Do numbers exist?

[Oreoracle]

No problem. I suggested Dedekind’s definition because you mentioned that the notion of an infinite set could be controversial.

At this point I just want to say that when I define the terms I use, it isn’t meant to be condescending. You seem to be familiar with all of this. I’m just defining terms for everyone’s benefit, including those without much of a mathematical background.

There are some folks in the mathematical community called “finitists”, and there are at least two varieties. Classical finitists will allow for countably infinite sets–that is, sets whose members can be arranged in a sequence–but reject larger sets. Ultrafinitists reject the existence of infinite sets altogether.

Recently I started reading a book by a mathematician named Norman Wildberger who proposed a method for doing trigonometry that requires only polynomial and rational functions. (The result is a system that lets you answer geometrical questions about objects that typically aren’t thought of as numbers in spaces typically not thought of as spaces.) Anyway, he doesn’t like infinite sets at all. I’m not sure that he rejects them, but he doesn’t seem to use them. For example, instead of talking about the set of all integers, he feels it is sufficient to characterize the integers by a list of properties so that he can decide whether a given object is an integer.

The main problem I see with his approach is what algebraists call an isomorphism. We like to avoid philosophical questions in math for the most part. Thus instead of saying that two sets of objects that exhibit similar properties are “the same”, we just say there is an isomorphism between them; a way of rewriting statements about objects in one set in terms of objects of the other set. A boring isomorphism would be to take every integer x and rewrite it as an ordered pair (x,0). Addition and multiplication are defined component-wise: (x,0)+(y,0) = (x+y,0) and (x,0)(y,0) = (xy,0). Note that addition and multiplication are still commutative and associative in this system, that the distributive property still holds, that (0,0) and (1,0) play the roles of 0 and 1, that everything still has an additive inverse (an opposite or negative), and that there is a one-to-one correspondence between these ordered pairs and the integers.

If you were just describing the integers in terms of their algebraic properties, no one would know which set of objects you were referring to because these sets (specifically, rings) are algebraically indistinguishable. So it seems that those who think like Wildberger would have to accept both 1 and (1,0) as integers. The issue is that unlike any pair of integers, I can’t just add or multiply these together. So I disagree with Wildberger’s approach on this matter and find it necessary to speak of the infinite set of integers.

I tend to think that he would be authorized to say that all numbers, except what we call “one”, are built, or created, but “one” is different. This must have a different mode of existence.

In the construction of the integers I described, the existence of these objects was implied the moment that we accepted the set-theoretic axioms used in showing how to construct them. The choice to accept those axioms was the creative act, in a sense. We create mathematics by adopting axioms, and we discover mathematics by proving theorems. That’s how I would put it.

They went on with “the Mayan numbers”, and one of those days I suddenly had the unpleasant impression that so far I just had been learning the husk, but that there was “something” behind it which remained mysterious to me: that “something” were the numbers, not the symbols and the rules to operate with the symbols (the husk), but the numbers. At that age I really could not say all this.

I think what you are describing here is much like what I referred to earlier. You realized that these number systems were all isomorphic to each other. The relationships between Roman numerals mimic the relationships between Arabic numerals. They are the same type of object in this sense, even though you can’t add or multiply a Roman numeral by an Arabic numeral for example.

In the case of imaginary numbers, they have been invented for certain engineering applications. So they do not exist.

On the contrary, use of the imaginary numbers can be traced back to at least the 16th century where they were used to make equations such as x^2+1 = 0 solvable, much like negative numbers were invented to make equations like x+5 = 3 solvable. They also have a very concrete geometric interpretation in terms of vectors and rotations. I certainly don’t have any qualms with the existence of complex numbers in general.

More questionable than their existence is their status as numbers. Unlike the real numbers, the complex numbers cannot be ordered since two dimensions are necessary to represent them. In the progression from natural to integral to rational to real to complex numbers, they are the first set that cannot be ordered, and order is typically something we expect numbers to have in everyday life. Then again, the integers mod n (the arithmetic of the numbers 0,1,2,…,n-1 where the numbers restart at 0 once you get higher than n-1) are considered numbers and these are used all the time to calculate the time of day in terms of the integers mod 12 (or the integers mod 24 in the military). Order is clearly not possible for these numbers.

 

[inocente]

On the contrary, use of the imaginary numbers can be traced back to at least the 16th century where they were used to make equations such as x^2+1 = 0 solvable, much like negative numbers were invented to make equations like x+5 = 3 solvable. They also have a very concrete geometric interpretation in terms of vectors and rotations. I certainly don’t have any qualms with the existence of complex numbers in general.

