Do numbers exist?

@Kielicious,

Some logics admit of modality, some don't. Ergo,

Quote:

No matter what kind of formal system you use, the same rules appear in all of the systems.

is false. QED. Can we get on with the thread now?

@jeeprs,

jeeprs;83990 wrote:

I say numbers are real regardless of whether they are perceived or not. The relationships that are expressed by numerical symbols have obtained since the moment of creation.

They are not, however, evident, until the intelligence develops to perceive them. So they don't exist in the same way that material objects exist.

---------- Post added 08-18-2009 at 10:15 PM ----------

this is why they are a problem for empirical philosophy - because they don't exist in the same way as an object, and yet they are also not simply a product of mental operations.

If the above is correct, it seems to imply that the quantity of actually existing numbers is infinite. This leads to the controversial question of whether it makes sense to talk of actual infinities.

@Theages,

Theages;84264 wrote:

Some logics admit of modality, some don't.

So you replace one statement by another.
Ok, at least it's your own words.
But i start wondering if this discussion is of any use.

@Exebeche,

Can we at least agree that some features (e.g. the law of non-contradiction) are common to all systems of logic, while others (e.g. modality) are not?

@ACB,

ACB;84325 wrote:

Can we at least agree that some features (e.g. the law of non-contradiction) are common to all systems of logic, while others (e.g. modality) are not?

Once again:

Exebeche;84168 wrote:

If this book is standard, it should be peanuts for you to explain why there is no circular reasoning.
I am still waiting for this explanation.

The point is that theage's post isn't even an answer to my question.
I demanded a reply that confutes my statement of Smith creating circular reasoning.
But theage picked just any sentence out of the context.
That's why i get the feeling that my conversation with the age is useless.
I realised that sometimes people are only in it for prooving that they know something better than others.
And not long ago i decided not to waste any time or energy on that kind of discussions anymore.
Let's see...

@ACB,

ACB;84284 wrote:

If the above is correct, it seems to imply that the quantity of actually existing numbers is infinite. This leads to the controversial question of whether it makes sense to talk of actual infinities.

So how about this:
Let's say a particular language is a particular system based on its own logic.
The grammar is the the formal system that reflects this logic, a description of its logic.
The words are the elements giving a shape to the principles.
(like the figures in a chess game).
Can we transfer this to numbers?
Let's say the universe is the system we observe and since the universe is more complex than a language the formal system to describe its logic is not as easy as a grammar book.
Mathematics is ONE description of the universe' logic, more or less like the grammar for language.
Numbers are elements of this logic, just like words (or chess figures).
Just like every word has its equivalent in reality, every number has its equivalent in reality.
Thus numbers don't have an ontological reality, but they are represantitives of quantities in reality.
Just like you can not really touch the progression of something you can not touch the quantity that a number is related to.
For example the aging that we all are subject to is something that takes place, however the aging is nothing we can touch. We can touch somebodie's skin and realize it has aged. But the aging itself is not attainable for us.
The aging might be subject to matter and energy, however it can only be described by the progression of the system, which itself is not energetical or material. It's time that causes a system's dynamics.
These relations are what is called ABSTRACT.
Abstract is pretty much anything we can not touch (or measure as energy).
That's why the existence of numbers is abstract after all.
Mathematics is the language of nature. Numbers are its words.
They remain abstract.
However reality is not only what we can touch (or measure as energy).
Reality is also defined by the relations caused by time (dynamic).
The relations exist.
Numbers exist in the same way as words. They are reflections of real relations.

@vectorcube,

Exebeche;84389 wrote:

Just like every word has its equivalent in reality, every number has its equivalent in reality

Well this is a version of 'correspondence theory', is it not?

There are a couple of problems with this idea though. The first is that there are absurd numbers, which obviously can't correspond with anything in reality, but which might nonetheless be correct in their own terms. IN fact there are entire realms of imaginary numbers which might be logically correct but not correspond to anything.

The second thing is that I sense a certain circularity in this argument. We use a number of objects to justify the reality of the concept of number. But then, how do we know the concept of number is accurate? Why, with reference to the number objects! It is tautological. There was a contribution hereabout the weaknesses of correspondence theory which quotes a number of sources in support of this criticism. Really it is a kind of naive realism isn't it?

