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I've searched for some time for a layman's introduction to the world of *actual* mathematical study. Some time ago George attacked an assertion I made about a constant such as PI, citing Stenger's view that physics is simply a product of language and constants only fill a gap. My argument was (and still is) that constants such as PI actually have an exact mathematical value, the hard part is communicating that value to a person not inclined to math. (Most people hate fractions, but in math fractions are *exact* values.) You've all probably heard the term "an approximation of PI is 3.14159..." What is confusing to most people is that the approximation is called such because it is a decimal interpretation of PI and not the ACTUAL value of PI. Adding to the confusion is the fact that PI is actually a non-terminating sequence--most people don't understand what an infinite series is. -The follow article requires no math skill at all, and discusses the history of mathematical study in a way that I think will bring everyone on board with how I think about things. -http://www.thewaytotruth.org/science/mathematics.html-Though the propaganda near the end is a direct assertion of a deity (and the rest of the website is an odd commercial for Islam) the meat of the writing is actually how most mathematicians view their science.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5754 views

xeno wrote: "Some time ago George attacked an assertion I made about a constant such as PI /// My argument was (and still is) that constants such as PI actually have an exact mathematical value"-It's true that pi has an exact mathematical value in Euclidean Geometry. But Euclidean Geometry is not the geometry of the "actual" world. It is a highly idealised system based on axioms. It provides a good approximation to the actual geometry of the real world for many practical purposes, but has been superseded by relativistic and quantum geometry for large and small scales. In Euclidean geometry for instance points and lines have no width and straight lines extend to infinity, and between any two points there is a continuum infinity of other points.-In reality the "perfect" circles of Euclid do not exist. The circumference and diameter of real circles can only be determined to within a certain degree of accuracy, and so their ratio (pi) is within similar error bars. -The article to which xeno links seriously misrepresents the Formalist approach. After all, Euclid was arguably a Formalist. There are also Intuitionist and Constructivist schools which have a lot going for them. It's true that most mathematicians take the Platonist view that mathematics can be treated as a world of "existents", but this is not the same as the "real" world of the physicists. It is an "ideal" world of forms. It "exists" in mathematicians' collective imaginations.

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GPJ

5625 views

xeno wrote: "Some time ago George attacked an assertion I made about a constant such as PI /// My argument was (and still is) that constants such as PI actually have an exact mathematical value"
> 
> It's true that pi has an exact mathematical value in Euclidean Geometry. But Euclidean Geometry is not the geometry of the "actual" world. It is a highly idealised system based on axioms. It provides a good approximation to the actual geometry of the real world for many practical purposes, but has been superseded by relativistic and quantum geometry for large and small scales. In Euclidean geometry for instance points and lines have no width and straight lines extend to infinity, and between any two points there is a continuum infinity of other points.
> 
> In reality the "perfect" circles of Euclid do not exist. The circumference and diameter of real circles can only be determined to within a certain degree of accuracy, and so their ratio (pi) is within similar error bars. 
> 
> The article to which xeno links seriously misrepresents the Formalist approach. After all, Euclid was arguably a Formalist. There are also Intuitionist and Constructivist schools which have a lot going for them. It's true that most mathematicians take the Platonist view that mathematics can be treated as a world of "existents", but this is not the same as the "real" world of the physicists. It is an "ideal" world of forms. It "exists" in mathematicians' collective imaginations.-You are correct that I was speaking of "pure mathematics," and I did forget myself in that, I apologize. You reference non-euclidean geometry, but its not as cut and dry as that. You are correct in maintaining Stenger's assertion that PI is only approximated in non-euclidean spaces, however it doesn't deal with the fact that the number represents a relationship that is independent of two observers. -How about e? This is a transcendental function that appears constantly in nature. Because it repeatedly appears are we to truly believe that it is a relativistic object? This sounds like a level of skepticism not seen since the Cynics.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5566 views

"Pure mathematics" is usually another name for the mathematics of real and complex numbers, otherwise known as "analysis", which involves infinity and limiting processes, like sums of infinite series. It is based on the same foundations as Euclidean Geometry.-You ask: How about e? -The function y = e(x) is the one whose rate of change at any point is equal to its magnitude at that point. That is dy/dx = y, and y = 1 when x = 0. This of course assumes continuous variation through real or complex values. It is thus obvious why such expressions often occur in natural processes.-In practice of course continuous variation does not occur. We tabulate values of y for distinct values of x and use finite processes to calculate them.-By the way, Scepticism, Cynicism and Stoicism are distinct philosophies, and my comments are nothing to do with Stenger. They are my own thoughts.

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GPJ

5589 views

Yeah, I'm definitely in the world of "pure" mathematics in the stuff that I've been studying. Obviously I'm still ignorant about a great deal of physics.-Your explanation on e is well, something I *should* have thought of, but I've been so concerned with theory that I haven't touched application outside of my domain. Most of my work with e has been in euler's formula and related derivations of sin, cos, PI, etc. -What about the important sequences, Fibonacci, Lucas, and of course the golden mean? I've argued previously that at least for the Fibonacci sequence, its pattern lies solely on the previous iteration, basically, take whatever was in the last two steps and add them. But the other two numbers found in nature I don't think can be as easily explained away as e or PI--I might as well mention Mandelbrot here as well.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5597 views

The Fibonacci sequence, and its variant the Lucas sequence, is essentially finite mathematics. Analysis only comes into it if you calculate the limit of the ratio of two successive numbers in the sequences, which is the golden ratio. All this means is that the successive ratios 3/2, 5/3, 8/5, 13/8 and so on are closer and closer approximations to the golden ratio, which is the irrational number (root-5 + 1)/2 = 1.61803.... Fibonacci ratios occur in nature in certain plants because of their regular process of growth. For me they have a greater "reality" than the golden ratio, which only has meaning in the fantasy world of infinitist mathematicians.-you wrote: "But the other two numbers found in nature I don't think can be as easily explained away as e or PI" which two numbers are you referring to? and why would I want to "explain them away"?

