Well, sure, but your choice of axioms is a bit like deciding to search for the Northwest Passage, sail up the Amazon, or go around Cape Horn. It’s not a choice as to which geometry is valid, amongst the one with one, or zero, or infinitely many parallels - they’re all valid. Your definitions, your axioms, decide which mathematics you may possibly see.
Nen - my comments on your first analogy: A tree exists whether or not a noun exists in the english language to describe it. Same with the value of pi.
my comments on your second paragraph: Again, the word “circle” does not exist without language, but my line drawn in the sand does. A ratio without language: I take two strings of equal length. I take one of the strings, fold it in equal halves, and cut it in the middle. I take the longer string, and I notice that by folding it twice it becomes equal to one of the shorter strings. The ratio is two.
This is certainly going “deeper” than I and my friends were, but that is probably better. I notice a number of points in here for the “discovered” option that I can’t come up with arguments for right away, but I’ll attempt to comment on what I can.
First of all, there seems to be a common theme here that we invented a mathematical “language” for interpretation. I agree with this (as do some of my friends). However, I believe that any number greater than “one” does not exist until I choose to see it as more than one. Instead of pointing and laughing at Tymp, I agree with him. I too see a bird, a bird, a leaf, a leaf, and a car… until I chose to see it as two birds, two leaves, and a car. Or 5 objects.
Cabbage and december mentioned other intelligent beings in the universe and if they would develop mathematics like we did. I agree that they would probably develop counting, but are they actually “discovering” it? Mathematics as we know it could simply be the most efficient way to evolve further. For example, different cultures on Earth concieved of the idea of ships to cross the water independently of each other – all of them came to the conclusion that it was the most efficient way to proceed further. I do realize that boyancy is an observable quality, but I am trying to keep the example simple.
Arnold Winkelried, your arguments on pi are the ones that stump me the most. At the risk of being totally off base here, I offer the following example. Assume for a moment that humans are the only intelligence race in the universe (or at least the most advanced). I will be optimistic here and assume that one day we will “invent” a form of complete mass to energy conversion, as in Einstein’s Energy = (mass)(speed of light)^2 equation. Assuming I remember correctly that this has not yet been found to occur naturally, did we invent this form of matter-energy conversion? Or did it always exist, and we just “discovered” it? Now, apply this to pi. Does pi exist until we invent a way for finding its value? This is speculation, though… I haven’t decided if it makes sense or not, even to me. I’ve said it like I thought it, but my mind has a “sticking point” when I try to apply it to pi.
Some of these arguments have knocked some support from my “invention” argument, but I still believe math is purely a concept of conscious thought. I do admit, however, that topics such as counting numbers and Euclidian geometry are harder to argue.
Well, clearly we disagree there. As far as I’m concerned the number of apples is the same as the number of motorcycles whether or not anybody decides to see them as such. Would you say that apples don’t have weight unless we decide to see them as such?
Psi Cop, this connects to something you just said…clearly, I think that numbers greater than one do exists before we “choose to see them”. Certainly if “one” exists, then “greater than one” exists, and the distinctions between “two”, “three”, and so forth are there because two apples are in a (pardon the language) one-to-one correspondence with two hats, but not in one-to-one correspondence with three hats.
The problem is, language doesn’t exist in a vacuum. All language, including the language of mathematics, is at its heart a way of describing the world around us, and for that reason it can’t exist (more accurately, can’t come into being) without the world around us. And while we invented the languages, we didn’t invent the things they describe. We invented the word “apple”, but we didn’t invent apples. We invented the word “red”, but we didn’t invent the colour red; things were emitting light at particular frequencies long before we came along. And in the same fashion, we invented the words for numbers, but we didn’t invent the property of physical things that those words describe, namely the property of being numerous.
Don’t let that stop you!
But how can any language come into being without describing some pre-existing phenomenon?
You seem to ascribe “reality” only to actual objects; hunks of granite and such. But objects also have properties, which they may share with other objects, irregardless of whether anyone comes up with a word or symbol for them, and those properties are just as real as the objects which possess them. We can confirm the existence of those properties through observation, and we can describe them through language, but the properties are separate from the names or labels we give them, and we don’t call them into being.