@vectorcube,

In fact I think this discussion is rather like Kant's critique of Locke. This quote from an essay of Emerson's, posted by Theatetus, is relevant:

Theaetetus;39376 wrote:

...the Idealism of the present day acquired the name of Transcendental, from the use of that term by...Kant..., who replied to the skeptical philosophy of Locke, which insisted that there was nothing in the intellect which was not previously in the experience of the senses, by showing that there was a very important class of ideas, or imperative forms, which did not come by experience, but through which experience was acquired; that these were intuitions of the mind itself; and he denominated them Transcendental forms.

@vectorcube,

vectorcube;83471 wrote:

Response from Peter Smith on August 13, 2009 in askphilosopher.org:

Here's a simple argument. (1) There are four prime numbers between 10 and 20. (2) But if there are four prime numbers between 10 and 20, there must be such things as prime numbers. (3) And if there are such things as prime numbers, then there must be such things as numbers. Hence, from (1) to (3) we can conclude that (4) there are such things as numbers. Hence (5) numbers exist.
Where could that simple argument be challenged? (2) and (3) look compelling, and the inference from (1), (2) and (3) to (4) is evidently valid.

So that leaves two possibilities. We can challenge the argument at the very end, and try to resist the move from (4) to (5), saying that while it is true that there are such things as numbers, it doesn't follow that numbers actually exist. This response, however, supposes that there is a distinction between there being Xs and Xs existing. But what distinction could this be?
Well, someone could use mis-use "exists" to mean something like e.g. "physically exists": and of course it doesn't follow from there being Xs that Xs physically exist. But it isn't physical existence that is in question when we ask whether numbers exist -- trivially, they aren't the kind of thing you can weigh or stub your toe on! It's granted on all sides that numbers are not physical things.

But once it is clarified that "exists" is not being used in a restrictive way to mean, e.g., physically exists it is very difficult to see what the supposed distinction is supposed to be between there being Xs and Xs existing. Certainly, few modern philosophers believe that there is such a distinction to be drawn. (In a slogan, they think that existence is indeed what is expressed by the so called existential quantifiers, 'there is', 'there are'.) So we'll set aside this challenge.

The other option is to challenge the initial assumption (1). But how can that be done? 11, 13, 17, 19 are the only prime numbers between 10 and 20, and so there are four prime numbers between 10 and 20. That's a simple truth of arithmetic, surely.

But ah, you might say, we need to distinguish: it's certainly true-according-to-arithmetic that there are four prime numbers between 10 and 20, just as it is true-according-to-the-Sherlock-Holmes-stories that Holmes lived in Baker Street. But it isn't really, unqualifiedly, true that there was a man called Holmes living there, and it isn't plain true either that there are four prime numbers between 10 and 20. And from the granted assumption that it's true-according-to-arithmetic that there are four prime numbers between 10 and 20 the most we can reasonably infer is that it should (given arithmetic is a consistent story) be true according to arithmetic that numbers exist. And that doesn't show that numbers really do exist.
This sort of "fictionalist" line that treats arithmetic claims as if claims within a story (the story of arithmetic) has its warm supporters. They will deny that numbers really exist, but at the high price of denying that arithmetical truths are plain true (a reversal, then, of the traditional philosophical ranking of mathematical truths as the most secure truths of all!). If you are not willing to pay that price, and are also not willing to play fast and loose with a supposed distinction between there being numbers and numbers existing, then the simple argument above for the conclusion that numbers exists will look rather compelling.

interesting question. numbers exist as long as we, humans, all agree on them. for example, money isn't real. it only exists in our reality. we make it real. if tomorrow, everyone didn't believe or accept in the notes we use for money then they would just be paper with funny drawings covering them.

@vectorcube,

The whole argument is about whether what you are saying is true or not. And I don't believe it is true. I think that numbers signify a reality which is not dependent on our belief or acceptance. It is not just 'made up' nor yet a convention. Yet numbers also do not exist without being counted. That is what is special about them. It is kind of magical, really.

@jeeprs,

jeeprs;84479 wrote:

The whole argument is about whether what you are saying is true or not. And I don't believe it is true. I think that numbers signify a reality which is not dependent on our belief or acceptance. It is not just 'made up' nor yet a convention. Yet numbers also do not exist without being counted. That is what is special about them. It is kind of magical, really.

The passage in red does not sit easily with the one in green (though I would not go so far as to say there is a contradiction). What exactly is the status of very large numbers that have not yet been counted? Do they have a kind of shadowy, potential existence? For the sake of clarity, I would prefer to say that at any given time there only exist (a) already counted numbers, and (b) rules for counting further numbers. Only when a number has actually been counted does it come into existence. So I basically agree with the passage in green, while being somewhat doubtful about the passage in red.

