In what sense numbers exist, if any? They were there and simply discover? Were there no matter if we thought of them? Some say that the numbers are entities that are in the real world, though not physical, supporting the possibility of defending the existence of other mathematical concepts absractos as sets, functions, and so on. But is it true?
Do Numbers Exist?
According to your disposition, you might have an immediate gut reaction to this question. My initial reaction (oh so long ago) was: “Of course numbers don’t exist. You can’t pick up the number 3 and throw it through a window.” That is, my intuition was that the only things that exist are the kinds of things that can be physically manipulated, and numbers, by almost every account, just aren’t this kind of thing.
To be clear about our terms, you can pick up numerals — that is, you can pick up concreteinstances of numbers, like the plastic number signs at the gas station telling you how much gas costs, or the printed numerals in a book, denoting page numbers. But you don’t, by virtue of tearing out page three of a book and tossing it out a window, throw the number 3 out the window, any more than you throw me out of a window by drawing a picture of me and throwing that out the window.
Numbers, if they exist, are generally what philosophers call abstract objects, and those who maintain that such things exist claim that they exist outside of space and time. If you’re like me, you shake your head at such talk. “Outside of space and time? What does that even mean? Gibberish!” If you are similarly disposed, you might be a nominalist (in case you’re accumulating self-descriptive philosophical terms), and you are part of a long, proud philosophical tradition that thinks that existence is the exclusive domain of the physical...
BY ALEC JULIEN
Follow this link to know more:
https://welovephilosophy.com/2012/12/17/do-numbers-exist/
The Pythagorean or Platonic reality of numbers has a long history in philosophy and mathematics. Physicists Max Tegmark and Roger Penrose, for example, argue for number as a fundamental substance, so you might try their books as well. In post-Kantian philosophy one might ask, can we validate the existence of anything "independent of mind." ~ Nelson Alexander
Philadelphia, PA
Dear De la Pena,
We are naively all inclined to agree that, say, "There is a number between 5 and 7." But what does this mean?
A good start on the question might be found in Paul Benacerraf, "What Numbers Could not Be," The Philosophical Review, 74, No. 1 (Jan. 1965), pp. 47-73.
http://www.jstor.org/stable/2183530?origin=crossref&seq=1#page_scan_tab_contents
H.G. Callaway
Responding to this, presumably because human life is determined only by conscience and thoughts that led to the arts and sciences, the numbers are not really exist based on art. Meanwhile, numbers are really exist based on scinence.
It would be better to give a sign, not a negative vote ... anyway numbers are placed by man to serve the man ... he / she who is to argue against it ... with respects for all to vote in favor of the vote against!
Yes sir, number exits. Number a symbolic representation of groups. So, it should exits.
I could cite hundreds alteranativa on the history of this question, but were stopped at what is key. When our ancestors have understood and have shared inert or living things, pointing out this is mine, you have this, it is that we have this, you have it, and so we have the first signs of the count ... and time after time after first began counting the fingers of the hand and the legs ... so "fairy story" ended ... ... I wish you luck in the count, particularly when you have dollars, Euro or other ...
Philadelphia, PA
Dear Nabab Alam,
A "symbolic representation" would seem to be something linguistic, a form of words or symbols, perhaps? Included here would be numerals, such as the symbols "1," "2," "3," etc. But if we are asking whether numbers (really) exist, then this is different from asking whether numerals exist. When you say "Number is a symbolic representation..." are you talking about the numerals, or things of a similar symbolic sort, or about the things which the numerals name or represent?
H.G. Callaway
In the book of S. Shapiro, Thinking about mathematics (a philosophic book from an analytic point of view) there is Chapter 8: Numbers exist. In the next Chapter 9: No they don't. Thus the first view is a realistic view, whereas in Chapter 9 we have a fictionalistic point of view. There is a new point of view expressed by F. Zalamea, in his book "Synthetic Philosophy of Contemporary Mathematics", where many polarities, like aposteriorism vs. empiricism, Platonism vs. empiricism etc, can be viewed in a more general pendulum-scheme that includes both poles as dialectical synthesis.
This question is inextricably linked with the general phenomenon of mathematics. The classic idea of it, which formed together with the development of philosophy, alienates the subject from the physical reality. Most people perceive everything connected with mathematics as the ideal world of Plato's ideas.