I’m no mathematician but to me Euler’s identity

(Please Note: This uploaded content is no longer available.)

proves that the square root of minus one is as real (or not) as zero, one, π and e, since any of them can be derived from the others.

The Wikipedia article from which I took the above typography quotes a lyrical Keith Devlin: “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence”.

 

[thinkandmull]

When you say the form of singularity is in something, what does “a thing” mean? Is a banana just the fruit, or the skin as well? If someone attaches a cape to his hat, is it two things or one? How can we know? This is something I’ve wondered about, and I’d like to find an answer from thomistic perspectives

 

[magnet]

When you say the form of singularity is in something, what does “a thing” mean? Is a banana just the fruit, or the skin as well? If someone attaches a cape to his hat, is it two things or one? How can we know? This is something I’ve wondered about, and I’d like to find an answer from thomistic perspectives

Is a mat of grass a single entity? Is a clump of Aspens a single entity or is it a collection of Aspens? Bermuda grass is notorious for sending out rhizomes and stolons that sprout new grass blades. Are these new blades, separate items or part of the parent plant?

 

[thinkandmull]

plato.stanford.edu/entries/medieval-haecceity/

 

[thinkandmull]

Is a mat of grass a single entity? Is a clump of Aspens a single entity or is it a collection of Aspens? Bermuda grass is notorious for sending out rhizomes and stolons that sprout new grass blades. Are these new blades, separate items or part of the parent plant?

Exactly. Its Buddhism vs Blessed Duns Scotus

 

[thinkandmull]

There may not be a form for an inn, because an inn can become a barracks. Since everything can be combined and decomposed, I think what makes something singular is its cohesiveness and its easy seperatibility from other things

 

[magnet]

There may not be a form for an inn, because an inn can become a barracks. Since everything can be combined and decomposed, I think what makes something singular is its cohesiveness and its easy seperatibility from other things

How about continuums that are defined as a set of entities?
How is the North Atlantic any different from the North Sea or the Caribbean Sea?
Is the Mississippi Valley different from the Missouri Valley?

 

[Oreoracle]

In the case of integers, a negative concept is useful in reference to zero. How can one have less than zero of something? So direction is also implied.

Also magnet, you may be interested to know that John Wallis, a famous 17th century mathematician, thought about negative numbers very differently. Consider what happens to the value of 1/x as you plug in smaller and smaller positive values for x. Its value increases without bound until we reach x=0 where perhaps we could say its value is infinity. Moving x further to the left along the number line results in plugging in negative values of x so that 1/x is negative. Thus, Wallis reasoned that negative values are greater than infinity! He also invented the number line in spite of this seemingly bizarre position.

The reason this seems bizarre is because he apparently ordered the real numbers one way (by the number line), but then used it to produce a different ordering. It is nonetheless a valid way to order the real numbers if you account for 0 somehow (you could consider 0 greater than all positives but less than all negatives, for instance). The catch is that numbers greater than 0 no longer have the properties typical of positives. For instance, -1 is greater than 0 in this ordering, so we might consider it positive, but (-1)*(-1) = 1, which would be negative in this ordering. So products of positive numbers would not be positive, a contradiction.

I gave this example to show that even the man who invented the number line thought your conclusion about direction wasn’t the obvious consequence. It goes to show how much we take ideas we consider to be common sense for granted. These ideas only seem obvious to us because we were introduced to them as children.

 

[JuanFlorencio]

In the construction of the integers I described, the existence of these objects was implied the moment that we accepted the set-theoretic axioms used in showing how to construct them. The choice to accept those axioms was the creative act, in a sense. We create mathematics by adopting axioms, and we discover mathematics by proving theorems. That’s how I would put it.

All your answer was very interesting; so, please excuse me if I take only this part.

It is not clear to me how the statement of an axiom could be a creative act. I cannot avoid to see it rather as a descriptive act. But you are talking about a system, the mathematics, not about isolated axioms; and so, I see that putting some axioms together to form a system can be understood as a peculiar kind of creation, or invention, or construction. Then, finally, as you say, the establishment of new relations through the proving of theorems can be seen as a series of acts of discovery. This needs a lot of further clarification, of course, but it seems to me that the chain of acts in the development of mathematics would be “descriptive acts”, “acts of invention” and “acts of discovery”, in that order. At the very foundation, it seems to me, there must be something that does not depend on our will, something that we just describe more or less skillfully.

What do you think?