In the same way as mass exists prior to the word “mass”. If I drop a cheeseburger, it doesn’t need language to go splat upon the ground. The set of planets
This is a subtle point, so I’m going to expand on Godel’s theorem a bit (note that I like to think the umlaut is implied, your spelling may vary…):
Godel’s theorem does not, strictly speaking, say that any mathematical system has undecidable propositions. What it says is that in any formal system X which includes the axioms of number theory, it is possible to write a statement S whose meaning is “Statement S cannot be proved in formal system X”. Therefore, S must be either true, and hence unprovable, or false, and hence provable.
Three things: first, note the use of the word “meaning”. Godel’s theorem hinges on how we as mathematicians interpret the symbols of number theory. That is, it draws a distinction between the symbols themselves and the real-world phenomena that those symbols describe. Second, the statement S is not necessarily undecidable, although if we were able to prove S that would indicate that X is not a good formal system in that it produces false statements; that is, statements whose meaning (after we interpret them in the same way we interpreted S) conflicts with observed reality. Finally, whether or not S is true or false doesn’t depend on which formal system we use to describe the integers; that only affects whether or not S is provable or unprovable. The truth or falsehood of S depends on a different criterion, namely the same criterion we can apply to any other scientific statement: whether or not S describes the real world to our satisfaction.
(in a Transylvanian accent) One, one person agrees with me, ha ha ha!
(Sorry…guess who my favourite Sesame Street character was…)
Finally, after saying all that let me put forth a slight disclaimer. A lot of modern mathematics is just pushing symbols around, and exploring the nooks and crannies of formal systems. (My own work is in the field of hyperbolic geometry and topology which, although it has applications in modern physics, is certainly not the most concrete of subjects.) This means that a lot of modern mathematics is, to paraphrase Tymp and Nen, a poetry appreciation contest guided (but not judged!) by aesthetic sensibilities. Nevertheless, that is not all there is to mathematics. The roots of mathematics are as firmly planted in the observed world around us as are the roots of physics, chemistry, or biology, and hence the subject matter of mathematics was discovered just as much as we discovered hydrogen.
ARGHH! I previewed that TWICE! That last sentance should read: “The set of planets between Earth and the Sun doesn’t need language to have the same cardinality as the set of planets between Earth and Saturn.”
I think the analogy to language is a poor one. We can ascribe different attributes to objects and actions and use different words for them, but I don’t see any room to do that with mathematics.
If I take a pair of apples (or apple, apple) and bring in another pair, I will always have two pairs of apples (or apple, apple, apple, apple). Even if I decide to say that apple, apple, apple, apple = network, that’s just another name. Another grouping of PC, PC, PC, PC has a fundamental analogue to my network. That can’t be changed. You could not have seen it any other way, once you saw it.
If I want to travel in a path such that my position equals its own rate of change, there’s only one (or one kind) of path I can travel on. If I put something with one physical quantity (say temperature, or number of electrons) next to another, and want to know how the distribution changes over time (diffusion is what I’m after here), there is no way that this will not include the ratio of Arnold’s string to the path it described in the sand.
This is, of course, only an applied mathematics (and I’m also ignoring the non-ideal situations of the physical world, which you may or not consider important to this debate.)
I think the invocations of alternative systems provide more support for the notion of discovery. And the other systems work out to be valid but occasionally incompatible. If one or more could apply to a situation, I would say that mathematics is invented. But the fact that only one kind of geometry applies to say, the Earth’s surface, and another kind to a flat plane, means that geometry is immutable and therefore discovered.
Math Geek, I do agree that concepts of greater than one do exist. However, someone could say “there is more than one object” instead of “there are three objects,” and still be correct. But, as I said, I have trouble deciding about counting numbers. At the risk of diverting this further from math, you are correct that we invented the word “apple” to describe the shiny red fruit that grows from certain trees, etc. However, you are working under the idea that we discovered math, like we discovered apple. In my “invented” premise, I think a better analogy would be using the word “love” or perhaps “happiness.” Until humans started feeling these emotions (and later naming them), did they exist? That is how I see math.