However, on closer examination of the subject, it appears that this is only an illusion of our perception of reality. Even theoretical physics does not consider question "why mathematics should describe nature?". It takes this as postulate, in other words, as axiom. Since all happens in the physical reality the model of which you want to get from this axiom, there is a fundamental problem.
Based on the nature of this illusion I can make a few important fundamental conclusions, which will have consequences for the physical picture of universe:
1. Your brain is a physical structure.
2. Implementation of mathematics (known to us) is a product of it physical activity. This is only a certain topologial relationships between groups of physical (information) structures.
3. Regardless of the type of specific information carrier it interactions are subject of common (physical) laws. It does not matter whether you consider the topological relations, between groups of neurons in the brain or between other objects in the Universe.
4. Hence an important conclusion that the considered topological relationships between objects are invariant with respect to all objects of physical reality.
5. Roughly speaking, cognition of reality is reduced to a study of the topology of some (terminal) manifold. This manifold directly related to the entropy.
(Mathematics is the strict formal system. The predictability of the appearance of symbols by using the language of mathematics is huge. The predictability of the appearance signs of alphabet with the message generation is directly related to the concept of entropy. This should be taken literally, this is not about the specific implementation but topological relationships.)
This equally applies not only to countable set (numbers from topic) but also to the continuum and irrational proportional relationships known as the "golden sections", infinity, etc. It just topology. It is global invariant.
It is for this reason we perceive mathematics as something that exists outside of reality. "Abstraction" which embodied in our formal implementation is inseparable from the "fabric" of reality. It shapes the reality. Perhaps this is the most actual fundamental question on the border of science and philosophy, which exists.
As I see it, numbers are cognitive structures --categories of thought-- that we use to make sense of the world. These structures are developed --in evolutionary terms and during the development of each individual-- through our action on the world. So numbers are a product of our neurological structure --which has been developed during evolution by adaptation to the environment-- in interaction with the structure of reality. Numbers, then, are "real" in this double sense.
Numbers exist not as physical objects but as concepts which are our own creations to represent or designate size, amount, length or duration depending on what we are dealing with, that are established based on predefined axioms of validity to certain effects. For instance when we say it requires 2 hours to cover a distance of 120 miles if we drive 60 miles per hour, the numbers are used to represent a right amount of a thing to effect some thing, the duration what we call 2 hours could have been named something else likewise 120 miles, similar to the cases in scaling and resealing sizes.
Numbers are descriptions of something exist, for example 4 pens, 5 ducks, etc.
The numbers are simply the representation of the amount of something physical, as functions represent the behavior of something, and thus, project their future or their past in time. The same can be said of the letters and the words, when they represent the beauty or a feeling.
No, they do not exist! Numbers are just an abstraction and they were invented not discovered to distribute some good or hunting or collecting fruits or goods.
Numbers are very usefull concepts for humans. We cannot do much without them. it enables us to organize our understanding of situations in the tangible world. When human began communicating , among the first gestures were probably some for small numbers. Prior to learn numbers in our childhood, we have some cognitive competence that allow us to keep track of a small number of objects and other animals have also these competences.
Numbers themselves do not have tangible existence but arithmetic has a logical structure which matches the structure of the exchange of goods (expressed in unit) that takes between human. Numbers and all the mathematical concepts are always used into a communication process and in science there is a metaphore being do between this mathematical communication and reality. Numbers thus metaphorically can be attribute to aspect of reality, but outside of this methaphor, they do not exist. We can say they exist by themself if we use metaphorically take our mathematical convention to exist by themself. But this is just a metaphor and we should not forget that in language we always play this metaphorical game which only exist into it.
Because as human language is so fundamental to our consciousness, it is impossible to imagine the world outside of this metaphorical world and it is so tempting to equate reality with its most more simple language metaphors such as numbers that we say that number exist , like in our mind. It is funny because we take numbers to be the quintessential of objectivity and they are only in our mind.
Our scientific model is objective not in the sense that it described reality as it is but in the sense that this mind model is a good and exact metaphor to some aspects of what is out there. We experience the world through our narrative mind or body interface and it is unconceivable (litterary) to see the world not through it. We cannot come out of our mind and so we have to see the world through our mind and mathematics is just one of our many language that allow us to do it collectively.
Hi sir,
Number exists in such way that it is a sense of the world and help us in organizing our knowledge, understanding of things and most important one is the analytical aspects.
Number (not the numeral) is an idea. If ideas exist, then numbers also exist.
