We could probably go on indefinitely offering examples and counter-examples to show that "twoness" does or does not exist independently of human observation and language. I think my example of the snowdrop and snowflake (yes, two flowers, but no, not two snowdrops) shows pretty conclusively that "twoness" requires a human act of association, but your helium v. hydrogen v. lithium makes a good case for your version. I was inclined to leave it at that, but out of interest I googled "Do numbers exist?" and found an article by Lee Lady: www.math.hawaii.edu/~lee/exist.html. I'm afraid I haven't had time to read and digest it fully, but I noted down the following: 
> 
> "The prevailing opinion among mathematicians, at least as far as I know, is that mathematics has to do with a man-made universe, a mental universe, completely separate from the "real world," whatever that may be. But it takes a highly intellectually sophisticated mind to think that supernovas and electrons are real but that numbers such as 6 and 59 are not."
> 
> This suggests he's on your side (except that he thinks the opposition is more widespread than you do). However, it may not be so, as you will see later from another quote.
> -In truth, mathematicians admit there's no existence proof for numbers, but the prof I pseudo-quoted told me something like "At the end of the day, and behind closed doors, few mathematicians feel they are studying something that isn't real." Having been exposed to this debate however, all the discussion happens in mathematical philosophy journals, and isn't typically broached in any class I've ever taken. -> ... Much more important to me is whether mathematicians are in a position to explain the mechanisms of life and the universe, as David suggests. And do their formulae imply a conscious intelligence at work (David's view), or a natural, unconscious order of things (George's view). Here is another quote from Lee Lady (but other passages in the article suggest he is not religious):
> -I view that David is correct as well. Insomuch as mathematical structure can be observed.-> "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."-To complete what I'd originally set out to do (and using the basic-chemistry example I began previously) if you have *any* kind of countable things, you can begin to build and infer about them. I undoubtedly recognize that the vast swath of mathematics deals with language, but all mathematics are logically built upon axioms for their structure. Yes, in many cases these axioms are either tautologies or improvable statements.-
As for what to do about the greater part of your questions, this short article will do: -http://en.wikipedia.org/wiki/Perspectivism-Again, a Nietzschean concept, but one that very clearly shapes our debate. It may well be that it is absolutely impossible to divorce yourself from a perspective. (Even agnosticism is a perspective.) Only by analyzing all competing perspectives can we possibly reach a truth, if one even exists.


> Though the existence of numbers is intriguing to me, a voice in the back of my head likes to remind me of the question "How is this question practical?" At least our other discussions--the three you mention above--all have real-world implications for everything we humans do!-What amazes me is the same thought as Einstein had. The universe and biology are both very comprehensible through math formulas. Did trees incorporate fractals? And so on.

Matt suggests that squirrels are intuitively doing physics when they leap from branch to branch. You might say that I'm also intuitively doing physics when I catch a ball (and I'm making a mess of physics when I drop it), or run across the road (hopefully before the bus can hit me). Even blinking can be broken down into scientific terminology, but it's only humans (or God, if he exists) that need and are able to do this. I can see that you're trying to draw a parallel here with numbers ... twoness exists just as squirrel-jumps exist just as the sun exists, and you've elaborated on this. You say: "To me, language doesn't actually enter into mathematics until you need to start doing operations with the numbers. "2" by itself is a property; an observation." I would say that "twoness" already requires an operation, but perhaps you can clarify this for me by dissecting an example, along the lines of your early man who put two rocks etc. next to each other. In my garden I observe a flower and another flower. With great pride I tell my wife that we have two snowdrops. I presume you would argue that the snowdrops have an objectively existing "twoness". However, as my wife knows all too well, I am an ignoramus. She points out that Flower A is a snowdrop, and Flower B is a snowflake. They are different. What, then, does the "twoness" relate to? Before there can be a twoness, doesn't there have to be a connection, an identity? How is that established? The flowers in themselves have no property of twoness. I must give it to them. Yes, there are two flowers. No, there are not two snowdrops. So is there an independent "twoness" or isn't there? It seems to me that if "2 + 1 means nothing", as you say, then "2" also means nothing until I have performed "an operation". 
> -You see, I look at this exactly the opposite way. Lets try this. Grab a pebble. Is it made of atoms? The knee-jerk response is "yes," but without sophisticated equipment, you're not going to be able to actually observe atoms. In my view, the "twoness" of your two flowers is just as much a property you can uncover as the fact that the pebble is made of atoms. The difference simply relies upon whether or not you observe the property. -Lets try perhaps a more difficult scenario. Helium is made of two and only two protons. What prevents us from stating that "twoness" isn't a real property in this instance? In my mind, since the atom is made of protons, and the defining characteristic of Helium is two protons, then the existence of "twoness" can be verified in this instance. Without the existence of "twoness," Helium wouldn't exist. If two protons are the necessary and sufficient conditions to make an atom of Helium, how can we assert that the "twoness" isn't a physical property? If its only one proton, it's hydrogen, and if its three, its Lithium. If there isn't some observable physical property of "twoness" then how can we base this upon fact?--> I'm in no position to judge the extent to which maths and physics overlap in extrapolating patterns from the natural processes of cause and effect. Nor do I know enough about the history of maths to comment on your statement that "all ideas in mathematics can trace their lineage back to the natural observation of "twoness"." You did say earlier that mathematical philosophers are still debating the existence of numbers, which suggests the issue is not so cut and dried. You also say numbers "might just be axioms that exist only because we need them to", but you're not convinced. I'm certainly not the person to convince you! As for the man-made formulae, they all provide terminology for existing objects or actions. I don't think we have any disagreement there, do we?
> -To be fair, *only* mathematical philosophers have that debate. By and large, most mathematicians (the prof I mentioned included) might not even have an opinion on the issue by the simple fact that well, since numbers work in the "real world," they obviously exist in some form, whether by necessity of language or by observable property. No real disagreement on how math is used in physics--they dialog and catalog real and existing relationships. -I think at this point, we can begin discussing how purely mathematical objects have found a foothold in our very real universe. (Not all objects are "numbers" per se, but I'll try to find a good way to keep that discussion at a "high level." (In computer-science, "low-level" means an increasing order of detail and mathematical description.) -> I should add that I greatly appreciate the trouble you're going to over this. I'm still unsure where it's heading, but then I'd say the same about life, evolution and the universe.-Though the existence of numbers is intriguing to me, a voice in the back of my head likes to remind me of the question "How is this question practical?" At least our other discussions--the three you mention above--all have real-world implications for everything we humans do!

