The Nature of Number

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Jonathan.s
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The Nature of Number

Post by Jonathan.s » Fri Aug 10, 2012 7:08 am

Hi all - new poster here.

One of the ideas I have been contemplating concerns the nature of number. Now, as we will see, I am not proposing to explain what number actually is. That, I believe, is actually a very difficult problem in its own right.

The points I want to make about number are simply that number is real, but that it is not material or physical in nature.

Why 'real'? Because (among other things), operations carried out on the basis of the accurate determination of numerical values have real consequences in the world. For example, if I mix the wrong amount of a substance, or calculate the wrong length of the beam in a structure, these mistakes have real consequences, such as explosions, or building collapses. Furthermore, while mathematical systems might in some ways be regarded as conventional or socially-detemined, the outcome of such calculations is not simply conventional. Number is not a 'social reality' or a fashion, again because the determination of the exact value of various measurements have obvious consequence throughout science, engineering, and commerce.

Now - why not 'material'? Well, whatever else number is, it depends on rational ability, the ability to count and to follow logical laws. (Note I am also "maths-challenged" - I was dreadful at school maths, so quite why I have become interested in this topic, I don't really know.) In any case, a number can be represented physically, obviously - by symbols, binary code, words, and so on. But the symbol is not the same as that which is signified. So what is signified? Well, crudely put, a quantity - but it might also be a relationship, or a ratio.

But the point about number is that it does exist anywhere 'externally'. It is not 'in the world'. It is only visible to a rational intelligence which is capable of counting. But to any minds capable of counting, a given number has the same value. (Although there are grey areas - perhaps this is something that can be said, at least, of real numbers and integers. 'God created the integers, all else is the work of man', someone said.)

Anyway, as I said, I am not going to try and explain what number is. Far greater minds than mine have come to grief on that topic. But I find the 'real but not material' nature of number very interesting. I also know from discussing this idea, that such an understanding is very unpopular in the current academe. It sounds awfully Platonist, and empiricism generally disdains Platonism. But there are many Platonists still around, even though they may not wear their heart on their sleeve.

Anyway, I will leave it there. I am interested to know whether anyone agrees, or disagrees, with the basic idea that 'number is real but not material'.

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Re: The Nature of Number

Post by attofishpi » Fri Aug 10, 2012 4:10 pm

Jonathan.s wrote:Hi all - new poster here.
I thought your name was Jonathan.

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Re: The Nature of Number

Post by hammock » Fri Aug 10, 2012 6:37 pm

Jonathan.s wrote:Anyway, I will leave it there. I am interested to know whether anyone agrees, or disagrees, with the basic idea that 'number is real but not material'.

"Real"(?) interpreted as useful meanings, purposeful / effective instructions attached to abstract quantitative symbols.

"Not material"(?) interpreted as generalization, not concrete instances except for representative symbols. Replace collections of specific phenomenal objects like 'a pebble, a pebble, a pebble' with counts or measurement values that groups of varying things have as in-common properties; I.E., remove empirical content of particular examples of such collections to produce a universally applicable concept. Over time, devise formal system or 'game' of rules, operations, and analysis performed on symbols that output new abstract products that may not always correspond (at least temporarily) to empirically observed entities or circumstances, as was the case with original derivative sources.

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Re: The Nature of Number

Post by ForgedinHell » Fri Aug 10, 2012 6:42 pm

Jonathan.s wrote:Hi all - new poster here.

One of the ideas I have been contemplating concerns the nature of number. Now, as we will see, I am not proposing to explain what number actually is. That, I believe, is actually a very difficult problem in its own right.

The points I want to make about number are simply that number is real, but that it is not material or physical in nature.

Why 'real'? Because (among other things), operations carried out on the basis of the accurate determination of numerical values have real consequences in the world. For example, if I mix the wrong amount of a substance, or calculate the wrong length of the beam in a structure, these mistakes have real consequences, such as explosions, or building collapses. Furthermore, while mathematical systems might in some ways be regarded as conventional or socially-detemined, the outcome of such calculations is not simply conventional. Number is not a 'social reality' or a fashion, again because the determination of the exact value of various measurements have obvious consequence throughout science, engineering, and commerce.

