Friday, December 30, 2005

Do numbers exist?

What would you say if I were to ask you, "do numbers exist"? You might say yes: we can count things so there is such a thing as number. On the other hand, you might say no. There are objects but there are no numbers; we make up numbers to make life easier.

This is a big ongoing debate in the philosophy of mathematics. What is maths? Nominalists say that its like a game of checkers. There are pieces (numbers) and there are rules (methods of calculation) but there is no meaning. This view is not widely accepted. If maths was just a made up game, then why does it have such useful applications in the real world? And why does each culture end up following the same rules?

A more plausible position is Aristotelianism. This view says that numbers exist as concepts. So we can't see them in the real world but they still exist because they exist in our minds. But what would happen if there were no people in the world, and no other intelligent life which could count? Would numbers still exist?

Aristotelianism is actually part of a bigger debate over the Universals. It tries to answer the following question: if all red things in the world were destroyed, would the colour red still exist? Aristotelianism says yes! The colour exists in our minds as a concept. But this faces the same problem as before. What happens if there are no minds to hold the concept? Does the concept die?

A popular position in the debate over the existence of numbers is platonism. This view holds that numbers and colours and all other properties (any adjectives you can think of) exist in a separate world - plato's heaven. So mathematics has an abstract existence. Unfortunately, this view is also problematic. If numbers do exist in this separate realm, then how do we know about it? Usually, if we know something, we had some sort of causal connection with it. For example, if I said that I knew there was a chair in the room, I would know because I saw it with my own eyes (or someone else who saw it told me about it). But in the case of abstract objects, we have no causal connection. Therefore, if numbers are abstract objects, we cannot know about them.

But we do know about them. We have an established discipline called mathematics. There is one more thing that is worth mentioning about platonism. It is called the Indispensibility Argument. It was formulated by Quine-Putnam and basically says that mathematics must exist because it is indispensible (it cannot be eliminated from) science. Field tried to show that Newton's gravitational theory could be proven without mathematics but it is doubtful whether he has succeeded in his endeavour. Even if he has, it is even more doubtful that he could achieve the same for quantum theory (our best theory of very small things). So it seems that mathematics must exist. But where?

Maddy has recently attempted to solve the problem. She suggests that mathematical entities have an abstract existence, but they do not reside in a separate realm. Rather, they are down here, with us. This solves the problem of our knowledge of mathematics. Maddy says that we know about mathematics because we have causal connections with the abstract objects. When we see three eggs for example, we can see the "threeness" as a set (group of objects). She attempted to prove this by showing that we have set-perceptive mechanisms. Also, once we understand the concept of number, we are able to deduce the rest through our use of logic.

So we have finally arrived at a view which allows for the existence of numbers as abstract objects and explains our knowledge of them. Please feel free to challenge this view...

29 comments:

Jeong said...

I think the Plato's Heaven concept works best.

The problem of how we get to know of numbers if they reside in another realm is resolved if you consider your mind to also reside in Plato's heaven.

- that raises the question of how we know about anything in the objective reality of the real world Out-There if our minds dont reside there: and I think we dont really know for sure.

It would seem that our bodies reside in the objective universe out-there and there is a causal relationship between the out-there and our senses. We hope that our sense data is not deceiving us on the truth of what's happening Out-There - but we cant escape the dubious but unignorable possibility that we're all in a dream, or we're living in the Matrix or we're a Brain In a Vat or Descartes' Demon is deceiving us etc.

Anonymous said...

numbers do not exist but when you apply a number to a solid object then the number does exist. eg: you can touch and feel solid objects you cant touch or feel numbers but numbers are used to count solid objects therefore becoming the object itself. eg:you can touch and feel apples you count 2 apples the apples then becomes the number 2 making the number 2 solid within the apples. if you look at numbers as a reference to an object then the picture becomes clear its just easier to use numbers rather than give each object an individual name eg: apple 1 becomes number 1 apple 2 becomes number 2 and so on. another way to look at it is think about your own name eg:peter, does peter really exist? you would say yes he does but really peter does not exist until peter is applied to a person just like the number being applied to the solid object. my conclusion is numbers do exist in our world because you can touch and feel them as the solid objects they are refering to just like your name refers to you.

Anonymous said...

I think this is a waste of time. Productivity is the answer. This is counter-productive. All we know is that we have numbers. Numbers are self evident. I believe they are given to us by a higher power and they are instilled within us. I think back to 1st grade. I was able to do square roots without even thinking about it. I believe the answer to the how is a higher power. I can't see how man could have organized such systems into place by his own power. The world would be chaotic without numbers. I believe we should stop worrying about nonsense and be grateful for math. These concepts are relativistic and self refuting.

Anonymous said...

Let me illustrate:
"Numbers (assumed to exist) don't exist" Thats self refuting and false.

Anonymous said...

