The Physicalization of Numbers – Can we conceptualize numbers as physical objects? — ucarr

No. They're only graspable by an intelligence capable of counting. They're intelligible objects.

Material Numbers – because a material object can hold a position, perhaps we can understand that any material object has a built-in property of number. The number property of a material object is its ability to afford a slot wherein a set of possible numbers gives value to its position within an array of other material objects in interrelationship. — ucarr

Did you ever play battleship? The numbers are part of the positional grid, not properties of the object.

Can someone please explain the OP to me ? I can't understand it.

I think the OP is referring to an argument where numbers are represented by a physical/material concept, not a metaphysical one. At least, he said Wittgenstein had some arguments in this points

Can someone please explain the OP to me ? — Hello Human

You pick up a rock & it weighs 1 pound.

You pick up another rock & it weighs 4 ounces.

The second rock weighs only 25% of what the first rock weighs. Holding each rock feels different because of their different weights.

Rock 1 pulls down on your left arm harder than Rock 2 pulls down on your right arm.

Weight, as you know, is a physical property of each rock. The weight of each rock gives you an impression of the identity of each rock.

I'm saying that another way to get an impression of the identity of the two rocks is by putting them into a line with other rocks & then counting up the total number of rocks.

Instead of the weight of the two rocks being experienced by you by holding them & feeling how hard they pull down on your arm, the number of the two rocks is being experienced by you by putting them into a line of other rocks & experiencing how the counting of the line of rocks changes after adding the two rocks.

With this idea, I'm just repeating to you things you already know.

What is slightly different here is how I'm asking you to look at what you already know.

Instead of looking at a number as a thing way over there, while a rock as another thing way over here, I'm asking you to look at a rock as being a physical number made of material we call granite or agate or diamond or (you fill the blank).

Are you saying the positional grid, a material thing, possesses the property of number?

No. They're only graspable by an intelligence capable of counting. — Wayfarer

What does an intelligence grasp when it counts?

I say an intelligence grasps a material thing, as when it counts a line of stones, en route to understanding numbers & counting.

Even a written number symbol, let's say, ink on paper, exists as a physical thing as, in our example, ink on paper.

I say an intelligence grasps a material thing, as when it counts a line of stones, en route to understanding numbers & counting. — ucarr

The stones are material - well, according to materialism - but the count, the quantity, is not. What the intelligence grasps is number. Ink on paper is physical, but what it signifies may not be. And the fact that the same information can be represented by different symbolic forms demonstrates that the meaning and physical representation are separate. And what about pure math? Nothing physical is involved at all in that.

You're barking up the wrong tree, wanting to justify the reality of number by saying it's material. I'm very interested in mathematical Platonism, and my view is in line with traditional platonism, i.e. numbers are real but not material. They can only be grasped by an intelligence capable of counting, but they are the same for all who count.

See the discussion in What is Math?

“I believe that the only way to make sense of mathematics is to believe that there are objective mathematical facts, and that they are discovered by mathematicians,” says James Robert Brown, a philosopher of science recently retired from the University of Toronto. “Working mathematicians overwhelmingly are Platonists. They don't always call themselves Platonists, but if you ask them relevant questions, it’s always the Platonistic answer that they give you.”

Other scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?

Note referece to 'the fear' - this fear is real and profound. If number is real but not material, then materialism is false, and as mainstream culture assumes that it's true - well, that's a big problem for it.

An alternative to the Platonist view is Aristotle's 'moderate realism'. It holds that numbers (etc) are real, but only encountered in relation to material particulars (although again this has a problem accounting for pure math in my view.) See Aristotle was Right about Mathematics After All.

See also The Indispensability Argument in the Philosophy of Mathematics.

Also Sir Roger Penrose, Is Mathematics Discovered or Invented?

It seems that numbers are generalized extrapolations from ( mostly visually ) encountered objects of the senses. Over there to the right I see two lamps together and over there to the left I see one lamp by itself. And there further left I see one wood heater. What does it have in common with the lamps? There is only one woodheater just as there is only one lamp to the left of where I sit.

In my fruit bowl I see three oranges. What do they have in common with the lamps? Taken all together there are three lamps just as there are three oranges. We see number everywhere in the environment. We don't see numbers, we don't see abstract ones or twos anywhere. We have developed a system of symbolic numerals to represent numbers; our recognition of those symbols and their relation to the number that is everywhere in the environment is the closest we can get to seeing an abstract number.