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GPJ

5605 views

The Fibonacci sequence, and its variant the Lucas sequence, is essentially finite mathematics. Analysis only comes into it if you calculate the limit of the ratio of two successive numbers in the sequences, which is the golden ratio. All this means is that the successive ratios 3/2, 5/3, 8/5, 13/8 and so on are closer and closer approximations to the golden ratio, which is the irrational number (root-5 + 1)/2 = 1.61803.... Fibonacci ratios occur in nature in certain plants because of their regular process of growth. For me they have a greater "reality" than the golden ratio, which only has meaning in the fantasy world of infinitist mathematicians.
> 
> you wrote: "But the other two numbers found in nature I don't think can be as easily explained away as e or PI" which two numbers are you referring to? and why would I want to "explain them away"?-Well, the Mandelbrot set is a bit more than a single number, but you pretty much covered the rest. And there's nothing wrong in my world with "explaining away." It means I've got nothing to argue with, which kinda sucks from the fun perspective, but it is what it is. -I do hate that you dismiss pure math like that, 99% of computer science both exists because of and relies upon many of these "fantasy objects." Fibonacci relations by themselves are responsible for some of the best search algorithms--there's other "fantasy numbers" that I'm aware of that also enable modern communication. Stirling and Bell numbers, and to be direct, Euclidean groups are also responsible for a computer's ability to construct 3d objects.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5616 views

xeno wrote: "I do hate that you dismiss pure math like that"-I don't dismiss pure maths only the infinitist type.-xeno: "99% of computer science both exists because of and relies upon many of these "fantasy objects."" -You won't find e or pi or root-2 in a computer anywhere, only approximations to them. If you blow up any computer graphics that appear to show circles, you will find that they are in fact polygons.
 
xeno: "Fibonacci relations by themselves are responsible for some of the best search algorithms"-That's probably quite correct, but it's finite maths, not analysis.-Stirling's numbers (of the second kind) count the number of ways of partitioning a set of n into k subsets. Bell numbers are the sum of these. But this again is finite maths.-I recommend the book "Concrete Mathematics" by Graham, Knuth and Patashnik.

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GPJ

5628 views

xeno wrote: "I do hate that you dismiss pure math like that"
> 
> I don't dismiss pure maths only the infinitist type.
> 
> xeno: "99% of computer science both exists because of and relies upon many of these "fantasy objects."" 
> 
> You won't find e or pi or root-2 in a computer anywhere, only approximations to them. If you blow up any computer graphics that appear to show circles, you will find that they are in fact polygons.
> -You are wrong here--you're confusing numerical computation with graphical representation. -PI, e, and several other constants aren't approximated, they use the raw Euclidean constants and hard code it into the math coprocessor on your Pentium or Athlon CPU. When PI is needed for a calculation it simply calls the hardware-coded value, rocks and rolls. This dramatically reduces calculation time at runtime. (2 opcodes vs. a chain of function calls if we're talking C.) -The computation of PI to an arbitrary precision is an open field within computer science, and if you did some digging you'd find that several big labs are also involved in calculating it out--including Intel. It is Euclid's PI that they are comparing their algorithms against. -http://www.cs.berkeley.edu/~ejr/GSI/2000-spring/cs267/assignments/assignment1-results/flab/-IN regards to computer graphics, they use matrix maths to represent 3-dimensional (or higher) objects. You are correct in the department of triangles--but that was a Euclidean result itself, right in the Elements. My overall point was that these matrix representations are still manipulating Euclidean objects in a Euclidean space. -This is irregardless of whether the graphics in question are rendered via shading, 3d projection, or raytracing. -> xeno: "Fibonacci relations by themselves are responsible for some of the best search algorithms"
> 
> That's probably quite correct, but it's finite maths, not analysis.
> -Not probably. 
http://comjnl.oxfordjournals.org/cgi/content/abstract/39/4/331
http://en.wikipedia.org/wiki/Polyphase_merge_sort-As you mentioned Knuth, you'd also be aware that in the book "The Art of Computer Programming" he discusses a tape-drive implementation of a mergesort algorithm that explicitly used Fibonacci. Fibonacci also works as a worst-case upper bound on the Euclidean Algorithm. -Also, tell me where finite maths end and analysis begins: You can take a derivative of the binomial theorem and derive further results relevant to "finite maths" and computing. When I took Combinatorics the professor consistently demonstrated how by using the operations of calculus we would regularly derive new results, and how they applied to computing. Combinatorics is supposed to be a "finite math," but in the words of that professor, "No mathematician can tell you what the difference is between discrete math and other math. The one who does wins a Fields Medal." Maybe one of you?" Probably not, I'll be 35 before I'm done with college. But I'd love to hear your answer!-Generally speaking, "finite math" tends to restrain itself to integer math, but the lines are far too murky to claim that with certainty. -> Stirling's numbers (of the second kind) count the number of ways of partitioning a set of n into k subsets. Bell numbers are the sum of these. But this again is finite maths.
> 
> I recommend the book "Concrete Mathematics" by Graham, Knuth and Patashnik.-Almost insulting: I start my Master's in Computer Science this summer. I've owned "Concrete Mathematics" since my first semester after calculus. I already have the background here. I'm not talking out of my ass.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5687 views