Now - why not 'material'? Well, whatever else number is, it depends on rational ability, the ability to count and to follow logical laws. (Note I am also "maths-challenged" - I was dreadful at school maths, so quite why I have become interested in this topic, I don't really know.) In any case, a number can be represented physically, obviously - by symbols, binary code, words, and so on. But the symbol is not the same as that which is signified. So what is signified? Well, crudely put, a quantity - but it might also be a relationship, or a ratio.

But the point about number is that it does exist anywhere 'externally'. It is not 'in the world'. It is only visible to a rational intelligence which is capable of counting. But to any minds capable of counting, a given number has the same value. (Although there are grey areas - perhaps this is something that can be said, at least, of real numbers and integers. 'God created the integers, all else is the work of man', someone said.)

Anyway, as I said, I am not going to try and explain what number is. Far greater minds than mine have come to grief on that topic. But I find the 'real but not material' nature of number very interesting. I also know from discussing this idea, that such an understanding is very unpopular in the current academe. It sounds awfully Platonist, and empiricism generally disdains Platonism. But there are many Platonists still around, even though they may not wear their heart on their sleeve.

Anyway, I will leave it there. I am interested to know whether anyone agrees, or disagrees, with the basic idea that 'number is real but not material'.
The two incompleteness theorems are Platonic. If man were truly the measure of all things, then we should be able to make up any mathematical system we want, which does not appear to be the case. I also find the Monty-Hall problem more significant than most who encounter it. To me, it appears to be more evidence that there is an underlying logical structure to the cosmos, and that the structure is mathematical. But, this is just the "Jew-boy" in me, who agrees with Einstein and Deutsch and Spinoza....

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Re: The Nature of Number

Post by Satyr » Fri Aug 10, 2012 10:30 pm

Jonathan.s wrote:Hi all - new poster here.

One of the ideas I have been contemplating concerns the nature of number. Now, as we will see, I am not proposing to explain what number actually is. That, I believe, is actually a very difficult problem in its own right.

The points I want to make about number are simply that number is real, but that it is not material or physical in nature.

Why 'real'? Because (among other things), operations carried out on the basis of the accurate determination of numerical values have real consequences in the world. For example, if I mix the wrong amount of a substance, or calculate the wrong length of the beam in a structure, these mistakes have real consequences, such as explosions, or building collapses. Furthermore, while mathematical systems might in some ways be regarded as conventional or socially-detemined, the outcome of such calculations is not simply conventional. Number is not a 'social reality' or a fashion, again because the determination of the exact value of various measurements have obvious consequence throughout science, engineering, and commerce.

Now - why not 'material'? Well, whatever else number is, it depends on rational ability, the ability to count and to follow logical laws. (Note I am also "maths-challenged" - I was dreadful at school maths, so quite why I have become interested in this topic, I don't really know.) In any case, a number can be represented physically, obviously - by symbols, binary code, words, and so on. But the symbol is not the same as that which is signified. So what is signified? Well, crudely put, a quantity - but it might also be a relationship, or a ratio.

But the point about number is that it does exist anywhere 'externally'. It is not 'in the world'. It is only visible to a rational intelligence which is capable of counting. But to any minds capable of counting, a given number has the same value. (Although there are grey areas - perhaps this is something that can be said, at least, of real numbers and integers. 'God created the integers, all else is the work of man', someone said.)

Anyway, as I said, I am not going to try and explain what number is. Far greater minds than mine have come to grief on that topic. But I find the 'real but not material' nature of number very interesting. I also know from discussing this idea, that such an understanding is very unpopular in the current academe. It sounds awfully Platonist, and empiricism generally disdains Platonism. But there are many Platonists still around, even though they may not wear their heart on their sleeve.

Anyway, I will leave it there. I am interested to know whether anyone agrees, or disagrees, with the basic idea that 'number is real but not material'.
Consider this...
Math is a language...and like all languages it refers to a mental abstraction.

What is a mental abstraction?
It is a model constructed using a priori methods combined with sensual interpretations.
A priori methods are established through trial and error - natural selection.
Upon this a priori groundwork sensual input is added.