To refute your nominalistic position...how many ways would you like for me to do this? A wonderful use of Psychological terminology here. I think you hit the nail right on the head. Nominalism fails based on the idea that if everything is also nominal, that would be concrete, and not abstract. The imagination that we have is nominal, however and can not be dismissed, but we create those images purposefully. The other naturally instilled ideas, motives, thoughts, beliefs, feelings, perceptions, insight, intuition, sensation ability, decision-making ability, judgment skills, common sense, discernment (or lackthereof for some people), knowledge, truths etc. (the complexity of the human brain, body, mentality, consciousness, spirituality and other components are just too formal to actually suspend to just a handful of concepts) are supportive of causalism. It is also important to notate that Causation is one of the Aristotelian logical terms that can not be refuted. A nominalist would state, "What if all human minds died?" for instance and the response can be one of several. "The human mind who conceived this idea would still be alive." Thats one. Or we can state, "Prove how a human mind can 'die'." Actual proof is the 2nd way we get rid of it (relativists and subjectivists don't rely on 3rd sources, if they do, they are using objective claims). A 3rd way is to show the 'all' in this statement to be self stultifying. Easily done by stating, "To prove that all human minds have died, you would have to know all humans. The concept is inherently false and delusional to assume." The 4th way is to show how it can be flawed literally. If we literally state that a human mind dies, the person's physical self has died. Inherently though, the generations that surpass the person's physical self live on and as such do other human minds (if 'proof' is needed, just direct them to historical accounts...thats self evident proof within itself, for if no mind existed, no books would exist.) The 5th way is to show how this mentality is extremely cynical. As we recall from Josh McDowell's book, Skepticism is self refuting. The concept itself is a negatively stated concept that is obviously skeptical. If we are to be skeptical of all things, that includes the existing subject to make the claim. As such, the concept fails even to the individual holding the concept. To these claims they may ask a question, "What if progress has never been made within the past and all of this is an illusion (the Zen Buddhistic approach)?" This exact statement is self stultifying. The language within the sentence has progressed beyond the past and therefore it is not illusionary to consider the progression. Relativists and Nominalists can not win. Just following the Socratic method (You ask the question you give the answer). Thats my answer on the matter (in case you need feedback if the question is asked again, take it or leave it of course).

Anonymous said...

What about minds that aren't human? With a universe as large as ours, there are almost certainly other minds on other planets, do they also know of numbers?

Anonymous said...

At 6:24 AM, Anonymous said...

"Let me illustrate:
"Numbers (assumed to exist) don't exist" Thats self refuting and false."

but surely "Unicorns don't exist", is this self-refuting?

Anonymous said...

"I think this is a waste of time. Productivity is the answer. ... I believe we should stop worrying about nonsense and be grateful for math."
Anonymous, 6:11 AM

The following is hinted at:
1. Mathematical theories (which assert the existence of numbers) are extremely useful for engineering, science and many other activities.
2. We should hold that numbers exist.

Although this reformulation of Pascal's Wager is at first compelling, it is a poor argument for the existence of mathematicalia. The argument hinges upon an ambiguous "should". We often use the term "should" with an epistemic tinge, ie. we use "We should hold that ..." when "It would be reasonable to hold that .. ." might be more illuminating (although less pragmatic). To be a cogent argument for the existence of numbers the above requires this epistemic sense of "should" in 2. But 2. does not follow from 1. when given this sense.

An analogy might make this clearer. Newtonian gravitational theories are extremely useful and have been used successfully for all sorts of fantastic engineering endevours. Yet we know, in light of general relativity, that the gravitational forces described in Newton's theory do not really exist. Newton's theory is extremely useful but we should not believe in the existence of all the entities it describes. The nominalist, rightly or wrongly, holds that the same is true of numbers: Although it is extremely useful to act as though numbers exist, really they are nothing more than useful fictions, like Newtonian gravitational force.

Anonymous said...

Science/physics has similar problems as mathematics - and not just the weird science of the very small and very large. Take magnetism for example, does it "exist"? I can not directly perceive it with my 5 senses. The best I can do is define what "it" does in the world (attract iron, spin a compass, etc).

I am new to this debate, so I guess spinning a compass is analogous to the "causal connection" mentioned above. So, does magnetism "exist" and if so, where? Do we have to limit causal connections to what humans via their 5 senses can determine? Are there other intelligent beings who have wider sensory bandwidths than us?

One of the problems I have with numbers is that it is fundamentally 1-dimensional (or maybe 2 if you treat imaginary numbers as phasors). Are there hyper-imaginary spaces that intersect our number line?

But to address the original post... I think numbers are as real as magnetism and dark matter. And there may be other "abstract" concepts out there that we don't have enough sensory bandwidth to establish a "causal connection" with (or at least not yet).

Anonymous said...

What is real?