xeno wrote: "PI, e, and several other constants aren't approximated, they use the raw Euclidean constants and hard code it into the math coprocessor on your Pentium or Athlon CPU. When PI is needed for a calculation it simply calls the hardware-coded value, rocks and rolls. This dramatically reduces calculation time at runtime. (2 opcodes vs. a chain of function calls if we're talking C.)"-I'm afraid I just don't believe this. You will have to explain how it's done more clearly. The link you gave admits that pi to ten places of decimnals is sufficient for all practical purposes.-I don't know why you should feel insulted when I recommend a good book. I'm glad you already have it.-xeno also asks: "Also, tell me where finite maths end and analysis begins:" -I'm just using analysis as a shorthand for mathematics based on the assumed existence of real infinities.

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GPJ

5703 views

xeno wrote: "PI, e, and several other constants aren't approximated, they use the raw Euclidean constants and hard code it into the math coprocessor on your Pentium or Athlon CPU. When PI is needed for a calculation it simply calls the hardware-coded value, rocks and rolls. This dramatically reduces calculation time at runtime. (2 opcodes vs. a chain of function calls if we're talking C.)"
> 
> I'm afraid I just don't believe this. You will have to explain how it's done more clearly. The link you gave admits that pi to ten places of decimnals is sufficient for all practical purposes.
> -All I can do is provide some resources:
http://kakku.wordpress.com/2007/10/26/super-pi-how-fast-is-your-processor/
For one to run on your own PC-http://findarticles.com/p/articles/mi_m1200/is_v131/ai_4702871/
For the supercomputing battles of computing PI-Or just google "PI algorithms" to find hundreds of others in multiple programming languages. In all instances, the accuracy of the algorithms are compared to the Euclidean ideal, otherwise how could they tell which one was more accurate than the others? -> I don't know why you should feel insulted when I recommend a good book. I'm glad you already have it.
> -That's why I said "almost." One cannot communicate tone over the internet!-> xeno also asks: "Also, tell me where finite maths end and analysis begins:" 
> 
> I'm just using analysis as a shorthand for mathematics based on the assumed existence of real infinities.-I'm not 100% sure of your math background, but what you call "Calculus," we call "analysis" as it pertains to analyzing the structure of functions, and since you can define anything as a function, you can't really separate "finite maths" from anything else.-[EDIT]
Or maybe you can consider that PI can be perfectly calculated when you assume a Euclidean space. Since we can do that in a computing environment, we can always get an arbitrary accuracy of PI. Does that help?

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5641 views

xeno wrote: "PI, e, and several other constants aren't approximated"-and now offers a programme to run on my computer. This states:
"it takes 37.510 seconds to calculate 1M digits of PI"
So it does not contain pi, it has to calculate it to an approximation. -xeno: "In all instances, the accuracy of the algorithms are compared to 
the Euclidean ideal, otherwise how could they tell which one was more 
accurate than the others?"-Sorry, there is no comparison going on here. The fact that one method of calculating pi is equivalent to another has to be shown by theory.-you ask: "I'm not 100% sure of your math background"-You may get a better idea from this section of my website:-http://www.mayhematics.com/m/m.htm-It's far from complete, and it's unlikely I will ever find time to complete it now, but the sections on Generalities and Numbers may help.

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GPJ

5656 views

xeno wrote: "PI, e, and several other constants aren't approximated"
> 
> and now offers a programme to run on my computer. This states:
> "it takes 37.510 seconds to calculate 1M digits of PI"
> So it does not contain pi, it has to calculate it to an approximation. 
> -Right, but all these different algorithms calculate the same exact identical value of PI! There is a standard, like a meter, that computer scientists use in order to verify the accuracy of a PI-computing algorithm, otherwise, you could just write a code that spits out a random sequence of integers, and call it PI. Obviously this doesn't happen, or nations and corporations wouldn't spend literally billions of dollars on computing a value deemed "approximation only." -You probably won't believe me still, but the only thing I can do is show you a snippet of assembler code that I use when I need PI in C programs. -
inline double PI()
{
 double x; 
 __asm
 {
 fldpi; 
 fstp x;
 }
 return x
}
 
Intel has a hard-coded value on x87 processors, the commmand "fldpi" takes the hard-coded value and pops it into the processor's L1 cache, fstp copies it to real memory. The value on Intel's chip is a 66-bit number, but since the memory address of a double is 64bits, truncation occurs but the value you get for PI is considered wholly accurate for the entire piece of memory, 64 bits or PI to 2^64 - 1 places. Assembly language is considered "machine language" so there is no faster way to get a value for PI. The value for PI that they get is done by the method I've talked about previously--asserting a "pure Euclidean space" and letting it fly. I don't know what else to tell you, this is everything I know! Short of getting an electron microscope and finding the binary representation of PI on the chip fab I don't know what to tell you. -To me, an approximation means that there would be some ambiguity, that if you tried to get the same result twice, there would be a variation. I showed you those other computer algorithms to demonstrate that there is no variation in this. All these various algorithms are computing the same value. Maybe you're familiar with Ramanujan/Hardy's formula for PI? One method for calculating PI that we're taught is an exact value for PI is Machin's formula, but it is an expensive call (even in C) so we use the above assembler method to get the same result. --
> xeno: "In all instances, the accuracy of the algorithms are compared to 
> the Euclidean ideal, otherwise how could they tell which one was more 
> accurate than the others?"
> 
> Sorry, there is no comparison going on here. The fact that one method of calculating pi is equivalent to another has to be shown by theory.
> -Again, all I can do is point you to where I've pointed you. All the algorithms you find "approximate" (in your words) the exact same carbon-copy value of PI. I've given you a snippet of assembler code and spoken as far as all my training in computers can possibly take me on this question. -> you ask: "I'm not 100% sure of your math background"
> 
> You may get a better idea from this section of my website:
> 
> http://www.mayhematics.com/m/m.htm
> 
> It's far from complete, and it's unlikely I will ever find time to complete it now, but the sections on Generalities and Numbers may help.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5685 views