In my opinion, things exist, but numbers are a human invention to count them.
numbers exist because we have 10 fingers and 10 toes and I think that numbers have been important since we see more than 1 of something. It is measurement in the skies and in our worlds. It is within our nature to count. Number of steps, number of stars. It is just a natural progression to language in my opinion. Communication tool.
Do Mathematical Entities Really Exist?
"A friend in the Philosophy Department at the University of Kansas once said to me that numbers do not exist. They are just as fictional, he said, as the character Frodo inLord of the Rings.
Certainly my own knowledge of philosophy is at best that of a dilettante. But I know enough to know for certain that on this matter he was wrong. I suggested to him that what was at issue was in large part how one defines the word "exists." But he immediately insisted that existence is a primitive notion which cannot be defined. I didn't want to argue that point further, but in this respect he was even more wrong than in his first statement. The meaning of "exists" is very much contextual. (And in fact, if the context is a discussion of Lord of the Rings, then there does indeed exist a character named Frodo, whereas in the novel there does not exist a character named McGruff the Crime Dog.)
The dispute over the existence (or reality) of mathematical entities is an example of what philosophers call the Problem of Universals, something which goes back as far as Plato. This is the question as to whether abstract concepts have some sort of real existence in the world, or whether they exist only in our minds. Like most philosophical problems, it seems to be more a question about language than a question about the world, although there are certainly philophers who would disagree with me in this respect.
I believe it was Kronecker who said, "The natural numbers were created by God; all the others are the invention of humans." I believe that most contemporary mathematicians would agree that Kronecker was wrong only in his statement about natural numbers; they too are the creation of human minds.
Certainly numbers do not have a tangible existence in the world. They exist in our collective consciousness. And yet they are not arbitrary products of our imaginations in the way that fictional characters are.
For instance, when a mathematician says that there exists a prime number which is the sum of two squares, his statement is not a product of his imagination. It is not a matter of opinion. The prime number 13, in fact, is the sum of 3 squared and 2 squared: 13 = 9 + 4. And when the Indian mathematician Ramanujan said to his fellow mathematician G.H. Hardy that 1729 is the smallest number that can be written as a sum of two cubes in two different ways, he was making a statement of fact:
1729 = 10³ + 9³ = 12³ + 1³.
The fact that no smaller number can be so written can be verified, with the help of a computer program or spreadsheet, by listing the values of m³ + n³ for m and n between 1 and 12 and seeing that there are no duplications smaller than 1729 in the list. (One can also note that if m³ + n³ is 1729, then one of these two numbers must be larger than 9 and, of course, no larger than 12. This leaves us with only a few possibilities to check.)..."
By: Lee Lady
http://www.math.hawaii.edu/~lee/exist.html
Do Numbers Really Exist?
"Here’s a nice, simple topic for first thing on a Friday morning: do numbers really exist? Well, do they?
As Mark Jago from the University of Nottingham explains in this video, it’s really quite hard to find out about what numbers really, truly are because they’re not physical, tangible things. Prepare to have your mind melted as he explains some of the ways people have tried to think about the very existence of numbers in the past."
Jamie Condliffe
http://gizmodo.com/do-numbers-really-exist-1731601472
Do Numbers Exist?
According to your disposition, you might have an immediate gut reaction to this question. My initial reaction (oh so long ago) was: “Of course numbers don’t exist. You can’t pick up the number 3 and throw it through a window.” That is, my intuition was that the only things that exist are the kinds of things that can be physically manipulated, and numbers, by almost every account, just aren’t this kind of thing.
To be clear about our terms, you can pick up numerals — that is, you can pick up concreteinstances of numbers, like the plastic number signs at the gas station telling you how much gas costs, or the printed numerals in a book, denoting page numbers. But you don’t, by virtue of tearing out page three of a book and tossing it out a window, throw the number 3 out the window, any more than you throw me out of a window by drawing a picture of me and throwing that out the window.
Numbers, if they exist, are generally what philosophers call abstract objects, and those who maintain that such things exist claim that they exist outside of space and time. If you’re like me, you shake your head at such talk. “Outside of space and time? What does that even mean? Gibberish!” If you are similarly disposed, you might be a nominalist (in case you’re accumulating self-descriptive philosophical terms), and you are part of a long, proud philosophical tradition that thinks that existence is the exclusive domain of the physical...