Teen with an IQ of 144 said...

I agree! What is real! i mean in a sense, they are a concept, but only a tool. Although they can often express more than words, i believe they are a transient factor to man's existence.

Anonymous said...

Unfortunately for this debate we have neglected a property of 'numbers', that is it has many facets that apply to many (abstract or not) aspects of life implying that to branch it into two directions (physical or concept) we would be foolhardy to attempt an answer that almost 'fits all sizes'.

Anonymous said...

Plato was my mum;)

Esra Atescelik said...

The numbers do exist. Yes, there are objects, but not physical. such as ideas, judgements, events. Also they can be counted and but they are not physical.

Anonymous said...

I believe numbers, concepts, and propositions exist in conceptual space

bortscher said...

My thoughts:

the brain works by connecting. Creating analogies, parallel systems of logic, comparing, finding relationships, etc etc etc.

Math is really just a system of universal logic, parallel to reality. Our brain can connect anything to anything... So if we establish a universal medium of logic that can be connected to anything, we can solve anything in reality.

The scientific method, 'what effect does the manipulated variable have on the responding variable?' can be expressed mathematically, as a function or relation. The manipulated variable could be the Y axis, the responding variable could be the X axis...

Now, in mathematical form, you can do ANYTHING with it. You can turn it into an equation, express it as a chart, put it into word form, put it back into the scientific method, etc etc etc....

Think about it... An analogy is like two parallel lines on a cartesian square. A similarity is like an intercepting point of two functions.

Math is about as real as the neurons in our brain.

How was math invented? Possibly through trial and error over a long period of time. Our minds can instantly recognize 1, 2, 3 or 4 objects instantly, even as infants. If you imagine yourself in that primal state, you would naturally be inclined to invent a 'system' that can reach out to numbers greater than four.
Perhaps after that, passing it down verbally through generations, someone wanted to expand the system and make it so any number could be expressed, no matter how small or large.
Math probably continually evolved, with the smarter people in society fine tuning it as it went on, like natural selection.

Diva4Jesus said...
This comment has been removed by the author.
Keysle said...

Numbers are just summarizations made by the human brain. The rules of the game evolved with the natural selecting pressure of usefulness.

Anonymous said...

Do numbers exist? I would have to say yes they exist but just because they exist doesn't mean they are 'real' They are not real just like money isn't real (the paper and print is real). They exist but only as a concept and when all living things that know/understand the concept die, so does the concept and therefore so do numbers.

Someone said numbers were given to us by a higher power...Maybe they is a God of numbers but then there'd have to be a God of language too and a God of...

Someone else said our world would be in chaos without numbers? Do fish/cats/dogs count? have any kind of numbering system? I think not but they don't live in chaos.

Anonymous said...

"my conclusion is numbers do exist in our world because you can touch and feel them as the solid objects they are refering to just like your name refers to you."

You can also calculate in your head. For example 5*7 = 35 the numbers don't need to refer to anything physical!!

Anonymous said...

As a boy I understood precious few things, some of which were trivial concepts of mathematics. As a man and a student of the universe I understand that even the number one is complex series of imaginary and real numbers, not oneness or conceptualized unity, the number 1 can be equated to the exact negative if Euler's unique and beautiful identifying characteristic... 1=-(e^ix)

Anonymous said...

The word number reference something cool. Ok, so remove number because it's man made and is just a word.

What we have left is how our brain operates, without this concept we can't think or move, everything would be static. We can't navagate in our memory, this concept is the foundation of our thinking, don't have the time to explain.

So basically infinity doesn't exist, what is infinity? it's our brain's limit plus one. We can only think within the limit.

John Gabriel said...

Actually, numbers do exist, have always existed, and what's more, numbers exist independently of the human mind or any other mind.

Real numbers do not exist, but all numbers up to the rational numbers have always existed.

See link for proof that real numbers do not exist:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409


http://thenewcalculus.weebly.com

Anonymous said...

Actually only numbers exist, any possible moment (state) of the universe that can be is a number itself, simply time we precieve as continous is the snapshots of these states that exist timelessly like frames in a movie rolls, we live in all those frames we just know the related frames at past (thus connected to them and this creates sense of time).

Anonymous said...

If numbers do exist in this separate realm, then how do we know about it?

The answer is, of course, given by Plato in the dialogue Meno. We remember them, vaguely and with all the confusion of the comments already posted here; but with strenuously application of intellect and virtuous will the possibility of rising up out of the Cave is there. All of us -- our real selves, which is to say our souls -- existed before we were born and will exist again after we die in different bodies. Before we are born as persons, our souls can see the Forms as they truly are. Number is the grand One, the purest of the Forms.

None of this is satisfying dogma to modern minds; but this is orthodox Platonic doctrine passed down through the Neoplatonists, including their Christian followers, Augustine et al.

What if it is true?

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