Anyone who’s taken a basic graph theory/combinatorics course knows that a planar graph has a vertex v with degree d(v)=<5. (This is a direct corollary of a theorem of Euler about the relation between vertices, edges, and faces of a polyhedral solid.)
The only really interesting thing I was able to prove in my dissertation was that either the graph had a vertex such that d(v)<5, or the graph had a vertex of degree 5 that was incident with four faces of degree 3. (Not a big improvement over Euler, but every little bit helps.)
Now, when did that fact come into existence? When I proved it? When the idea of a planar graph was first defined? In the mind of God, before the foundations of the Universe?
It seems inconceivable to me that this result only came into existence when I ‘invented’ it. Planar graphs have always been bound by this criterion, whether anyone knew it or not.
So was it ‘invented’ by implication when planar graphs were originally defined? And was that when people began studying planar graphs per se, or when they began considering the geometric properties of general polyhedra? Or was there some earlier embryonic moment at which the field first became latent in seemingly unrelated work? As one idea evolved from another, when did the pile of theorems inherent in the field of planar graphs get ‘invented’? Did the ancient Greeks look at geometric solids in a way that could have been said to ‘invent’ planar graphs and the theorems thereon?
That should help explain why I have trouble with the ‘invention’ approach.
But I would like to add one comment: only here, of all the places I’ve ever been (outside an actual math department, that is) could I imagine this debate taking place.
panamajack, ask any mathematician whether or not mathematics is a language and I’m sure they will not assert that the comparison between mathematics and language is an analogy–mathematics is a language just as is an program or dialect.
A tree does exist because it possesses physical properties. The value of pi does not exist without a formal system to define because 1) pi does not exhibit any physical properties, and 2) pi is solely defined by the construct.
The problem is that you still utilize the mathematical construct, namely the element deemed “two,” to describe this thing called “ratio.” Surely, your line in the sand exists, but the ratioe does not. A ratio is an element of the construct. Defining the ratio (stating that the value is two) again refers to the construct. This example does not escape the bounds of language.
Naturally, it can.
Obviously, properties of an object to not appear at our whims, but one must differentiate between intrinsic properties and perceived properties. One can perceive two people standing next to each other; and therefore enumerate two people, yet the number two is intrinsic to the set of two people. Physically speaking, there is a flux in density. There may be, and most certainly are, properties which we have yet to observe, but these properties must be primary in nature, not secondary.
In regards to your response to Tymp, although it was direct at him, I feel compelled to respond. Certainly, apples have mass irregardless of one observing that fact. But the difference lies in the distinction between physical properties and mathematical properties. Apples possess the physical property of mass. How does an apple posses a mathematical property, namely quantity?
As I stated before, what you are doing is discovering; however, the thing you are discovering is a property of the construct we have invented to describe other things. One may invent a system to count, which leads to arithmetic, which leads to calculus, et cetera; however, these modifications are simply realizations afforded by an analysis of the construct. The inherent properties of the construct came into existence at some point when the construct was modified in such-and-such a way such that d(v)<5 is necessarily true of the construct. Still, you are only defining the construct, not a physical property. You may be modelling a physical property, but the mathematical property is simply a model of the physical property. One may delve into linguistics and discover nuances of phonetics, morphology, syntax and semantics, but such endeavors have no bearing on physical entities. Such discoveries are pertain solely to the construct; although they may lead to insight pertaining to the subject of the model, in and of themselves, they possess no physical properties.
I bite into an apple and I encounter sweetness or tartness. Until it hits my tongue, the apple has no flavor at all. Rather, it possesses only an assortment of chemicals that can be described as flavor upon interacting with my own chemistry.
Light reflected from the apple strikes my eye and I encounter red or green. Until the light is reflected and received by my eye, the apple has no color at all. Rather, it possesses only an arrangement of molecules and a relationship with light that can be described as color upon interacting with my own physical being.
A function of my being is to separate or group encountered objects by color and taste.
There is no function of my being that separates or groups encountered objects by number. Unless, of course, one suggests that an aptitude for mathematics which resides in consciousness is a natural organ which some people discover and others do not. While that is a really cool concept, I can’t quite swallow it. Therefore, I suggest that mathematics is a tool invented to ascribe unnatural properties to encountered objects.