BY ALEC JULIEN
Follow this link to know more:
https://welovephilosophy.com/2012/12/17/do-numbers-exist/
Since we all learned numbers, they exist in that sense of being a mathematical concept. The question is not if numbers exist as mathematical concepts. THis is not controversal but simply an evidence. The issue that is controversial is ''Did numbers exist in Nature prior the existence of humans and animals and life? It is obvious that human uses numbers and some animal behavior studies sudgest that animals have some cognitive concepts of number. Were there any use of numbers, trace of this concept in Nature previous to life? Yes it is essential for us to use numbers to understand the early universe. But this is a use in our understanding and we cannot (in my opinion) attribute this essential element in our understanding to be actually existing in Nature. If two stars orbit around each other. It is a binary pair. It is essential for us to use the number 2 to describe this situation. This is not controversal. But is 2 really be part of this situation?
There are five apples on the table in front of me. Does this factual state of affair implies that the number five exist in this situation. For a bacteria living on the table, this state of affair does not exist. Only certain type of animal can distinguish this table and five distinct apples there. So for these animals, there are something in them corresponding with five. But this is a state of affair that only exist relatively/within these forms of life.
Not being one among the great mathematical thinkers, I will just add my two cents.
The question, to me, is similar to the philosophical question of "If a tree falls in a forest and no none is around to hear it, does it make a sound?" We tend to think of things in a egocentric way, as if we as the observer influence the outcome just by being present at an event. The answer is if you drop 100,000 steel balls on a sheet metal table and each time a sound occurs, then why would a human not being present affect the outcome?
Numbers. Mathematicians have said for years that everything in the universe can be described with mathematics (numbers). If X+Y+Z describes each and every thing, then in seems to be apparent that those truths, those numbers, exist as a descriptor of all.
Natural numbers exist since are the result of counting objects; it were defined by Peano, who postulated the number one and the operation "successor of" to construct the sequence of natural numbers. Thereafter, integers are the result of the operation c = a - b, where, a and b natural numbers, given that, c, could be zero or a negative number, i.e. none natural numbers. In addition, rational numbers are the product of the division operation; that is, a rational number is represented as follows, r = p/q, where p a nd q are integers. Meanwhile, real number are a result, on the one hand of the Pythagora's Theorem, since some square roots can't be represented by a rational number, for example the (square root of 2) = 1.4142135623.... can't be represented by r = p/q, on the other hand the number Pi = 3.1416.....is also a real number. Finally, complex numbers resulted from the (square root of −1) = i, cant be represented by any real number; hence, z = a + i b where a and b are real numbers is a complex one
Natural numbers exist since are the result of counting objects; it were defined by Peano, who postulated the number one and the operation "successor of" to construct the sequence of natural numbers. Thereafter, integers are the result of the operation c = a - b, where, a and b natural numbers, given that, c, could be zero or a negative number, i.e. none natural numbers. In addition, rational numbers are the product of the division operation; that is, a rational number is represented as follows, r = p/q, where p a nd q are integers. Meanwhile, real number are a result, on the one hand of the Pythagora's Theorem, since some square roots can't be represented by a rational number, for example the (square root of 2) = 1.4142135623.... can't be represented by r = p/q, on the other hand the number Pi = 3.1416.....is also a real number. Finally, complex numbers resulted from the (square root of −1) = i, cant be represented by any real number; hence, z = a + i b where a and b are real numbers is a complex one
Dear Colleagues,
Good Day,
Please, see this interesting Youtube, entitled "Do numbers EXIST? - Numberphile"....
https://www.youtube.com/watch?v=1EGDCh75SpQ
as per my opinion number of fictional please go through the article attached here with
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.
http://platosheaven.blogspot.com/2005/12/do-numbers-exist.html
Numbers exist as well as language and mind. They are objective. Who is not sure? May be he is not sure in his personal existence?
Dear Ricardo De la Peña Sir,
I believe it was Kronecker who said, "The natural numbers were created by God; all the others are the invention of humans.Certainly numbers do not have a tangible existence in the world. They exist in our collective consciousness.
Numbers really exist & if not how can we show our existence !!. In this line the numbers is playing an very important part .Numbers as an area in mathematics ,as it covers an area of scientific technological development .
Numbers is alarming ,counting ,days of 365 which becomes an important year for every human beings & also for trade & industrial sector of the world .
In this line we cannot omit & ignore numbers in our life .
This is my personal opinion
Do Numbers Exist?