What is a number without a material referent?

It's just another material thing, but unlocateable.

Number = position. Only material things can have position.

What is counting without a world of material referents?

It's just a series of neural networks oscillating.

In a world without spacetime, do numbers have any meaning?

What is a number without a material referent?

It's just another material thing, but unlocateable. — ucarr

It's not a material thing. It's an idea. You know what a number is, because you're h. sapiens. But your dog, if you have one, does not.

You should read some of those refs I gave, or come with some other argument, because this one's a dud.

Looks like you see numbers as I do.

I didn't say an abstract conception of a number is a material thing. I implied it is a sign that has a material referent.

I'll go to your references. Will you go to my world devoid of spacetime and think about the role of numbers there?

Will you go to my world devoid of spacetime and think about the role of numbers there? — ucarr

There's not a lot to go on based on what you've said, but if by that you mean: are numbers real in the absence of reference to space-time?, my response would be again: 'well what about pure mathematics'?

(I hasten to add, I'm appallingly bad at mathematics, unlike several accomplished contributors here, but I don't think the point I'm trying to make necessarily depends on that.)

I recall an anecdote I read decades ago about Arabs who used to play chess whilst riding camels across the desert - without a board. That level of mental acuity and recollection also far exceeds anything I'm capable of, but I remember thinking at the time that it was a sterling example of the power of abstract thought.

Yes. Several philosophers agree with you.

One, Descartes believes quantity is proper of physical multiplicities in the sense of extension:

8. A thing that has a certain quantity or number isn’t •really distinct from the quantity or number—all that’s involved is •distinctness of reason. [See 1:62.] There is no real difference between quantity and the extended substance that has the quantity; the two are merely distinct in reason, in the way that the number three is distinct from a trio of things. ·Here’s why they have a distinctness of reason·: Suppose there’s a corporeal substance that occupies a space of 10 ft3—-we can consider its entire nature without attending to its specific size, because we understand this nature to be exactly the same in the whole thing as in any part of it. Conversely, we can think of •the number ten, or •the continuous quantity 10 ft3 , without attending to this particular substance, because the concept of •the number ten is just the same in all the contexts where it is used, ten feet or ten men or ten anything; and although •the continuous quantity 10 ft3 is unintelligible without some extended substance that has that size, it can be understood apart from this particular substance. ·And here’s why they aren’t really distinct·: In reality it is impossible to take the tiniest amount from the quantity or extension without also removing just that much of the substance; and conversely it is impossible to remove the tiniest amount from the substance without taking away just that much of the quantity or extension. — Descartes

From the second part of Principles of Philosophy.

And in fact Avicenna had a similarly veined argument, a sort of identity between quantity and ostensibly the extension it "models," to forward against the atomists (not the same as the Greek tradition, rather, atomists only in the mereological sense) of his era. That is, because, Aristotle's critiques, which more primarily concern the broader metaphysical doctrine rather than the specific atomistic thesis, Epicurus had developed atomism a tad bit more and the Arabic Kalam tradition had numerous proponents of atomism (who were only considered with the thesis of a discrete space rather the Greek doctrine in full). Since they'd take the notion of discreteness to be a literal representation of space, Avicenna argued that the functionality of the Pythagorean Theorem in physical space, and its utter incompatibility with discreteness even at a level of approximation, disproved atomism.

Finally, in more recent literature, material numbers have been a charge utilized to dispense away various logicist accounts of the philosophy of mathematics. This is known as the "Julius Caesar problem," namely that in accounts where numbers are objects, the principles to which we designate numbers can't tell us when exactly something is or isn't a number. What I mean is, take something like Hume's Principle, we wouldn't know when to identify an arbitrary object, i.e. Julius Caesar, with a number.

The number of a material object is then a kind of measure of the built-in positionality of a material object.