What are sensual interpretations?
Sensual interpretations are simplifications/generalizations of sensually perceived phenomena, experienced through an intermediate medium, like light, air, matter.
The mind is accosted by stimuli, through the sense organ, which are then gathered and processed, finding patterns in them.

What are patterns?
Repetitive, consistent activities, behaviors.
It is this repetition which makes them comprehensible.
The more sophisticated the mind is the more refined patterns it can perceive or become aware of.

Now what is a numerical value?
It is based on a simple on/off neurological mechanism which creates a dualism. Upon this dualism all human conceptions are built.
From it we get good/bad, threat/no threat, here/there, now/before or after, God/Satan, I/Other...etc.
We also get the absolute states of something and nothing...here "thing" is presupposed as a given self-evident, static truism.

To the some-thing - the ambiguity is evident...we juxtapose its negation, the no-thing.
Thingness remains common and is part of the "logic".
It is a useful presupposition, to say the least.
It offers a center where non is evident.
Math is the most abstract of all languages, perhaps only music is more so.

And as the most abstract it is also the most useful. It can refer to anything at any time on any level.
One world can be reduced to one continent, then to one area, then one forest, then one tree and then one molecule on the tree and so on.
It can also go the other way...one world can turn to one solar system then to one galaxy then one cluster of galaxies then one universe and then, perhaps one cluster of verses and so on ....

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Jonathan.s
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Re: The Nature of Number

Post by Jonathan.s » Fri Aug 10, 2012 11:27 pm

ForgedinHell wrote:The two incompleteness theorems are Platonic.
Indeed.
Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason.
from here.
Satyr wrote:What is a mental abstraction?
It is a model constructed using a priori methods combined with sensual interpretations.
A priori methods are established through trial and error - natural selection.
Upon this a priori groundwork sensual input is added.
I don't believe 'natural selection' provides an explanation for the emergence of reason. I accept that rational ability must have evolved, but to attempt to explain it in terms of adaptive necessity, conflates two different types, or levels, of explanation. It is a category error. Perhaps the way I could put it is that we evolve the ability to see reason. But what is seen by reason already exists.

I can understand that a creature can learn, through stimulus and response, quite intricate behavioural patterns, such as those seen in symbiotic relationships. But I don't see how we can use this mechanism to account for the ability to perceive logical necessity, and the like. There is a radical difference in kind between any number of stimulus and response patterns, and the ability to say 'this must be the case'.

In fact I think this is a big gap in our current explanatory models of the nature of intelligence. We assume that it is something that can be understood through the lens of evolutionary biology, but I don't necessarily agree. This is discussed at some length in Thomas Nagel's The Last Word.

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Re: The Nature of Number

Post by Satyr » Fri Aug 10, 2012 11:41 pm

Then consider yourself a religious mind.

Necessity, of any kind, is founded on experience.

But you are entitled to your faith.
As such, the others will be on your own level of thinking.

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Re: The Nature of Number

Post by ForgedinHell » Sat Aug 11, 2012 12:22 am

Satyr wrote:Then consider yourself a religious mind.

Necessity, of any kind, is founded on experience.

But you are entitled to your faith.
As such, the others will be on your own level of thinking.
I thought that "methinketh" fellow, Hume, showed that necessity can never be deduced by observation?

If there is an ordered structure to the cosmos, and that ordered structure is mathematical, then there is logic embedded in the cosmos. If that is true, then how is that religious? It would simply be the discovery of what nature is. If one assumes before hand that such is the case, then that may be religious; however, given the evidence that we have to date, there is a lot of empirical evidence to support the position, although it has not been "proven," it is still a position that may be taken purely within the realm of science.

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Re: The Nature of Number

Post by Jonathan.s » Sat Aug 11, 2012 12:42 am

I don't think the idea of 'a greater order' is necessarily religious, but it will often be considered that way. And why? I think that the answer to that question is historical. It goes back to the debates between the scholastics and the nominalists. The nominalists were the predecessors of the empiricists. What 'nominalism' means is the rejection of the idea of universals, and the Platonist notion that number is real (see for example Platonism in Metaphysics.) So naturally any such ideas a pre-existing order or objective intelligibility is categorized as 'religious'. This is one of the reasons why today's empiricists and anti-Christian polemicists generally deprecate ideas such as 'natural order' on the ground that such ideas 'sound' religious or evoke a particular kind of religious worldview. (Again, that Nagel book, in particular the essay Evolutionary Naturalism and the Fear of Religion, is pretty good on all this.)