MATT has commented on David's reference to fractals: "This ID argument operates under the notion that since we have to be intelligent in order to unravel the nature of nature, then nature itself must have been designed by intelligence. This parallels arguments used by dhw."
> 
> Let me once again register a protest against all your "parallel" arguments in which, as always, you insist on using the modal auxiliary "must". I don't recall even David using a "must", and I certainly haven't and wouldn't. My own argument is that the combination of materials necessary to create life and the mechanisms of reproduction and evolution is so complex that I find myself unable to believe in chance as a possible explanation, and therefore cannot discount design. If you want to pin me down to figures, I'd say 50/50 for each theory. So please banish this "must have" from your vocabulary! (At least now I reckon you will have learned something from me, because I'll bet you didn't know "must" was a modal auxiliary!)
> -Maybe it's just me, but when I hear the delineation "chance" OR "design," the structure of the claim is such that it will be one or the other and not both. I realize that you're a "fence-sitter;" (a good place to be) but you often argue the devil as do I. To me, a parralel argument isn't identical it is similar. Two lines can be parallel but reach different points. You don't use "must," and David hasn't publicly said "must," but it is implied via David's argument and my understanding of Adler's techniques. David clearly doesn't believe in chance, and since "OR" is exclusive--if it's not chance it "must" be: ___________! David has mentioned that he doesn't enjoy using a negative argument, but he does. So a "must" is a fitting word here. -I've mentioned before that there is a false dilemma, and it leads to logical hell. If we say "chance" AND "design," this technically covers all possibilities to the point of possibly being a tautology, but if we take similar interpretations as David's, that's precisely where the argument leads us--a creator that tinkers and surprises itself sometimes. But at least in your primary theses on this website, I don't recall you restating the hypothesis as such, it is still "chance" OR "design," something as difficult to prove, especially since OR is exclusive. So I recognize as still that you're "in league" as much as I am, but I see no reason to eliminate "must."

MATT (bold type mine): Prior to writing a symbol that represents the number "2," early man had a concept of "twoness," where you could put two rocks, two dogs, two people, and say two apples next to each other and were able to reason that all these different objects had a commonality--the aforementioned property of "twoness." So, for certain, mathematics had its birth in the very real world, by observation of a natural phenomenon--"twoness." So in that light, I would ask you if you think numbers are real--like the sun, or are they purely imaginary?
> 
> My (tentative) suggestion was "that numbers, quantities and shapes are part of the language we use to systematize our observation of nature, whereas the laws of physics, though we express them in words, operate actively and independently of our observation." I can't see any difference between us here, if you accept that the symbol "2", like the word "two", is part of language. 
> -It is on this point that I stopped. "two," and "2" are both a part of language. But my more subtle point is that they represent a real property. The Sun would exist just the same without a human to name it "Sun." The same thing that goes for "twoness." We need to come to some kind of terms or common ground here before we continue--you say other things here of merit, but I feel that we need some kind of agreement that the concept of "two" is as independent a fact as the existence of the sun, atoms, and molecules. -You seem to be asserting that two is a concept that requires language. I disagree. But everything I discuss about math will stem from this topic.-> You wrote on 3 March at 15.35, "I rate all "truths" by exactly that criteria: Would it exist independent of human existence?" Your starting-point above is early man's concept of "twoness", which already depends on human existence. If there were no humans, there would be no concept of "twoness", but there would still be a sun. I'm not enamoured of the real v. imaginary dichotomy ... the word "football" represents something real, but if there were no humans, there would be no football. I wonder if a better alternative might be natural and man-made. I would (again tentatively!) suggest, in opposition to the title of this thread, that language is a man-made system used to represent all aspects of the world we live in, and mathematics is the form of language that represents numbers, shapes, quantities etc. If there were no humans, there would be no language, including maths, and no "concepts", but the natural objects and processes represented by language would still exist.
> 
> In view of the convoluted relationship between physics and maths, and to accommodate David's belief in design and related "math truths", I'd be interested in an answer to the question with which I ended my last post: "Could one perhaps say that mathematical calculations are needed to measure or predict the physical effects of physical causes, and that without humans such calculations are not needed and cannot be made, except by a possible God?" However, I don't want to drag you into a discussion that may not lead anywhere, and I only entered it myself out of curiosity about certain claims that were being made. So do feel free to end it if it seems unproductive.-I fully intend on dealing with everything you've said, it will be very fruitful, but any disagreement we have will hinge on this notion above.