http://download.intel.com/design/PentiumII/manuals/24319102.PDF-On page 250 there's reference to the assembler commands I was discussing. I'm a little off on the architecture--the constants aren't hard-coded on the math coprocessor at all, they're hard-coded on the main part of the CPU. The FPU it references is the logic that handles floating-point operations, and as you can see more than just PI is represented as hard-coded objects in the FPU.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5720 views

Perhaps I should open my own discussion forum on finitism, since I don't want to monopolise too much of dhw's space on this only loosely related topic.-It seems from your links to various wikipedia pages that there are getting on for an infinity of varieties of philosophy on this subject! My own views are not represented there. I don't deny basic logic.-Your statement that algorithms for calculating pi always produce the same result is incorrect, it depends where you stop the calculation. You have to stop it at some stage. Any summation of an infinite series up to a certain value, for example Taylor's or Maclaurin's series, ends with an error term.-In the same way calculating successive ratios of numbers in the Fibonacci sequence doesn't give you the exact golden ratio no matter how far you go.-It may be useful to recognise different categories of "existence" for numbers. Thus numbers you actually write down, say 12345, exist within the current context. Numbers such as pi, e, root-2, i, can be said to exist symbolically since we have a symbol for them, though they cannot be written down in the same complete way as 12345 can.-This idea of different forms or degrees of existence may have wider applications to the agnosticism debate. For instance, in what sense do fictional beings like Sherlock Holmes or Pagasus exist? Do legendary figures like Robin Hood, King Arthur or Jesus have a greater claim to existence? Can gods be said to have subjective existence, though not founded on objective evidence? Just a passing thought.

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GPJ

5653 views

George,-I need a little more help, it appears to me that your argument is still semantic. The way I read your post here, it's like significant digits. You always cut off at the final term--your error term. We have nothing to disagree with.-But the number is *still* considered accurate up until the point you terminate the sequence. This is why the hard-coded number on Intel chips is a 66-bit instead of 64 bit number, so that when the value is truncated on the copy command, the value is wholly accurate for the entire length of the bit string. -If all these different algorithms stop at the same point, and have the same digit at that point, then they are wholly identical--they are calculating the same number. -When I did work in chemistry, we kept 4-digit precision which means that the third number after the decimal was considered suspect. In this case (PI) whatever we call the final digit we call "suspect" but it still means to me that we can calculate PI to an arbitrary precision.-> This idea of different forms or degrees of existence may have wider applications to the agnosticism debate. For instance, in what sense do fictional beings like Sherlock Holmes or Pagasus exist? Do legendary figures like Robin Hood, King Arthur or Jesus have a greater claim to existence? Can gods be said to have subjective existence, though not founded on objective evidence? Just a passing thought.-This is a good point. Some of these figures have archaeological evidence to support their existence--at least King Arthur and Jesus. Jesus is referred to in some Roman historical documents, and in general, there hasn't been many cases I can recall where such a rich mythology was built around person(s) who didn't exist at all. To the extent that we can, we seek evidence of the existence of historical figures. But again, its up to an individual to determine to what extent and to what weights these evidences provide, as historical records are always woefully incomplete. We know almost nothing of my 'nymsake Xenophanes of Colophon, yet we don't declare him nonexistent. Even the Personage of Socrates as a historical figure is fought over in some circles. In the end, some level of practicality must be brought in on the subject: Does it really matter?

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5649 views

If pi is written in the intel chip as correct to 66 bits that's still way off being "exact". After all pi is a transcendental number that is expressed by a nonrecurring infinite decimal. Am I right in thinking that 66 binary digits is only equivalent to about 20 decimal digits? So there are still infinity minus twenty digits to come! If the algorithms always cut off at the same point to ensure they agree that's a bit of a cheat. Even with an accuracy of 66 bits I'm sure there are calculations that can be devised that will give an incorrect result because this is not sufficiently accurate. How about the 1000th root?

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GPJ

5708 views

If pi is written in the intel chip as correct to 66 bits that's still way off being "exact". After all pi is a transcendental number that is expressed by a nonrecurring infinite decimal. Am I right in thinking that 66 binary digits is only equivalent to about 20 decimal digits? So there are still infinity minus twenty digits to come! If the algorithms always cut off at the same point to ensure they agree that's a bit of a cheat. Even with an accuracy of 66 bits I'm sure there are calculations that can be devised that will give an incorrect result because this is not sufficiently accurate. How about the 1000th root?-A 64-bit (double) number has a sign bit, so 2^11 possible numbers, then 52 bits for the fraction, 2^52 possible numbers. Here is where my programmer thinking masked my mathematical thinking. ...you're right. The total translation in precision only comes out to about 16 places on a 64-bit number. This was my fault for not slowing down and remembering my hardware classes. Computers think only in integral terms, you need algorithms to represent floating points. This is why floating point addition isn't necessarily commutative--you're liable for rounding errors. -But I still must ask the question--if its already predetermined that you're only going to approximate PI, how on earth can you reasonably say that any PI algorithm is accurate? Like this paper suggests:
http://mathdl.maa.org/images/upload_library/22/Hasse/00029890.di991740.99p0456b.pdf-If you use this to calculate PI to one billion places, how is anyone supposed to believe that the algorithm is actually doing what it says it does?-The only thing that makes sense is to construct a Euclidean only space and then calculate under those restrictions in order to get a standard to measure by.-(I wasn't suggesting that they were measuring algorithms by the Intel number, only trying to make the statement that good programmers don't "calculate" PI they just make a hardware call for it.)