Dear Sir,
I thought that the relation between the numbers and the existence of God may be same that if we take the numbers into the various subjects structure of the numbers are changes ,humans are developed the various numbers and used it for there purpose ,Numbers are exist but changes by subject to subject in the same way concept of God changed by religion to religion but it exist.
Numbers exist as part of our language, which is the médium to access the real world.
I like the Barbara´s answer. "Existence" is a philosophical concept that might has several interpretations. The objective existence is that of the physical objects, independent of the human concience (atoms, stars, waves, etc.) The subjective existence is that of concepts, feelings, etc. They are relations that create the brains of animals; and also in many occasions are accepted for wide group of animals, as the human been concept of number. The third type of existence is the pragmatic one: just if it is useful, it exist.
From these ideas, I accept that numbers exist "subjectively" Of course, concepts can be valid or invalid; and certain humans accept validity (existence) of some concepts; but other humans reject these same concepts (for example, God) In general, the humans give subjective existence to concepts associated with objects of objective existence. In these cases, the concepts are scientific. Otherwise the concepts are pseudoscientific or speculative.
The concept of number has subjective existence because it is based on the objective existence of setsof physical things. For example, the number 2 is the generalizarion of all the couples: couples of atoms, couples of persons, couples of bean grains, couples of stars. All the cultures of humanity accept the number 2 in the same fashion; and I beleive in that human beens from other planets or galaxies should accept the number two in the same way. It has an subjective existence, independentely of the development nivel. The culture of Incas in America accepted the numbers in the same way that in China, but they didn´t conceive the wheel.
The concept of number is so real that many animals accept it. A lion attacks a cow only if it is isolated. A dog female look for the third "kid" if she sees only two. The woolfs only attack with the cooperation of several "colleagues"
Do Numbers Really Exist?
Jamie Condliffe
Here’s a nice, simple topic for first thing on a Friday morning: do numbers really exist? Well, do they?
As Mark Jago from the University of Nottingham explains in this video, it’s really quite hard to find out about what numbers really, truly are because they’re not physical, tangible things. Prepare to have your mind melted as he explains some of the ways people have tried to think about the very existence of numbers in the past.
http://gizmodo.com/do-numbers-really-exist-1731601472
Dear Colleagues,
Good Day,
"Do complex numbers really exist?
The concept of mathematical numbers and "existing" is a tricky one. What actually "exists"?
Do negative numbers exist? Of course they do not. You can't have a negative number of apples.
Yet, the beauty of negative numbers is that when we define them (rigorously), then all of a sudden we can use them to solve problems we were never ever able to solve before, or we can solve them in a much simpler way.
Imagine trying to do simple physics without the idea of negative numbers!
But are they "real"? Do they "exist"? No, they don't. But they are just tools that help us solve real life problems.
To go back to your question about complex numbers, I would say that the idea that they exist or not has no bearing on whether they are actually useful in solving the problems of every day life, or making them many, many, many times more easy to solve.
The math that makes your computer run involves the tool that is complex numbers, for instance.
I really do not like questions, the answer of which demands to present a whole philosophical position. There are the realists which answer, YES there exist! On the other hand there are "fictionalists" which deny the existence of numbers. They claim that numbers have the status of Tom Sawyer! There are books which name and use "numbers" the same way that the author of Tom Sawyer was introduced by his author. There is a way to construct a "pendulum" out of any polarity, e.g. Platonists-empiricists, and we may have any position in between! So that we may have that numbers exists 1/3 of times and 2/3 of time fictionalist!
Hazim,
''Do negative numbers exist? Of course they do not. You can't have a negative number of apples.''
Of course negative number exist. This is called a debt. If you do not believe it, and make is zero, they will put you in jail. Maybe you will believe in negative numbers then.
Costas,
''They claim that numbers have the status of Tom Sawyer!''