"A fractal requires a programmer in order to be generated, therefore fractals require intelligence to build!" They then extrapolate that to nature at large, especially life. 
> > 
> > A flaw--I can find more, but I will use one and only one word for my refutation:
> > 
> > Snowflakes. 
> 
> My thinking is more on the point that math exists somewhere, whether we develop it or not. (A point made by many philosophers, and others). Fractal formulas are fractal formulas with or without plant and tree dendrology as a human study to find them or compare them. I agree with you that 'math truths' exist whether we are lead to them by the study of the universe, nature or whatever. What is extraordinary is that plant growth follows those fractals. Why are they built into nature and expressed in the DNA? That is the point the ID folks raise. Certainly many organisms don't follow fractals. Snowflakes look like that because of the arrangement of water molecules. No fractal here, but a crystalization of a pattern. On the other hand coastlines can follow fractal patterns. I read "Chaos", by James Gleick , mentioned in the ID website, years ago. I think it is very important reading for anyone interested in our discussion.
> 
> Apropos of all this view is Dean Overman's "A Case Against Accident nad Self-oganization", 1997, a carefully explained discussion of the very tight design of the universe, discussing most all of the parameters that exist in very defined limits.-All taken. Though I will state that snowflakes are considered fractals, using Koch's snowflake I can create a "unique every time" snowflake using his recurrent relation. There's no formula to predict a snowflake, but they do always conform to the hexagonal structure pertubated up from the chemical structure of ice molecules. From utter simplicity you get a structure of roughly 10^18 molecules that is unique every time. If you break that down to structural components you would have 10^3 hexagonal rings. Hmm. Sounds like a thesis project...


> > I wonder whether it might be a fair test to ask whether something would continue to exist in the absence of human beings. I suspect that the laws of physics would manage to carry on pretty well without us. How about maths?
> 
> The math formulas in nature are truly amazing. The following is a discussion of fraactal formulas in nature. They are there with or without us. Studies of dentrology show that fractals describe branching patters as shown in the following essay from the dreaded ID website:
> 
> http://www.uncommondescent.com/biology/formulas-and-forms/#more-12123-Their argument here is truly one I've seen oft-repeated here. "A fractal requires a programmer in order to be generated, therefore fractals require intelligence to build!" They then extrapolate that to nature at large, especially life. -A flaw--I can find more, but I will use one and only one word for my refutation:-Snowflakes. -Okay, I'll explain a bit more. This ID argument operates under the notion that since we have to be intelligent in order to unravel the nature of nature, than nature itself must have been designed by intelligence. This parallels an argument used by dhw. I could rephrase this exact same argument to state "Because the universe cannot be described simply, it must have been created by intelligence." A further refinement: "Because some problems about the nature of the universe stump our best minds, it must have been created by a smarter mind." A final factoring: "Because we can't explain the universe, it must have been God." All of these arguments are identical. And fallacious. -One of the great insights of chaos theory is that it is incredibly difficult to observe the origins of something when you're caught in the midst of the system. Or, when you're in the middle of a "chaotic" phenomenon, you have no purchase because everything looks chaotic. In my mind, this is where we're at in terms of studying life. I stress again, that I think it's a wrong path to start with life as we see it now, and try to roll it back to the beginnings. In my mind, this is why ID fails, because it's arguing in effect, that the existence of a car itself is an argument for the existence of a human being. -The *right* path is to put as much effort as possible into creating life *in any way we can.* Shapiro and David are both right in stating that we won't know which method is the "right" one, but once we have "A" method for creating life, this gives us the purchase we desperately need to try and solve the problem of origins. (It will be another beginning itself!) Shapiro's skepticism is valid but really does nothing more than state "abiogenesis won't be the end of faith."-The fractal argument itself would have merit if there were no self-assembling systems at all in the universe, but we know there are, so the conclusion is held back by the existence of a mere snowflake. No, its not as complex as life, but it means that we're examining a question that turns on a degree between life and nonlife, which at present isn't exactly a solved problem.