However, this bit of your post, I don't understand. Perhaps instead of the very extension of the object itself, you're referring to spatial or temporal location, if I understand you correctly. If so, this is an example of an extrinsic, relational predicate (or an impure property). There is debate about whether these are truly proper to the objects they designate or not, but here, the quantity would moreso be a reference of things like physical distance (presumably "material"). The way this would be treated would depend on if the underlying metaphysics for space & time is absolute or a container, i.e. Platonic or Newtonian, versus if they're reductionist, like Leibniz, Hume, or Einstein (how they are treated right now in modern physics!) So there's that.

Proceeding from here, perhaps we can characterize math as a property of material objects inhabiting the neighborhood of epiphenomenon.

While I can tell you that it's unlikely you'll be able to account for all of mathematics this way, I shall inform you as to a view known as structuralism in the philosophy of mathematics, where mathematical ideas are really just structures that are instantiated by various things. So physical systems, given satisfying conditions, can instantiate mathematical structures just as ideas in our heads can. However, there are various forms of structuralism with various committal status, and I'm not sure how you would tame those with a materialist doctrine (not saying they're incompatible at all, rather just making a comment WRT my knowledge on the subject).

Are you saying the positional grid, a material thing, possesses the property of number? — ucarr

The positional grid is not a material thing, it is an abstract. And numbers are also abstract ideas that have physical representations or icons so that dumb people like most of us can use the concepts.

Just like an word does, they represent ideas. Writing words and numbers down does not make them physical objects, it just makes it easier to transmit ideas.
Try playing telegraph with a few friends using a complicated idea, see if the idea can be passed on accurately.

Will you go to my world devoid of spacetime and think about the role of numbers there?
— ucarr

There's not a lot to go on based on what you've said, but if by that you mean: are numbers real in the absence of reference to space-time?, my response would be again: 'well what about pure mathematics'? — Wayfarer

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics.

...the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

...presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference than a rigid subdivision of mathematics. -- The Apple Dictionary

It seems to me -- especially in light of paragraph 3 above -- that pure math, in order to be intelligible beyond the circular reasoning of logical truth by definition, must trace back to material objects interrelated.

...almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. -- The Apple Dictionary

...intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles

Logic is continuity, which is to say, interrelationship, rooted in inference. Would anyone have any notion of continuity & interrelationship between material things without firsthand experience of a spacially-extended, material world that affords empirical experience?

Pure math, and all other forms of signification, once uncoupled from empirical experience, become unintelligible.

Numbers, uncoupled from interrelated material objects, become random, unable to signify anything intelligible.

Abstract thought is non-specific WRT our material world; it is not uncoupled from our material world.

So physical systems, given satisfying conditions, can instantiate mathematical structures just as ideas in our heads can. — Kuro

Abundant thanks with much gratitude to you, Kuro. Your input here is tremendously substantial and, I presently like to think, more encouraging than otherwise. There's much in your input I must study further. If warranted, I hope additional input from you is forthcoming.

The positional grid is not a material thing, it is an abstract. — Sir2u

If I remember Battleship correctly, there is a plastic platform full of holes, the grid where a player's battleship moves to various positions.

Writing words and numbers down does not make them physical objects, it just makes it easier to transmit ideas. — Sir2u

No argument with you here. Yes, number symbols & words are signs that refer to material things.

I'm saying that number symbols refer to & derive meaning from material things whose set of attributes includes one particular attribute I call number. All of this verbiage is an attempt to say material objects are numericalizable because they have a built-in property of being movable, which is to say, positionable.

Pure math, and all other forms of signification, once uncoupled from empirical experience, become unintelligible.

Numbers, uncoupled from interrelated material objects, become random, unable to signify anything intelligible. — ucarr

These are very rigid statements that are beliefs, not facts. You should indicate as such. Should a philosopher state their beliefs as facts?

These are very rigid statements that are beliefs, not facts. You should indicate as such. Should a philosopher state their beliefs as facts? — jgill

The key word in my statement is signification. I'm tempted to argue that my claims are true by definition, since a sign without a referent is like a material object without elements or compounds. I sense, however, that is a weak argument.

As for my etiquette as a person making claims (you compliment me with the title of philosopher), your response exemplifies what it denounces.

Saying,

These are very rigid statements that are beliefs, not facts. — jgill

Is like saying,

This statement is false.

What fun is philosophy without bold claims subject to refutation?

By the way, what is your refutation of my bold claims?