Unnatural? Yes, because the only purpose of enumerating and describing encountered objects using my invented math organ is to understand their relationships with me, each other, and the things I want to do with them.
Will the apple fall from the tree? Yes because it possesses a mass that will interact with the earth. How fast will it fall? Now the math organ comes into play and I use invented terms to describe the velocity and acceleration of the apple. How can I accurately strike Nen in the head with this apple from this distance? Blood flows to the math organ and I use invented terms to describe acceleration and trajectory. Will it knock him unconscious? Now I use invented terms to describe the force of impact. I have not discovered the terms I use. I have discovered that I can heft this apple that has just fallen to the earth and, by throwing it, knock Nen out of his chair. Mathematics has played no role in this other than me attempting to describe the whole affair in a humorous account of Nen’s misfortune.
Arnold scratches in the sand and proclaims triumphantly, “Circle!” I scratch in the sand and declare the same. Arnold responds, “No, you idiot. That’s elliptical. A circle has a ratio blah, blah …” and I tune him out while drawing more pictures in the sand. Pi does not exist until Arnold wants to describe the difference between his sand scratching and mine.
This theorem of which you speak was discovered, but the concept of mathematics in which it resides and upon which it s dependant was invented. I have discovered that the word “syzygy” looks really cool and describes something interesting that has always existed. However, my language was invented.
Nen: Apples possess the physical property of mass. How does an apple posses a mathematical property, namely quantity?
Do apples posses the physical property of volume? Or is volume a mathematical property? Does a mountain posses the physical property of height? Is the distance travelled by the earth around the sun in one revolution a physical or a mathematical property?
I believe that the example of the string cut in half does escape the bounds of language (to use your expression). I have two physical objects (the half-lengths of string) that when put end to end match the length of the full length of string. This is a physical property, in the same way that one tree being taller than the other is a physical property of the tree.
Tymp: you give the example of an apple having no flavour until you bite into it. What that means is that the chemical reaction between the apple and your taste buds does not happen until they come in contact. Nonetheless the chemical reaction can be predicted even if you do not eat the apple. That would be like saying “the sun didn’t shine until the first man looked at it.”
You also give the example of circle versus ellipsis. To prove that my drawing in the sand is different than yours, all I have to do is challenge you to reproduce your drawing by using the string attached to the stake and drawing by dragging a stick at the end of the string. You will be unable to do so. As far as pi not existing until I say the magic word “ratio”, I will reply that pi has not been calculated yet. Nonetheless the value of pi will not change whether or not I try to determine the ratio.
I see it this way: We look at the world around us and notice certain relationships (Hey! apple, apple can be put in correspondence with motorcycle, motorcycle!) We invent mathematics to deal with this. (Poof - integers.) Because of Goedel’s theorem(s), we know there will be ambiguities in the result. (Math Geek - IIRC there are two of them - and incompleteness theorem and an undecidability one. I was probably confusing the two in my post. You described the incompleteness theorem - do you remember how the undecidability theorem goes?) We can then look to the world to see which version works (for a particular situation), or treat both (all) possibilities mathematically, i.e. without worrying if it describes the real world.
Hmm - it seems to me that the mathematical system is invented, but then individual theorems are discovered within the system. Does this work?
So much to catch up on. I hope people will forgive me for just hitting a few highlights:
Mathematics is a language. That is not a metaphor.
Most languages (if not all) have syntactical elements that correspond to material observations. That does not mean languages are discovered.
Setting aside for the moment the question of whether pi has a reality divorced from mathematical expression, if it did that fact would haqve no bearing upon whether mathematics was discovered or invented. Iron, caron, tungsten, manganese, etc. all exist in nature. We invented steel.
Returning to the question set aside, pi does not exist in your line & circle reality. Pi is the ideal ratio which your example will only aproximate.
Physical objects have properties which we use mathematics to describe. They also have properties which we use English to describe. We discover the properties and invent the languages to express them.
Planar graphs were invented. They have certain properties. All of those proerties were established at the time of invention (by definition). One of those properties was discovered many many years later by RT (woohoo! You da man ;))
The set of two apples does not exist until the mathematical construct of sets is invented. Neither does the identification “apple” exist until it has been invented. The individual objects exist, but our grouping them together as “apples” is a human invention.