 

[thinkandmull]

Magnet, my opinion is that the form of everything something can be is in something. So a cup is a drinking aparatus, a book holder, and everything else it can be used for. Or instead it is just its physical aspects and its uses are just its uses. I don’t know which way is best to see it. But a number would exist in the sense of something that is singular. But the singular must be a single solid things. A country or valley can be seen to be many counties of valleys, or just one. Its relative. There has never been a proof of Platonists theory. Never been a refutation, never been a proof. But I think math is best undestood by psychologism. Take a line segment, or rather two of them, one longer than the other. One is longer, until you put the longer one on an angle and draw a line point to point between the segments. So our minds are imperfect in its mathematical skills. The best we can do is understand that things are either fluid or not

 

[thinkandmull]

youtube.com/results?search_query=banach+tarski

 

[Oreoracle]

It is not clear to me how the statement of an axiom could be a creative act. I cannot avoid to see it rather as a descriptive act.

…]

This needs a lot of further clarification, of course, but it seems to me that the chain of acts in the development of mathematics would be “descriptive acts”, “acts of invention” and “acts of discovery”, in that order. At the very foundation, it seems to me, there must be something that does not depend on our will, something that we just describe more or less skillfully.

I’m not sure exactly what is meant by “descriptive” here, but let’s talk about something straightforward such as definitions. Consider possible “descriptions” of lines. The Greeks conceived of lines in the context of what we now call constructive geometry, the primary concern of which is to build complicated structures from lines and circles given a list of rules for manipulating those lines and circles. “Line” was taken as a primitive; a term too fundamental to define. (Euclid did attempt a somewhat lame definition by invoking the idea of dimension, but we now consider dimension a more sophisticated concept.) The closest we get to a description of lines is the list of things we can do with them. For instance, we can connect any two points with a line segment, then we can extend any line segment in either direction indefinitely according to Euclid’s postulates. Thus a line is determined by a pair of points.

That is not how students are taught about lines in geometry today. Today we teach analytic geometry, where our space comes equipped with coordinates. A line in the Cartesian plane is defined as the set of points (x,y) such that ax+by = c for some real numbers a, b, and c. This is an algebraic definition, so only axioms of algebra are needed here. It no longer needs to be assumed that there exists a line passing through any pair of points. Using this definition, that can be proven algebraically.

Personally, the latter definition seems less “descriptive” to me in the sense that it doesn’t seem immediately obvious why a line is an interesting object to study. But we see that whether a statement is a descriptive act as opposed to an act of discovery depends on your starting point. Here, starting with only the rudiments of geometry forces us to describe lines in terms of their properties. Starting instead with facts of algebra lets us discover properties of lines after defining them in a clever way.

 

[JuanFlorencio]

I’m not sure exactly what is meant by “descriptive” here, but let’s talk about something straightforward such as definitions. Consider possible “descriptions” of lines. The Greeks conceived of lines in the context of what we now call constructive geometry, the primary concern of which is to build complicated structures from lines and circles given a list of rules for manipulating those lines and circles. “Line” was taken as a primitive; a term too fundamental to define. (Euclid did attempt a somewhat lame definition by invoking the idea of dimension, but we now consider dimension a more sophisticated concept.) The closest we get to a description of lines is the list of things we can do with them. For instance, we can connect any two points with a line segment, then we can extend any line segment in either direction indefinitely according to Euclid’s postulates. Thus a line is determined by a pair of points.

That is not how students are taught about lines in geometry today. Today we teach analytic geometry, where our space comes equipped with coordinates. A line in the Cartesian plane is defined as the set of points (x,y) such that ax+by = c for some real numbers a, b, and c. This is an algebraic definition, so only axioms of algebra are needed here. It no longer needs to be assumed that there exists a line passing through any pair of points. Using this definition, that can be proven algebraically.

Personally, the latter definition seems less “descriptive” to me in the sense that it doesn’t seem immediately obvious why a line is an interesting object to study. But we see that whether a statement is a descriptive act as opposed to an act of discovery depends on your starting point. Here, starting with only the rudiments of geometry forces us to describe lines in terms of their properties. Starting instead with facts of algebra lets us discover properties of lines after defining them in a clever way.

But…, are we talking about the same thing here or is it just another “isomorphism”? If it is the same thing, then both the line we draw passing through two points in the space, and the equation, are different representations of it. Which one represents it better? Whatever the answer, both are a kind of description of the object we call a “line”, and the starting point for both is unique.

If it is not the same thing, but something similar, then we don’t have to compare their “descriptive value”.

As for the interest that the equation might or might not awaken in us, I think that depends on the context in which the intellectual life of each one of us develops; but that is something else. To me, the relevant point in relation to the thread is the kind of independence that some mathematical objects exhibit and which is shown by the fact that we always have to start with something which is given. That is associated to the “descriptive acts” which I have mentioned before without providing specific examples.