What interested me about platonist notions of number in the first place, was that number (and other 'intelligible objects') does not exist on the purely phenomenal level. The very word 'exist' is incorrect, in relation to things like logical laws, number, and so on. (I don't know what other word ought to be used, but 'subsist' might be one.) And that struck me as extraordinarily significant, from a philosophical point of view.

So, I don't necessarly agree that these questions are within scope for the natural sciences. Science assumes mathematical reasoning to be true. Using mathematical reasoning, science has obviously been able to discover an astonishing range of information. But the question 'what is the nature of number' is a different order of question, to the many things we can discover using mathematics. That is why I posted this in the metaphysics forum!
Last edited by Jonathan.s on Sat Aug 11, 2012 12:46 am, edited 1 time in total.

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Re: The Nature of Number

Post by Satyr » Sat Aug 11, 2012 12:45 am

The only "order" is in your head.

Jew-Boy...you have no offered a valid definition for your basic scripture model: the #1.
As such everything you say or have said should be taken as the ramblings of a religious fanatic who can only defer and except himself and hide behind "authorities"..."so and so said it".
The mind is an ordering tool ..small as yours might be.

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Re: The Nature of Number

Post by Satyr » Sat Aug 11, 2012 12:48 am

Big Bang: Theoretical point of near-absoluteness - singularity...absolute order.

Reality: a towards increasing entropy, disordering....away form order or this theoretical near-point which was obviously not complete.
Reality is constant (inter)activity.

1 = singularity, absolute, order.
0 = negation, absolute nil, disorder.

Both binary notions with no reality outside the human mind: a method of understanding and orienting and categorizing.

Of course you can baptize anything with a name.
You can take this disordering and turmoil and name it "perfect"...but what do you feel.
You are a part of existence...do you feel perfect? Do you not feel need?
Is not need the sensation of lack?

How can the perfect lack?

The problem is solved by the religious fanatic:
It's an illusion....the senses evolved not to aid us but to trick us. the world is a test, a staging point, limbo.

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Re: The Nature of Number

Post by ForgedinHell » Sat Aug 11, 2012 12:56 am

Satyr wrote:Big Bang: Theoretical point of near-absoluteness - singularity...absolute order.

Reality: a towards increasing entropy, disordering....away form order or this theoretical near-point which was obviously not complete.
Reality is constant (inter)activity.

1 = singularity, absolute, order.
0 = negation, absolute nil, disorder.

Both binary notions with no reality outside the human mind: a method of understanding and orienting and categorizing.

Of course you can baptize anything with a name.
You can take this disordering and turmoil and name it "perfect"...but what do you feel.
You are a part of existence...do you feel perfect? Do you not feel need?
Is not need the sensation of lack?

How can the perfect lack?

The problem is solved by the religious fanatic:
It's an illusion....the senses evolved not to aid us but to trick us. the world is a test, a staging point, limbo.
Your first assumptions regarding the Big Bang are way off base. Of course, I have never known an anti-semite to be scientifically literate.

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Re: The Nature of Number

Post by Satyr » Sat Aug 11, 2012 12:58 am

As always a declarative statement with no substance.
Thanks for being predictable, JEW.

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Jonathan.s
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Re: The Nature of Number

Post by Jonathan.s » Sat Aug 11, 2012 1:01 am

I was rather hoping that this forum might be an outpost of civilization. It seems such places are hard to find. :(

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Re: The Nature of Number

Post by Satyr » Sat Aug 11, 2012 1:04 am

Jonathan.s wrote:I was rather hoping that this forum might be an outpost of civilization. It seems such places are hard to find. :(
Are you lamenting an ideal that exists in your mind, wanting to reveal how you do not belong because you are "above" it?

Culture is created, boy....not found.
What have YOU contributed to its realization?

You come here, amongst swine and buffoons expecting what?
I am Satyr...creature of the wood, worshiper of Dionysus, the god of wine.
I dance amongst the pigs and frolic with the bovine.

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