The laws of physics are discovered. The language of physics is invented. (Come on – you knew that was coming.)
dixiechiq, for which side are you presenting evidence? The fact that the two geometries are incompatible means that there are situations where one or the other will simply not work in the ‘real’ AKA physical world, even if they both have applications to other parts of it.
Nen wrote *
I think I was a little too brief earlier. I agree that mathematics is a language. Where I think the analogy breaks down is in what it describes. A spoken/written language is often, maybe necessarily, overlapping and indistinct. Dictionaries are going to be flawed from the start.
Mathematical languages may be expanded, and different mathematicians may provide indistinct explanations of not-well-understood concepts, but it is not the same. Nothing, as far as I know, can be thrown out in mathematics unless it involves faulty reasoning. Additional geometries do not negate Euclidean geometry. Mathematical languages are indeed more similar to computer programming languages (maybe that should be the other way around). Just as one program instruction will always perform the exact same way, one mathematical concept will always describe the same thing.
Consider comparing the language word “chair” to the mathematical term “circle”. Now, someone might call just about anything a chair. It might have legs & a back, or it might not; it might be low to the ground, or they might lean on it; it may or may not even support their weight. What some call a chair some might call only a stool; no consensus is reached.
Once we define the circle, though, it could not be otherwise that the ratio of its circumference to its diameter will be pi. This relationship is what is unchangeable about mathematics (and why we are saying it is discovered).
I think what I’m trying to say (and it was probably said already) is that while we invented the terms, we did not invent the relationships. It is the unchanging status of those relationships that I’m calling mathematics. Maybe you think this is an unfair definition. But I would challenge anyone to generate an entirely different kind of mathematics. If mathematics is invented, that should be possible, or why is it not?
I noticed this when about to submit (from Spiritus Mundi):
6) Planar graphs were invented. They have certain properties. All of those proerties were established at the time of invention (by definition).
But, as RT mentioned, what about the relationship they have with polyhedra? And what about the relationship polyhedra with other parts of geometry? I don’t think planar graphs could have been invented any other way.
FWIW, I am an instrumentalist when it comes to Science, but that’s a different debate.
Hmm. . .Tymp, how is it that my misfortune, in this case, projectiles directed at my head, tend to be the examples of your points? All I did was point and laugh.
Yes, this truth was discovered, but this is a discovery about the construct. I can’t assert when it had been true. Mathematical languages evolve over the years just as does any other language. I’m not sure at which point in the evolution Fermat’s Last Theorem became a truism. The point is, there was a point in time somewhere before the proof was postulated that is was possible for it to be true, yet, the possibility is limited by the evolution of the construct.
Well, Arnold, I was really hoping the debate wouldn’t lead to a metaphysical analysis of the nature of the space-time continuum, but I’ll give it a shot. This is going to get dicey, so I’ll start with the easy stuff.
I’ll defer to Spiritus Mundi’s explanation in item #8. “The laws of physics are discovered. The language of physics is invented.” The statement does apply to Newton’s laws of motion, yet the ramifications differ from what you state. One discovers things about physics. One discovers things about the language which was invented. This does not mean that physics is invented–just the language of physics.
I disagree, and my response is directly related your discussion with Tymp. Objects have innate physical properties. A tree has height. The sun has mass–and radiates as energy and mass are equivalent. These properties are primary properties. The act of observing introduces other properties. A tree does not have a property of being taller than another tree. The tree has the property of occupying space (let’s neglect a definition of space for the moment). One tree’s occupation of space is independent of another tree’s occupation of space. Observing or comparing to objects reveals a ratio between the two heights. This ratio is not inherent to either tree. This ratio is an element of the construct we utilize to express the difference between the two trees.
Similarly, the apple has a specific molecular structure. This structure is intrinsic to the apple–it is a primary property. But the perception of the apple, whether it be sight or taste, is a secondary property. The apple is not red, but it’s particular structure dictates that when light reflects off the apple, the wavelength of light perceived correlates to the color red. Although this perception can be predicted, “redness” is not an intrinsic property of the apple. “Redness” is simply and auxiliary property as a result of perception–just as is the concept of “ratio.”