Regarding the passage in red, you could argue that numbers exist independently if they represent some actual physical quantity in the world or universe. But things get more complicated if you have to include numbers representing such things as possible permutations rather than just physical objects.

As far as 'counting' is concerned, I think this term needs to be construed broadly as meaning that a specified number exists if it could be reached in a finite number of steps, even if these steps are not literally counted one by one. But even this principle is problematic beyond a certain point, because mathematicians sometimes use numbers so large as to be unimaginable, so that the idea of counting up to them step by step cannot be even vaguely grasped.

These are just some of my thoughts. I have no dogmatic views on the matter.

@vectorcube,

Jeeprs hit it pretty well i think.

The quantities we describe with numbers (those objects involved) exist independently of any counting/calculating we do. But that's not to say that 'numbers' have any similiar objective existence. Numbers is a concept we made up to represent varying quantities - that's all - they're representations. Does "fast" really exist? Does "green"? Does "fat"? It's a correlation to a particular aspect of the general; in this case, the quantities involved.

These concepts, terminologies and systems we construct help to solidify ideas in our minds. To me - and I'm no expert mind you - asking "do they really exist" is a question that doesn't fit it's subject/object.

My View... thanks

@jeeprs,

jeeprs;84399 wrote:

Well this is a version of 'correspondence theory', is it not?

There are a couple of problems with this idea though. The first is that there are absurd numbers, which obviously can't correspond with anything in reality, but which might nonetheless be correct in their own terms. IN fact there are entire realms of imaginary numbers which might be logically correct but not correspond to anything.

The second thing is that I sense a certain circularity in this argument. We use a number of objects to justify the reality of the concept of number. But then, how do we know the concept of number is accurate? Why, with reference to the number objects! It is tautological. There was a contribution hereabout the weaknesses of correspondence theory which quotes a number of sources in support of this criticism. Really it is a kind of naive realism isn't it?

Actually i reject naive realism; if you want to call it a realism it would be critical realism - however not in the sense it is defined on the english wikipage.
The german wiki defines critical realism like this:
'Critical realism is a philosophical, in particular cosmological and epistemological position which assumes that there is a real world which corresponds with our perceptions, but (as oppose to naive realism) by the means and ways of human perception it is not immediately and directly recognizable to what extent its appearance caused by human processing matches it. ' [...matches reality (my own translation)]
In other words we carry images of reality which are distorted reflections as oppose to perfect reproductions.
As you can see this is pretty close to some kinds of constructivism. All of this is of course based on the preassumption that an objective reality exists (I am open to critics of this preassumption, however for this discussion i take an objective reality as given fact).
The way i see it is like this: Imagine the inside of our skull is a cinema screen.
Whatever we hear or see or perceive in any other way from our environment flows into this complicated system of mirrors which is called our brain.
Finally the mirrors produce a reflection that is projected on the inside of our skull.
We believe we see the outside world but actually all we ever perceive [edit: observe] is the projection on the inside of our skull.
The objects in this projection correlate very well with the outside reality (at least most of the time), that's why we have this deep trust that we DO see the outside world.
But still we only experience reality from the projection on the inside of our skull.
Since material things are reflected very adequately in this system, the different projections inside the skulls of the individuals differ only slightly at least in terms of material reality.
This is why they agree that they all see the same reality. What they can not see is what the differences in other projections look like.
So as you see i am pretty much on the side of constructivists.
Every individual holds his own projection of reality, which is the individual's construction.
(Actually i believe that the individual has little freedom of shaping his construction, but let's not get to complicated.)
Khethil kind of took the words out of my mouth, only he does it in a more sophisitcated way, while i try to master philosophy in the most simple words and pictures.
One of the major features of our brain is to recognize relations (might have something to do with the necessity of pattern recogntion).
However it does not only recognize them, it also creates them.
For example when a child says 'table' it refers to eating and a flat surface. You may not see the table, but when your child puts his toy dishes on a rock and starts having a picknick you realise what makes the rock a table.
A table is pure matter, consisting of atoms, no matter if you use an artificial table or a rock. The only thing that makes it a table is you when you connect things like eating and a flat surface.
On the inside of your skull there is a table, while outside there's only matter and energy.
Does the table exist? Has the rock always been a table when your child calls it such?
The fact that nobody else has seen the table before, simply makes your childs reality more individual and less universal.
Some observations seem to be totally universal, e.g. Newton's laws of mechanics and his axiom that time and space are completely fixed.
Then Einstein comes and shows his laws were not as universal as we thought.
The questions wether numbers exist or not leads to the question "Does anything abstract exist?"
The question about numbers is more interesting because the existence of numbers seems somehow solidely proven because they seem to affect reality in a foundational way.
Apart from the fact that our mind creates distorted reflections of reality another one of its interesting qualities is to show that reality is constituted by more than matter and energy: Non energetical relations.
What makes a particular monkey an alpha mail? This is something that can not be expressed [edit: described] physically.
You have to take into consideration the relation of the animals. In such a relation system a monkey baby can be as important as the alpha mail or any other participant.
Some observations are more universal than others (ignoring the fact that the word 'universal' is absolute).
A number is a reflection on the inside of our skull representing a relation that has its equivalent in reality, just as a child's table is a reflection on the inside of his or her skull representing a relation that has its equivalent in reality.
The number itself does not have an existence.
The relation however does.
Numbers are equivalents of relations in reality that are so logically reduced that they seem to fit every reference system.
That's what makes them appear universal.
And the more universal an observation is, the more it is accepted as real.