--
\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5768 views


> This is a good point. Some of these figures have archaeological evidence to support their existence--at least King Arthur and Jesus. Jesus is referred to in some Roman historical documents, -The only roughly contemporaneous mention of Jesus is by Josephus, who was born about the time Jesus was crucified. The entry in Josephus' history is though by many to be a forgery. The first gospel was written about 40-60 years after Jesus' death.

5700 views


> > This is a good point. Some of these figures have archaeological evidence to support their existence--at least King Arthur and Jesus. Jesus is referred to in some Roman historical documents, 
> 
> The only roughly contemporaneous mention of Jesus is by Josephus, who was born about the time Jesus was crucified. The entry in Josephus' history is though by many to be a forgery. The first gospel was written about 40-60 years after Jesus' death.-While I actually remember that (from some far corner of my mind), how large of a consensus is there that Jesus was not a historical figure?

--
\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5591 views

Please could you keep discussions of whether Jesus existed to the thread I have opened on Categories or Degrees of Existence? This thread is about mathematics. OK, I started this digression. I apologise.

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GPJ

5628 views


> > > This is a good point. Some of these figures have archaeological evidence to support their existence--at least King Arthur and Jesus. Jesus is referred to in some Roman historical documents, 
> > 
> > The only roughly contemporaneous mention of Jesus is by Josephus, who was born about the time Jesus was crucified. The entry in Josephus' history is though by many to be a forgery. The first gospel was written about 40-60 years after Jesus' death.
> 
> While I actually remember that (from some far corner of my mind), how large of a consensus is there that Jesus was not a historical figure?-I'm not denying he existed, just pointing out how tenuous any true facts about him are.

5617 views

NOW the whole thing you talked about in dealing with "infinitists" makes sense. Though from what I've seen this seems a wholly semantic argument. -http://mayhematics.com/m/m3_order.htm-I did some more digging (because I was unaware of this debate) and found some more interesting things. -http://en.wikipedia.org/wiki/Infinitism
http://en.wikipedia.org/wiki/Finitism-In terms of mathematics it appears you argue that since we cut off counting at some point then we can consider the system "finite." What then, is the true purpose of mathematical induction when it always asserts that if a statement is true for n + 1 then it is true for all n? Or do you simply prove for n - 1 and then assert for all numbers n? How on earth could you tell the difference? -Doing more digging it appears that Finitism is based on constructivism. -http://en.wikipedia.org/wiki/Mathematical_constructivism-So according to this, an induction proof proves nothing. Constructivists must not have many friends at all! Especially considering that the law of excluded middle is pretty much a cornerstone inside of Computer Science. Digging deeper I find that I'd like a bit more justification on why Finitism is "correct."

--
\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5657 views

http://www.boo.net/~jasonp/pipage.html-hit f3 and search "wrong"-How could he tell what terms were wrong if there wasn't something to compare it to?-(There are plenty of other interesting algorithms for PI here as well.)

--
\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

5680 views

This question has been burning away at me for the past week, so I took the time to drop in on one of the head honchos of mathematics at UNO. His response goes something like this: (paraph)-PI exists if Euclidean Geometry exists. One of many interesting geometric results is that you can create a euclidean space in hyperbolic geometry, and vice-versa. Just because we don't live in a Euclidean 3-space is not an argument for the non-existence of Euclidean 3-space. Have you asked him if sqrt(-1) exists? -And a quote: "One thing that you might bring up is the fact that while mathematical objects are a consequence of their axiomatic basis (language and the rules of the language), they are not time-dependent. Nor are they culturally dependent (relativism). So I wouldn't consider them to be purely imaginary."

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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MATT, quoting a UNO mathematician: One thing that you might bring up is the fact that while mathematical objects are a consequence of their axiomatic basis (language and the rules of language), they are not time-dependent. Nor are they culturally dependent (relativism). So I wouldn't consider them to be purely imaginary.-Since the peak of my mathematical career was a pass at O-Level, I'm going to limit myself to what seems to me a link to our more general discussions. Under "Back to Irreducible Complexity (Part Two)" on 21 February at 20.49, you wrote: "all things we view as causes and effects are as such because we build them to appear that way." I pointed out that this negated about 90% of science, to which you responded: "Yes, I especially mean that for science. Everything we know, we know by language. Everything we know by science as well." I would say the laws of science were no more "time-dependent" or "culturally dependent" than the laws of maths, and if you're going to argue that causes and effects are the result of our own constructions, this seems to me far more apparent in maths than in the natural sciences. At least we can say that thanks to the law of gravity, when I fall off a ladder here on Earth, I go downwards and not upwards, regardless of the language in which we describe the process. I suppose you can argue that taking three bricks away from five would leave two even if we didn't have words to describe the change, but in terms of cause and effect, I'd have thought there was rather more solid "reality" behind my fall than there is behind the mathematical concept. If we're going to talk in terms of reality v. imagination, 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?