For the sake of ease of argument. I would assert that displacement, whether it be one, two or three dimensional, is a physical property. Our language to describe it is mathematical. Yet again, I state that we have defined a construct, or mathematical model, to speak of a physical property.
Okay, now we come to the tricky part–defining dimension. There are two extremely bizarre phenomena we experience: one is space and the other is time. Unfortunately, we can’t grab hold of a piece of time and analyze it, likewise with space. One can’t grab hold of a magnetic field, but one can observe its effects on objects. How does one observe the effects of space? How does one observe the effects of time? (We may think we experience aging, but do we really? Show me that bit of time from a few moments ago. Show me that bit of time a few moments in the future. Until we can establish more than one point in the continuum, we can’t truly define the continuum–we can’t give credence to it’s reality except on a moment by moment basis). We construct devices to measure spatial and temporal displacement, but these measurements are that of auxiliary properties.
Unfortunately, I must state that the space-time continuum is a construct. It is the vehicle through which consciousness is born; enabling us to perceive. I think I just opened up a big convoluted and self-contradictory can of worms. Egads.
Methinks so as well, thanks for the elaboration.
The term “chair” is a very loose term, thus many thing can be included in its category. The term “circle” simply has a more precise definition. Were we to more rigorously define a set of objects with another word, say “Lazyboy,” the relationship that this-chair-is-a-recliner-produced-from-this-particular-manufacturer is unchangeable. Did we discover Lazyboys?
Firstly, entirely different kinds of mathematics have been invented. Calculus and Riemann geometry are examples, but perhaps this is not what you mean since they share a basis with other mathematical languages.
Secondly, I don’t disagree with your assertion that we didn’t invent the relationships. We discovered the relationships, but these relationships are elements of the construct we invented.
Yeah, the first one says there are undecidable propositions, the second one says that, working within a formal system (that includes arithmetic), it’s impossible to prove the consistency of the formal system.
What I was about to post was along the lines of panamajack’s post. I was going to make a distinction between the language aspect of mathematics, and the mathematics that is studied by that language. For example, the Peano axioms of arithmetic were mentioned earlier; these were, I’ll grant, invented a little over 100 years ago. Obviously, the study of arithmetic goes back thousands of years before that. What was invented was the language used in the study of arithmetic, formally by Peano, less formally by those before him. In either case, however, it seems quite clear to me that the observable properties of the counting numbers dictated the choice of the axioms, not the other way around. Again, this indicates a discovery to me, not an invention.
The point being that it’s impossible to change the properties of counting. Quantity is certainly a physical property, as is the property that x has a greater quantity (of whatever) than y. If a given pile of apples has more apples in it than another given pile, that fact is independent of any language being used to describe it, in fact, it’s true even if there is no language to describe it. Is there a minimum possible quantity of apples? Of course, and whatever words or symbols you use to denote that fact will have the same meaning as the English word “zero”. Assuming you have some apple(s), again, is there a minimum possible quantity of apples you have? Again, of course, and call it whatever you want, the labeling you give it will be equivalent to what we call “one”. And so forth.
We invent the words, numerals, symbols to describe the properties of the quantities of apples we observe (discover), and those discovered quantities must necessarily be the counting numbers.
You know I agree with you, Nen, but I would like to take this one step farther.
I think that the laws of physics are invented just like the language used to describe them. I argue this back and forth with engineering students all the time. I believe that one cannot presume the truth of a set of laws merely from their ability to predict and explain perceivable phenomena.
Our systems, of course, are limited by our perception. So for all we know, the fact that the laws of physics actually work could be mere coincidence. The laws of physics are useful because they work, to be sure. But it would be arrogant to presume that they are true just because they work, for it implies that our current perception of phenomena is so total that only our explanation could possibly hold under all conditions.
Aristotle thought that understanding all matter as composed of five elements was useful for explaining and predicting phenomena. And for all intents and purposes, he was right. We simply don’t know how much farther along the progressive spectrum we are. Sure, we can explain a bit more than Aristotle could, but we still can’t truly see the “bigger picture.”