Many people knows Tom Sawyer but so far nobody create a Tom Sawyer device acting by itself in the world. The reality of Tom Sawyer stay in our memory and act on our own character. It is a good prototype for characters and thus is a social character reality. Numbers are special fictions, that live in our mind like all fictions in company of Tom Sawyer but they allow us to point to very simple aspects of our fictional perceptual world such as the number of objects, etc. Our perceptual fiction points to more primitive aspects of our interaction with the world than the societal fiction of our interaction with other human beings. Animal experiement has demonstrated that some primitive notion of number also belong to the perceptual fictional apparature of their nervous system. They may not have Tom Sawyer type of character in their nervous system but surely have some protope fictional character of behavior of their fellow animal in the case of mammals which are animal interacting in families. We can probably go down the biological tree of life down to cellular life and still find some trace of symbolic numbers being actively used. But I do not thingk that carbon atom have any internal active structure which behave like a number and so at that point I would say that numbers cease to exist prior to life. The fact that we humans need number to describe carbon atoms, the number of their atomic orbits for example, is not a testimony that number exist there. It only tell us that some things are discrete, that we can count them or that other aspect of that reality we can quantify with probabilities are usefull for our interaction with carbon atoms.Absolutly nothing can be said for these type of atomic interactions without numbers. That is in not way says that number exist at this level of reality. We cannot exclude ourself from a discourse on atoms because our quantum physics do not describe this world in itself but how we interact with it. How could number belong to a world of atoms which is not a world we can say anything since our only physics is not about atoms alone but about atoms interacting with macroscopic systems such as us. So the problem of existence is not unique to numbers but even for individual event and atom. We cannot cut the relational from the discourse. Extract the narrator and get a God's eye view of a platonic world.
''
A variety of research has demonstrated that non-human animals, including rats, lions and various species of primates have an approximate sense of number (referred to as "numerosity") (for a review, see Dehaene 1997). For example, when a rat is trained to press a bar 8 or 16 times to receive a food reward, the number of bar presses will approximate a Gaussian or Normal distribution with peak around 8 or 16 bar presses. When rats are more hungry, their bar pressing behavior is more rapid, so by showing that the peak number of bar presses is the same for either well-fed or hungry rats, it is possible to disentangle time and number of bar presses. In addition, in a few species the parallel individuation system has been shown, for example in the case of guppies which successfully discriminated between 1 and 4 other individuals.[1]
Similarly, researchers have set up hidden speakers in the African savannah to test natural (untrained) behavior in lions (McComb, Packer & Pusey 1994). These speakers can play a number of lion calls, from 1 to 5. If a single lioness hears, for example, three calls from unknown lions, she will leave, while if she is with four of her sisters, they will go and explore. This suggests that not only can lions tell when they are "outnumbered" but that they can do this on the basis of signals from different sensory modalities, suggesting that numerosity is a multisensory concept.''
https://en.wikipedia.org/wiki/Numerical_cognition
Dear Louis,
I think you did not understant me completely. Fictionalism takes a mathematical book as a "fiction"! Mathematical entities in The Book, are very similar to Tom Sawyer! What you call "Numbers are special fictions", may refer to Platonism, at least not so strict. In fictionalism the conceptions of the author exists first, but the "entities in the story" become existent until the book-fiction is written.
Fictionalism is a nominalist (or anti-realist) account of mathematics in that it denies
the existence of a realm of abstract mathematical entities. It should be contrasted with
mathematical realism (or Platonism) where mathematical statements are taken to be true, and, moreover, are taken to be truths about mathematical entities. It also lends itself to a very straightforward epistemology: there is nothing to know beyond the human-authored story of mathematics.
See e.g. http://plato.stanford.edu/entries/fictionalism-mathematics/
I forgot to say that I am not a fictionalist!
Let me refer to an example from S. Shapiro, Talking About Mathematics, p. 227:
“We begin with Field's view, which he calls 'fictionalism'. The idea is to think of mathematical objects as being like characters in fiction.
The number three and the empty set have the same status as Oliver Twist. There is a sizeable philosophical literature on the semantics of fiction, which we can thankfully avoid here. At least prima facie, one can take fiction seriously (so to speak) without being committed to a fictionalist 'ontology'. Few philosophers are tempted to think that Oliver Twist exists in the same sense as, say, the White House does.”
Numbers and relational properties is the same, Stephen.
Numbers is really existing ability (possibility) to distinguish and calculate.
If numbers don't exist, then You, Costas, does not exist as unity (number one).
Natural number can be viewed as set. Does not it exist?
Dear Eugene,
Did I say that numbers do not exist? I just mention fictionalists that believe so! Personally I believe that there exist!
Costas,
Aristotle constructed on Plato but he had to modify it. One of these modification was to say that concrete reality is first and forms second. There is a concrete Socrate before the universal Man. There is no realm of desimbodied forms. All of them are embodied. Yes the forms do exist, but are embodied. Understanding for Aristotle is not a contemplation of the universe of forms directly by the mind not through the senses. For Aristotle, perception is the reception through sense organs of the forms of things, he called these images in the case of vision. And the process of perception was one where the forms of the objects embodied/actualized themself into the receptive organ and understanding is the continuation of the actualization of the object's form up to the rational level of actualization.