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> 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

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> > 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.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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"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.

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"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...

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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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!)-I agree with you that by far the best approach to the problem would be to find a method for creating life, but quite apart from the fact that it won't provide definitive proof of anything, we can't lose sight of life as we see it now. As I keep stressing, it's not just life we have to create - it's also the mechanisms for evolution, which brought out of nothing every single sense, faculty, organ, variation that we know of, culminating (so far) in human consciousness. It's not enough to create a living organism unless that organism can evolve. It may be that a method will actually be found in your lifetime, but I can't see it happening in mine, and we can only base our beliefs on what we know now. In the light of current knowledge, David and George put their faith in design and chance respectively, and there wouldn't be much of a discussion if they didn't! But faith it is, because at present there's no other way to bridge the gaps.

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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."

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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MATT, on the subject of chance v. design: 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:_________! [...] So a "must" is a fitting word here.-You are as stubborn as me! I can't speak for David, but here's the result of your logic, applied to my agnosticism. I don't believe in chance. Therefore I must believe in design. But I don't believe in design. Therefore I must believe in chance. And so now you have me believing in both. Withdraw your "must", or I shall find an effigy of Nietzsche and stick pins in it.-The combination of chance AND design, in the sense of "a creator that tinkers and surprises itself sometimes" is quite a different matter. Here design would refer to the origin of life and the mechanisms of evolution. What happens after that is a mixture of the two, depending on what you think were God's reasons for creating life. In the "brief guide" I've simply speculated on the various possibilities. In other words, the alternative (chance OR design) refers only to the origin.

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MATT, quoting a UNO mathematician: One thing that you might bring up is the fact that while mathematical objects are a consequence of their axiomatic basis (language and the rules of language), they are not time-dependent. Nor are they culturally dependent (relativism). So I wouldn't consider them to be purely imaginary.
> 
> Since the peak of my mathematical career was a pass at O-Level, I'm going to limit myself to what seems to me a link to our more general discussions. Under "Back to Irreducible Complexity (Part Two)" on 21 February at 20.49, you wrote: "all things we view as causes and effects are as such because we build them to appear that way." I pointed out that this negated about 90% of science, to which you responded: "Yes, I especially mean that for science. Everything we know, we know by language. Everything we know by science as well." I would say the laws of science were no more "time-dependent" or "culturally dependent" than the laws of maths, and if you're going to argue that causes and effects are the result of our own constructions, this seems to me far more apparent in maths than in the natural sciences. -I think it would be fair to file a disclaimer that I also recall saying "I'm willing to entertain the notion" or something of that nature, as that particular discussion was in my experience with the Buddhist perspective on life. You're actually dangerously close to what I consider to be my basis for what I believe are "objective truths," namely, constructs that exist independent of an observer. -"At least we can say that thanks to the law of gravity, when I fall off a ladder here on Earth, I go downwards and not upwards, regardless of the language in which we describe the process. I suppose you can argue that taking three bricks away from five would leave two even if we didn't have words to describe the change, but in terms of cause and effect, I'd have thought there was rather more solid "reality" behind my fall than there is behind the mathematical concept. If we're going to talk in terms of reality v. imagination, 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?-
And now we're not "dangerously close," we're on the head. I rate all "truths" by exactly that criteria: Would it exist independent of human existence? From what I know of that prof, he would say "In maths, especially." (I'm not so sure.) For the very structure of our universe as we have unraveled it, we have done so via mathematical objects. An atom is an atom, whether or not we humans exist. Obviously, the only maths we can be sure of, are those maths where we've observed them in nature--the fact that our universe isn't one of a (purely) Euclidean nature is what allowed us to dig into deeper geometries, and it is these geometries that String Theory bases some of its claims upon. (n-dimensional spaces is a fancy word for "Building a bigger cube around another cube." -Things such as the Fibonacci relation occur naturally and this gives credence to number theory as something that studies "real objects," even though the concept of a number itself is something that tends to be quite abstract.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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I've taken a dangerous dive into what for me are the deep and mysterious waters of mathematics. I suggested that the laws of physics would operate independently of human beings, but expressed my doubts about maths. 
Matt: "For the very structure of our universe as we have unraveled it, we have done so via mathematical objects. An atom is an atom, whether or not we humans exist." Perhaps I ought to duck out of the discussion at this point, simply because this is such foreign territory for me, but I'll press on because you will put me right if necessary and I will have learned something.-The moment you talk of atoms and other natural objects (David mentioned trees), I think of the natural sciences, and although of course maths comes into these in one form or another, I don't see maths as laws governing the natural processes of cause and effect, but as patterns extracted from those processes. Is it wrong to define maths as the study of numbers, quantities and shapes? If it's not, then I would suggest ... very tentatively! ... 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. Light travels at 299,792,458 metres per second, there are 2240 lb to the British ton, snowflakes are hexagonal, but these are all man-made formulae. Their artificiality is shown by the fact that the US ton is 2000 lb! Physics, chemistry, biology and the related sciences all deal with actions and interactions that would go on even if there were no humans around. The numbers, quantities and shapes of natural objects would still be there, but I see these formulae as a human description of the results of the physical processes, as opposed to the processes themselves. Is this nonsense?-In your response to David's post on fractals, you have interpreted his statement that "the math formulas in nature are truly amazing" as support for ID. This suggests that for David there is an intelligence applying mathematical principles TO nature (= God), whereas by rejecting ID you are suggesting that there is an intelligence formulating those principles FROM Nature (= man). If I'm right about the latter, can you argue that maths exists independently of man? -*** I see from the latest posts that David agrees with you that 'math truths' exist, and he refers us to Dean Overman's discussion of "the very tight design of the universe". Am I looking for too rigid a distinction between maths and physics? 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?