If I remember Battleship correctly, there is a plastic platform full of holes, the grid where a player's battleship moves to various positions. — ucarr

Before that was invented we used to play on 2 pieces of paper. Draw lines with a pencil to form a grid, letters on the top numbers down the side. But they decided to make easier and dumb down the people. But they both are only physical representations of concepts. The location C12 is not a physical thing, ask any kid that learns about map coordinates or Excel in school.

Yes, number symbols & words are signs that refer to material things. — ucarr

Could you then please indicate to me where I can appreciate a nice looking number 69. I doubt that you can.


This is the image of a Greek road sign. Is it a physical number? No, it is only the paint on a piece of metal that has been manipulated so at to form what is commonly accepted as the visual representations of the numbers 6 and 9. The paint is physical, the metal is physical but the idea is still an abstract.

Numbers do not refer to physical things either, they refer to the quantity(another abstract) of physical objects.

I'm saying that number symbols refer to & derive meaning from material things whose set of attributes includes one particular attribute I call number. All of this verbiage is an attempt to say material objects are numericalizable because they have a built-in property of being movable, which is to say, positionable. — ucarr

645,798,845,635,345,665,332,313,464,655,876,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

If numbers derive meaning from material things, then the number above either has no meaning or a meaning that is unknown. Which do you think it is.

Numbers do not represent objects they specify the quantity of objects, the length of object, the weight of objects. But not the objects themselves.

I recall an anecdote I read decades ago about Arabs who used to play chess whilst riding camels across the desert - without a board. — Wayfarer

I'm reading this book on Mathematics, The Art of More it's called, by a Michael Brooks. It gives an account of how expert abacists can work sums with imagined/simulated abacuses. I find that quite fascinating, don't you? Do you think I can simulate a fully operational computer in my brain; you know, turn the idea of computer simulation on its head by reversing the roles - instead of a computer simulating a mind, why not let the mind simulate a computer for a change. Do you see any relevance/importance of this in the philosophy of mind?

I know Michael Brooks' writing, but hadn't heard of that book - looks very interesting though. I'm not sure of the feasibility of simulating a computer in one's mind - you would have to have quite an extraordinary mind to do that, something which I lack.

Pure math, and all other forms of signification, once uncoupled from empirical experience, become unintelligible. — ucarr

Each day recently arXiv.org has received about 200 research papers in math. Many of these are "uncoupled from empirical experience", yet thousands of math people find them intelligible. However, the general public will not.

Numbers, uncoupled from interrelated material objects, become random, unable to signify anything intelligible. — ucarr

A good calculus student might disagree.

But they both are only physical representations of concepts. — Sir2u

The connection you name above goes in both directions.

Concept - Philosophy - an idea or mental picture of a group or class of objects formed by combining all their aspects. -- The Apple Dictionary

Consider a) Mental representations of material objects; b) material representations of concepts

Which comes first?

Even when we form concepts of mental things i.e., concepts of concepts, the line of reality traces back to material objects within our empirical world.

Pure math is a language about how the language of math works logically.

What is logic? It is an examination of the continuity that connects events of our lives into an intelligible narrative. A narrative is intelligible when a group of people all recognize noteworthy actions & reactions of humans bound together within cause-and-effect relationships.

Apart from our conscious experiences within our daily world, logic has no intelligible meaning or value.

Since math & logic are interwoven into our daily experiences, a study of how math works logically, pure math, likewise is intimately interwoven into our daily experiences.

You should ask a pure mathematician whether their mind enters an immaterial realm while they're working.

Each day recently arXiv.org has received about 200 research papers in math. Many of these are "uncoupled from empirical experience", yet thousands of math people find them intelligible. — jgill

As you may have seen in my statement to Sir2u above, pure math is concerned with the innate workings of the language of math itself. Any brief survey of the sciences with show you just how worldly is math in application to many, many real world events. An examination of how this language works by its own lights is thus an examination of our real world, although not directly.

I know Michael Brooks' writing, but hadn't heard of that book - looks very interesting though. I'm not sure of the feasibility of simulating a computer in one's mind - you would have to have quite an extraordinary mind to do that, something which I lack. — Wayfarer

Good to know. Simulation, aka...acting: Data (Star Trek), Robocop, Terminator, etc. humans acting like computers (AI).

Humans can act like computers but can computers act like humans?