In that version of fictionalism that I described, I took this Aristotlean version of Platonism that I think we could called an Embodied Platonism. Here the numbers are fictions or embodied rational forms in the imagination of the organism. It is not anti-realism at all in the version here presented. It is a Aristotlean nominalist. This version of fictionalist can also be a form of realism when the fictions are derived/abstracted from real experience. It would be hard to denied the fact that numbers are necessary for the description numerous aspects of experience as live by organisms and so it is normal to expect that the processus of biological evolution to have embodied them in organic systems, to be used as useful fiction of their interaction.
The so-called platonic world of forms is not into an etheral realm but active in more or less elaboration version in the organisation of all forms of life. In the case of humans , the culmination of the development of the mamalian imagination the capacity to of this fictional realm tightly tigh to the interaction of the body liberate itself and could interact with itself , to self-contemplate and the history of humanity is the exteriorization of this into external stabilized forms of art. Our nervous system is our reality engine and numbers with all we can think or interact are there and the mathematician , the painter, the author are working there. They cannot work without a concrete activity of expression with other of their fiction their subconscioius extract and fashion in this reality engine. Plato and Socrates were saying: learning is a kind of remembering; this reality engine realm is our memory and our culture is its expression and our collective attempt to remember/create who we are.
Dear Louis,
I agree with you! I did not support anything different than that!
All these philosophical notions ("exist", "forms" and so on) can not be strictly defined, Louis, as well as in Math "set", "number", "probability" and so on. Even Math can not be made purely formal system as was shown by Geodel and others, not saying about philosophy. But, nevertheless, thank You for your deep insight into philosophy.
According to ancient greek philosopher Plato mathematical concepts like the numbers exist. The realism in mathematics is a generalization of Plato´s ideas.
Do Numbers Exist?
BY ALEC JULIEN
According to your disposition, you might have an immediate gut reaction to this question. My initial reaction (oh so long ago) was: “Of course numbers don’t exist. You can’t pick up the number 3 and throw it through a window.” That is, my intuition was that the only things that exist are the kinds of things that can be physically manipulated, and numbers, by almost every account, just aren’t this kind of thing.
To be clear about our terms, you can pick up numerals — that is, you can pick up concrete instances of numbers, like the plastic number signs at the gas station telling you how much gas costs, or the printed numerals in a book, denoting page numbers. But you don’t, by virtue of tearing out page three of a book and tossing it out a window, throw the number 3 out the window, any more than you throw me out of a window by drawing a picture of me and throwing that out the window.
Numbers, if they exist, are generally what philosophers call abstract objects, and those who maintain that such things exist claim that they exist outside of space and time. If you’re like me, you shake your head at such talk. “Outside of space and time? What does that even mean? Gibberish!” If you are similarly disposed, you might be a nominalist (in case you’re accumulating self-descriptive philosophical terms), and you are part of a long, proud philosophical tradition that thinks that existence is the exclusive domain of the physical.
However, your nominalism begins to run into problems pretty quickly. Never mind numbers. What about things like, say, novels? What exactly is the novel The Catcher in the Rye? It’s not any of the particular instantiations of it — it’s not the copy on your bookshelf; it’s not the copy on mine. All of the print copies on the planet could be eradicated and still the novel could be able to be said to exist. Is the novel the original manuscript sitting in a safe somewhere? But that could be burned and you could still argue that the novel exists. But if the novel itself is not identified with any of its particular instantiations, then the nominalist is in a bit of a quandary. On this perspective, the copies of the novel are instantiations of the novel itself, and the novel itself is seeming to be something abstract — something non-physical.
So the idea of something somehow existing outside space and time is suddenly not as absurd as it may have seemed. What about numbers, then? Of course there are disanalogies between numbers and novels. Novels are invented by humans, while, on most views of the subject, numbers exist whether or not humans ever happened to discover them. But, putting such differences aside for the moment, perhaps the existence of novels as abstract objects gives us some traction to say that numbers exist as abstract objects.
https://welovephilosophy.com/2012/12/17/do-numbers-exist/
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...
http://platosheaven.blogspot.com/2005/12/do-numbers-exist.html
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