanced mathematics.

But the philosophy of mathematics can far exceed the layman's understanding and interest. As far as I'm concerned (a laymen), math just works. It's the way I feel about a lot of things in science, I don't need to understand the intricacies of different concepts, but the application of those concepts are evident and pervasive in everyday life. A layman doesn't need or care to know what gravity is, but the applications of the theory are fundamental to our way of life.

 

[warpus]

Integers and rational numbers (fractions) are natural constructs though. (Negative numbers represent debt, fractions represent ratios of lengths or division of goods).

What do you mean natural constructs?

 

[ParadigmShifter]

In that negative numbers represent debt (or going below the zero of a scale, e.g. temperature).

Fractions used when cutting cakes or pies into equal sized pieces.