@ACB,

ACB;84486 wrote:

The passage in red does not sit easily with the one in green (though I would not go so far as to say there is a contradiction). What exactly is the status of very large numbers that have not yet been counted? Do they have a kind of shadowy, potential existence? For the sake of clarity, I would prefer to say that at any given time there only exist (a) already counted numbers, and (b) rules for counting further numbers. Only when a number has actually been counted does it come into existence. So I basically agree with the passage in green, while being somewhat doubtful about the passage in red.

Well the point I am working on is that numbers don't ever 'exist' in the strict sense. They express a reality about existence, or the structure of existence. That is why logic, numbers and rationality are very difficult to contemplate; they are the very fabric of existence, and we use them to think. So when we come to give an account of what they are, it is like the hand trying to grasp itself - it is a problem of recursion. (Of course the pre-moderns would attribute rationality and lawfulness to the action of divine intelligence which created both the world and us. But this perspective may not be available any more.)

Since going down this route I have done a bit of tentative reading of 'philosophy of mathematics' - it is a difficult subject and I don't think I will ever master it in its modern form. I am basically going with an intuition in all of this, which is by no means fully articulated or worked out. I think it all goes back to Platonism, or perhaps even Pythagorism. It is the idea of the Form, of which number is a simple example.

So the question as to where numbers exist is like the one usually asked about Platonism, 'where do the forms exist' and again the answer is - they don't exist, instead they are expressed, imperfectly, by existing things. Hence 'transcendent forms'. So the realm of forms is not 'anywhere' - there really is no such place and it can't be conceived or thought about, perhaps again because of the problem of recursion. I am beginning to suspect that this idea that science has hit on that all these specific ratios and numbers - actually a very small number of them - had to be 'just so' in order for the universe to evolve - isn't this the same idea? According to Plotinus, existence 'emanates from the One' according to these forms and ratios (which is where the whole idea of rationality comes from).

You see, this is very basic to Western philosophy, prior to the empiricist revolution. I feel that the West has lost something of great importance in going down this naive realist route. This is because scientists wanted to throw out metaphysics and start again with 'what is really there'. And what they have discovered - and this should be a source of acute embarassment to them - is that nothing is 'really there'. Reality is not what you see when you look out the window. And I think any decent medieval philosopher, and even the early moderns, could have told them this centuries ago. Hence all the angst and throat clearing about quantum mechanics.

So, unlike Exebeche, whose approach I nevertheless very much appreciate, I think I am heading for 'critical idealism'. But I have much more study and contemplation to do yet, there are only small parts visible at the moment. So it is not a conclusion, but a beginning, I am actually just beginning to discover philosophy through this topic. I will keep working on it.

@jeeprs,

jeeprs;84574 wrote:

So, unlike Exebeche, whose approach I nevertheless very much appreciate, I think I am heading for 'critical idealism'.

I am just trying to find out the meaning of 'critical idealism' and i end up being more confused than before.
Can you help me out, what's the meaning of 'critical idealism'?

@vectorcube,

I will be developing that idea, but have to do some more reading first. Briefly, as with all idealism, it posits 'intelligence' or 'consciousness' as the first cause or ground of being, however it avoids solipsism by realising that consciousnesss is a collective reality, not the activities of an individual mind (e.g. 'pan-psychism'). In this philosophy 'consciousness' is actually another word for 'spirit' but spirit is not a word I like much because it has connotations of 'substance' or 'ethereal reality'. This kind of thinking has caused a lot of problems in Western philosophy. I am working on a non-dualist approach but it is quite difficult to articulate.