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dhw,-The relationship between physics and math is heavily convoluted, and indeed is the course of study for an entire branch of mathematical philosophy. -Descriptive mathematics is the kind of thing you're talking about, the differences between tons for example, is an example of a declared unit of measurement, but doesn't really say anything about numbers--its a use of numbers. Lets go back a few thousand years. -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?

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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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. -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.

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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.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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MATT: The Sun would exist just the same without a human to name it "Sun". The same thing that goes for "twoness". [...] 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.-We're on very slippery ground here, and I'm not treading with any confidence. However, it's pleasing to hear that I'm saying some things "of merit", and I'm happy to blunder on if you think the discussion will be fruitful!-You are actually asking me to agree with you before you deal with my reasons for disagreeing with you, which I think is a little unfair, but on a certain level I can do so. Perhaps that's what it boils down to ... levels of existence.-As I see it, certain types of reality like the sun, the law of gravity, the mechanics of heredity are independent of human observation, and language merely represents them. Unlike the sun, "twoness" requires human intelligence to observe, identify and describe it. "Twoness", as you have said yourself, is a concept. In your example, you talked of early man establishing "a commonality" between pairs of rocks, dogs etc. It's this need for man's interpretation that in my view distinguishes the reality of "two" from that of the sun, which didn't require early man to establish anything. I accept that "twoness" exists in nature (this is the level on which I can agree with you), but it can only assume reality/meaning/ significance/substance as a concept because man has observed connections and given them articulate form. In other words, without the language of maths, the CONCEPT of "twoness" (as opposed, let's say, to the STATE of "twoness") would not exist, in the sense that we would not be able to experience it, discuss it, develop it. In fact, I'm not at all sure that it's possible for any concept to exist without language. -Even while I'm writing this, I'm aware of David looking disapprovingly over my shoulder, because I'm ignoring design, and therefore ignoring the "math truths" which he has discerned in the universe. If God designed the universe, I would have to say that mathematical concepts exist independently of man's observation and his language. This, however, is the same as saying that the concept of God depends on man unless God exists.-I'm also aware of George looking disapprovingly over my other shoulder, but that might be my imagination.

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dhw,-The existence of numbers is actually still something of debate among mathematical philosophers, so I won't claim to have an answer. But we both represent different sides of the argument. For sake of clarity I will define how I'm going to talk about this. I know I called "twoness" a concept, but I don't know how to describe a property sans language. I'm trying to say that "twoness" is a property that is observable, and exists outside of language. -Mathematical language is how humans reason about mathematical objects. -Mathematical language requires the ability to observe numbers.-I would be a little more willing to agree that the concept of "twoness" requires language if it weren't for the fact that we know that several kinds of other animals can also observe this same fact--without evidence of any kind of mathematical or other language that approximates human language. For example, Crows on average can count up to 4. I view this as significant because if the numbers 1, 2, 3, 4 were products of language only, than it shouldn't be possible for animals that do not have language to observe the property of "twoness." -{.}
{..}
{...}
{....}
{.....}- My position is that the existence of numbers (I have not yet addressed mathematics!) is no less a physical reality than the existence of any of the other objects we are discussing. We need language to talk to each other about numbers, but we don't need language to observe them. -Maybe a better way would be to ask you, how could "twoness" NOT exist? If the symbols "two" and "2" represent something that doesn't exist outside of language, then how would we prove this? -We'll give this one more shot, but if we still disagree, lets just agree to disagree and start discussing the rest of what you have to say?

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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MATT: I know I called "twoness" a concept, but I don't know how to describe a property sans language. I'm trying to say that "twoness" is a property that is observable, and exists outside of language.-That actually sums up the situation very neatly, and seems to me to coincide with my statement in my last post: "I accept that "twoness" exists in nature (this is the level on which I can agree with you), but it can only assume reality/meaning/ significance/ substance as a concept because man has observed connections and given them articulate form." By articulate form, I mean the language of maths, which almost ties in with your statement that "we need language to talk to each other about numbers, but we don't need language to observe them." Almost. But I don't think it's just a matter of talking to each other about them. We give this meaningless "twoness" a substance that it doesn't have in itself. -You go on to ask how could "twoness" NOT exist? That is not what I'm arguing, as my earlier statement makes plain. The difference between us comes out most clearly through your wanting to put numbers on the same plane of reality as the sun. I like your reference to crows, though. Crows and chimps can count and use tools too. You're right, they can perform such simple intellectual, conceptual tasks without our language (usually when it involves obtaining food), so I can shift a bit further in your direction there. But even with crows and chimps (which are capable of thinking as well as observing), I still can't put numbers on a par with the independent, objective reality of the sun. Maybe what I'm about to say anticipates what you intend to move onto later. If we go from your simple "twoness" to, say, 0.75, or Â½, or 2 x 1 = 2 we can certainly leave the crows and chimps behind. At this level, can we talk of numbers existing independently of human intellect and language, like the sun? I'd say this illustrates how the so-called "property" of your numbers depends on us for its substance etc. -We give names to everything we observe and everything we invent, and it may be that maths is a unique combination of the two: part observation, part invention. I say "unique" because I've been trying to think of a parallel, but can't. Earlier in this discussion, George suggested a link-up with legendary characters (part fact, part fiction), and it would certainly be more in line with our general theme if we could broaden the discussion in this way. However, with mathematical calculations one is able to test the accuracy of conclusions, which we generally can't do with history ... particularly ancient history. -Before we move on, I need to stress again that I'm not arguing from any fixed position. This is a subject about which I know nothing, and I've only entered the discussion because I felt certain statements should not go unchallenged. Ideally, you need a "mathematical philosopher" to take you on! I'm willing to learn, and I'll continue to challenge you, but on the understanding that my arguments are improvised and I have no idea where they're heading.