 

[magnet]

Magnet, my opinion is that the form of everything something can be is in something. So a cup is a drinking aparatus, a book holder, and everything else it can be used for. Or instead it is just its physical aspects and its uses are just its uses. I don’t know which way is best to see it. But a number would exist in the sense of something that is singular. But the singular must be a single solid things. A country or valley can be seen to be many counties of valleys, or just one. Its relative. There has never been a proof of Platonists theory. Never been a refutation, never been a proof. But I think math is best undestood by psychologism. Take a line segment, or rather two of them, one longer than the other. One is longer, until you put the longer one on an angle and draw a line point to point between the segments. So our minds are imperfect in its mathematical skills. The best we can do is understand that things are either fluid or not

My point is that numbers can denote quantities of discrete objects. But an ocean is not a discrete object. The Pacific Ocean is an artificial idea, not a real object. So are the Atlantic, Indian, Arctic, and Antarctic Oceans.

Numbers can denote relationships between two discrete objects as long as they are expressed as exact multiples of each other. However mathematical concepts such as pi, and e can never be exactly denoted by numbers. Neither can Infinity.

 

[Oreoracle]

But…, are we talking about the same thing here or is it just another “isomorphism”?

I think it’s more accurate to say the two concepts are in some sense “isomorphic”, yes. For example, there are a minority of mathematicians who would readily accept the properties of lines proposed by the Greeks in their constructive geometry but are skeptical of the existence of the real numbers. To them, the existence of the algebraic versions of lines would seem more dubious or at least appear to be the stronger claim.

Another example of this sort of thing is the rise of constructivism in mathematics. Constructivists dislike proofs by contradiction. Such proofs have the form “Premise 1: A or not A (Law of the Excluded Middle). Premise 2: Not not A. Conclusion: Therefore, A.” They have made it their goal to reconstruct as much of existing mathematics as possible without ever using the Law of the Excluded Middle. Thus their system is classical logic and set theory minus that one axiom. Clearly anything you can prove without that axiom can be proven with its added help, so constructivist mathematics is just a subset of classical mathematics.

So here’s where it gets a bit strange. To the classical mathematician, the constructivist versions of classical theorems are equivalent to the old theorems since classical logic subsumes the logic of constructivists (called “intuitionist logic”). To the constructivists, classical logic relies on stronger, more dubious claims than they would like, so these theorems may not be equivalent from the constructivist’s point of view.

The lesson here is that when you claim two types of objects, statements, etc., are the same in mathematics, you must do so from the perspective of an axiomatic system that is capable of expressing both of those ideas. If your system is too narrow to directly compare the two, then they cannot be the same as far as mathematics is concerned. And there is no absolute perspective from which to view every system, at least not in mathematics. So all claims are system-dependent.

 

[JuanFlorencio]

I think it’s more accurate to say the two concepts are in some sense “isomorphic”, yes. For example, there are a minority of mathematicians who would readily accept the properties of lines proposed by the Greeks in their constructive geometry but are skeptical of the existence of the real numbers. To them, the existence of the algebraic versions of lines would seem more dubious or at least appear to be the stronger claim.

Another example of this sort of thing is the rise of constructivism in mathematics. Constructivists dislike proofs by contradiction. Such proofs have the form “Premise 1: A or not A (Law of the Excluded Middle). Premise 2: Not not A. Conclusion: Therefore, A.” They have made it their goal to reconstruct as much of existing mathematics as possible without ever using the Law of the Excluded Middle. Thus their system is classical logic and set theory minus that one axiom. Clearly anything you can prove without that axiom can be proven with its added help, so constructivist mathematics is just a subset of classical mathematics.

So here’s where it gets a bit strange. To the classical mathematician, the constructivist versions of classical theorems are equivalent to the old theorems since classical logic subsumes the logic of constructivists (called “intuitionist logic”). To the constructivists, classical logic relies on stronger, more dubious claims than they would like, so these theorems may not be equivalent from the constructivist’s point of view.

The lesson here is that when you claim two types of objects, statements, etc., are the same in mathematics, you must do so from the perspective of an axiomatic system that is capable of expressing both of those ideas. If your system is too narrow to directly compare the two, then they cannot be the same as far as mathematics is concerned. And there is no absolute perspective from which to view every system, at least not in mathematics. So all claims are system-dependent.

That seems interesting…, because if you look, for instance, at the way in which Dedekind, Peano and Frege develop their systems up to a certain point when they are able to say “these are the natural numbers”, it would seem -if what you say is correct-, that, even though what each one of them present certainly has a similar structure, it cannot be the same thing, because their systems are different. Either all of them are wrong or only one of them is right.

 

[didymus]

Thanks for all the responses, esp. the book recommendations and links.