I am waiting for a couple of books I have ordered from Amazon and will have more to contribute later.

@vectorcube,

My sense of it would be numbers exist in the realm of the mind (idealism). In the same realm or manner that logic and reason exist. A lot of mathematics is reducible to logic (although Russell and Whitehead failed in their effort to reduce all of mathematics to logic). All of these constructs number, logic and reason are not experience or sense dependent they are inherent features of the mind. Numbers (mathematics) somehow inherently expresses the structure of the universe. The concept of number is not at all dependent on having a material object to correlate with.

In fact the case could be made for a form of idealism, the structure of material objects is dependent on the ideal forms of number as previously suggested. (Plato's notion of forms or eternal objects suits the concept of number quite well). I am a theist of sorts so I have my own notion of where the forms come from but to say that numbers do not exist because they are products of the mind instead of products of matter is to look at the problem backwards. IMHO

@Khethil,

Khethil;84489 wrote:

Jeeprs hit it pretty well i think.

The quantities we describe with numbers (those objects involved) exist independently of any counting/calculating we do. But that's not to say that 'numbers' have any similiar objective existence. Numbers is a concept we made up to represent varying quantities - that's all - they're representations. Does "fast" really exist? Does "green"? Does "fat"? It's a correlation to a particular aspect of the general; in this case, the quantities involved.

These concepts, terminologies and systems we construct help to solidify ideas in our minds. To me - and I'm no expert mind you - asking "do they really exist" is a question that doesn't fit it's subject/object.

My View... thanks

Awesome, as I was reading the thread I was thinking about how I was going to say what you just said, and now I don't have to. Its a good thing too because this is more concise than I probably would have managed.

Now then..

I view mathematics as a kind of philosophy. The reason it is so successful is because we can clearly and precisely conceive of the ideas involved (those concerning quantity,) which allows us to examine the relationships between them with greater regularity.

Even though we can only form clear and functional ideas of limited quantities, the principles we form from examining the relationships between these ideas can be successfully applied to larger quantities without the need to properly conceive of them. This transition between knowledge and belief is common to any pursuit of truth, but it is hardly noticable in mathematics thanks to how solid our conception of its principles are. In fact I am sure many who read this would deny that such a transition between knowledge and belief really occurs.

I think any branch of philosophy could achieve this level of precision and applicability if only the ideas which it reasons concerning could be conceived as distinctly, and the relations between them defined as clearly.

[edited for redundant language]

@vectorcube,

Well Bertrand Russell would agree with you there.

Quote:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture.

I think this was a major factor behind the rise of analytical philosophy, Russell, Frege, and those kinds of thinkers. But the problem is, for me, they lost the mystical element which was a crucial element in Platonism. They had no spirituality about them. I like the way that Whitehead ended up developing the ideas but have yet to read him. But from what I understand I think he is probably the best representative of philosophical idealism in the Western tradition in the 20th Century.

@jeeprs,

jeeprs;84632 wrote:

Well Bertrand Russell would agree with you there.

I think this was a major factor behind the rise of analytical philosophy, Russell, Frege, and those kinds of thinkers. But the problem is, for me, they lost the mystical element which was a crucial element in Platonism. They had no spirituality about them. I like the way that Whitehead ended up developing the ideas but have yet to read him. But from what I understand I think he is probably the best representative of philosophical idealism in the Western tradition in the 20th Century.

Yes, everyone should read Whitehead. Process philosophy and process theology, a philosophy which trys to integrate science with values, religion and aesthetics. The terminology is dense and difficult but the concepts are very insightful. Whitehead once said that all philosophy is merely commentary on Plato. Whitehead much preferred Plato an idealist to Aristotle who was more empirical.

---------- Post added 08-20-2009 at 10:32 PM ----------

I think the question about "do numbers exist" hinges on ones concept of what it means to "exist". Empiricists and materialists are probably going to argue numbers have no "real" existence. Those of us who are idealists, rationalist or believe in transcendental forms of existence are going to argue the affirmative. Thus the answers given are dependent on more basic philosophiical assumptions or metaphysical speculations about "existence" ontology.

Personally I think empiricism and materialism give a much too limited and very incomplete view of reality and experience. I do not think one can merely brush aside the elegance, beauty, power and explanatory value of mathematical presentations of nature. For me it almost seems mathematics preceeds nature not the other way around.

Philosophy Forum

Do numbers exist?

Quick Links

My Account

able2know