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dhw, -Alright, we'll move beyond the existence of numbers, though I think much of what I have to say might be automatically shot down just by virtue that I think numbers to be an observable physical property. -We'll start with the question you were most interested in:-"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?" -The difficult part of this deals with mathematics that outlines relationships. For example, it's largely accepted that if a tree falls in a forest, it does indeed make a sound. -I think that indeed, it *is* possible to say what you ask here. Things get incredibly muddied when you deal with creatures that intuitively make calculations... for example, when a flying squirrel leaps from one branch to another, it makes a series of difficult calculations. Namely, it calculates how much energy it needs, its trajectory, and it obviously adjusts for wind-speed and other factors as it makes its jump and glides to the next tree. -Isn't the squirrel doing physics? -Going back to one of your other points, I think its necessary for me to explain more of where I'm coming from.-"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."-There's something that troubles me here, and I'll try to name it. 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. But 2 and 1 (2 + 1) is no longer an observation. Just like when we observe the sun, I submit that when we start describing the sun we begin building a structure of language around it. -2 + 1 by itself means nothing. What does it mean? Are we saying "Two, and then another," or "Another representation of the number 3," or are we meaning this to represent a relationship, such as "two apples and one orange?" -So, in my view, the generic concept of "twoness" is simply a natural property, only when it is taken into language to perform some kind of operation does it "become language." Or, maybe more succinctly, Two is a real object but the moment you reason about it, you are dealing with a man-made language. I don't see how to differentiate between "2" and the Sun. -
"I don't see maths as laws governing the natural processes of cause and effect, but as patterns extracted from those processes. Is it wrong to define maths as the study of numbers, quantities and shapes? If it's not, then I would suggest ... very tentatively! ... 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. Light travels at 299,792,458 metres per second, there are 2240 lb to the British ton, snowflakes are hexagonal, but these are all man-made formulae."-This is something you said earlier that I wanted to discuss. The first bolded statement, I would say that it is physics that does this, not maths. Physics uses mathematical language to describe the universe. But sometimes it is instead inspired by "pure" math. String Theory is a prime example of this. They took ideas from Riemannian and other differential geometries to explain the world. But all ideas in mathematics can trace their lineage back to the natural observation of "twoness." Perhaps numbers might just be axioms that exist only because we need them to--but I'm just not convinced of this.-[EDIT]-Also, when looking at what you discuss as "man-made formulae," lets dissect this a bit more. Metres per second is a language reference, but light travels at a fixed and constant velocity, whatever we call it. The British ton is more arbitrary, but snowflakes being hexagonal is yet another observable property. I'm not trying to say that "six" is a governing law behind snowflakes, but that snowflakes conform to a general semblance of a pattern. -[EDITED]

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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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". -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?-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.

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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!

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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> 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.

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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.
 
There's no disagreement between us on the importance of maths or its relation to our real world. Obviously physics and man-made activities such as engineering and architecture depend on it, and I'll take David's word for it that "the universe and biology are both very comprehensible through math formulas." The reason why I've challenged you is your claim that "twoness" is as real as the sun, and all mathematics trace their lineage back to the natural observation of "twoness". However, you also write: "a voice in the back of my head likes to remind me of the question "How is this question practical?" ... and a similar voice in the back of my head is asking whether it really matters whether twoness does or does not exist independently! 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 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."

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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.

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\"Why is it, Master, that ascetics fight with ascetics?\"

\"It is, brahmin, because of attachment to views, adherence to views, fixation on views, addiction to views, obsession with views, holding firmly to views that ascetics fight with ascetics.\"

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MATT: 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."-Apart from people who study fantasy, I don't suppose many folk devote their lives to studying something they think is not real. But in this case, since maths is an integral part of so much that is demonstrably real (e.g. physics, architecture, engineering), I'm still not convinced that the philosophical "is it or isn't it?" matters very much.-You referred me to a wikipedia article on Nietzsche's perspectivism, which doesn't seem to tell us more than that all views are subjective. In my subjective view, the Schacht interpretation of Nietzsche's aphorisms is, if anything, considerably less comprehensible than the aphorisms themselves, and in the preceding analysis I can't see any difference between concepts defined by the circumstances surrounding individuals and peoples, and concepts evaluated according to culture and context.-You yourself wrote: "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."-This is a much clearer line of argument than the wikipedia one, and I think you should forget Nietzche (and Schacht!). If by "truth" we mean unsolved mysteries like the existence of God, the nature of consciousness, the origin of life, then I'd say there has to be an absolute truth. Whether we are equipped to find it, I don't know. If we mean the material truths of our current world, I think many of them are accessible to science without the interference of perspective. If we mean non-material "truths" ... ethics, aesthetics, philosophy ... then in my view there's no objective truth, no matter how many perspectives you analyse. As for your parenthesis, I agree completely ... agnosticism is indeed a perspective, though I'd go so far as to say it's a more comprehensive one than theism and atheism, since it allows for both.

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