I couldn’t understand most of the proofs I’ve seen for 1 = .999… But I haven’t understood a proof yet that I didn’t see a fallacy in.
So, since there are non-fallacious proofs, you’ve simply not understood one of them. The relevent question then is why you didn’t understand them.
2^2 apples is 2 sets of 2 apples. Well, that’s 4 apples. 1^3 apples is one set of (one set of one apple)) which is just one apple. But now what about 2^0? Zero sets of two apples? Well, that is one apple.
Now this makes absolutely no sense and has no relation to reality. So how do we as philosophers go about handling it? How do we reconcile the need for concepts to be tied to reality with the existence of mathematical concepts that have no tie to reality?
Since it resembles nothing I’ve ever read in mathematics, why should I be concerned about it at all other than as a gross misstatement of an ordinary mathematical concept? Where did you ever see a text in mathematics state a “mathematical concept” that “2^2 apples is 2 sets of 2 apples” or that “2^0? [is] Zero sets of two apples [which] is one apple”?
When dealing with simple exponentiation on natural numbers, say j and k, a correct and ACTUALLY proffered mathematical concept is that k^j is the number of functions from j into k (where j and k are natural numbers construed to have j and k number of members, respectively (note that is NOT a DEFINITION, thus not introducting a circular definition)).
With your apples, 2^0 apples is 1 apple, since 2^0 = 1, since there is exactly one function from 0 (the empty set) into 2, where 2 is construed to have two members such as does {0 1}.
Or, we may simply take heed of the recursive definition:
k^0 - 1
k^(j+1) = (k^j)*k.
Thus, focusing on natural numbers, exponentiation may be defined in at least two equivalent different ways:
(1) Recursively:
k^0 = 1
k^(j+1) = (k^j)*k (where ‘*’ stands for multiplication)
(2) k^j = cardinality of {f | f is a function from j into k}
By the way, both of those methods can be justified both in ordinary informal mathematics and in an ordinary set theoretical formalization. However, as an historical note, if I’m not mistaken, it was Godel (using the Chinese Remainder Theorem) who proved that exponentiation can be defined in a system as spare as (what we now call) first order Peano Arithmetic.
Or, we may simply take heed of the recursive definition:
Let me fix up your typing:
k^(j+1) = (k^j)*k.
let j=-1
k^0 = k/k = 1
Which is what I suggested a few pages ago. All that crap about mapping functions or whatever is waaay too abstract to minimally address this problem.
Let me fix up your typing:
k^(j+1) = (k^j)*k.
let j=-1
k^0 = k/k = 1
You didn’t “fix” my typing, since my typing was correct.
Also, your formulation is not the method of recursive definition on NATURAL numbers, as I specified that my formulation pertains to the NATURAL numbers (extending to the integers in general is another matter).
Again, the standard, and a correct recursive formulation of exponentiation on natural numbers (as can be found in many of thousands of basic texts on the subject) is:
k^0 = 1
k^(j+1) = (k^j)*k.
All that crap about mapping functions or whatever is waaay too abstract to minimally address this problem.
My purpose was not necessarily to “minimally address” anything but rather, as I alluded, to provide a CORRECT understanding of the COMBINATORIAL significance of exponentiation on natural numbers, especially in view of some bizarrely incorrect interpretations that had been posted in this thread. What you refer to as “crap” is quite basic mathematics, correct as I typed it, and serves the aforementioned purpose - understanding exponentiation in a general sense and as a first step toward extending to integers, rational, and reals, as anyone may even better inform himself (past, e.g., “mapping functions” [redundant] or whatever") by looking in the early chapters of many a textbook in undergraduate mathematics.
Please, if there is some MATHEMATICAL error you think you have found in my posts, then I am happy to correct myself or to explain why the content is not correct, as the case may be. But I hope (can’t predict) that I would have the restraint to leave unanswered further gratuitously insulting characterizations (such as " crap") of correctly stated mathematics.
P.S. For that matter, as anyone who has even a smattering of knowledge in just beginning undergraduate combinatory analysis would know, the notion of exponentiation as the number of functions from one set into another provides the basis for the calculation of many even basic problems as applied to probability, odds, and statistics used for everyday physical sciences. So the notion is hardly “waaay to abstract” toward understanding such questions as why k^0 = 1, but is indeed fundamental to understanding a motivation that has practical application beyond merely stipulating that 1 is the value in the base case of some “arbitrary” [my double quote marks] recursive definition.
The language used is inaccessible, to put the case mildly. You may as well have written it in sanskrit.
The language used is inaccessible, to put the case mildly. You may as well have written it in sanskrit.
Mathematics is a technical discipline that has a technical vocabulary than can be simplified only so much. May I suggest that if you wish to understand mathematics, you should “learn Sanskrit”. The language is NOT inaccessible. It is accessible to those who wish to learn it and take the time and effort to do so. I was able to teach myself calculus at the age of thirteen and set theory at the age of fourteen (that was many years ago). I taught myself mathematical logic at the age of sixteen.
Bob Kolker
Not quite. The rigorous theory of real numbers was developed from the middle of the 19th century onward. Not all mathematicians were happy about it.
Please see: http://en.wikipedia…opold_Kronecker
Leopold Kronecker was of the opinion that all of mathematics should be based on the integers. It was he who said “God made the integers. The rest is the work of man”. Kronecker did not accept Dedikind’s construction of the real numbers (Dedikind Cuts) nor did he accept Cantor’s theory of transfinite numbers and the theory of sets. Other mathematicians insisted on strictly finitary constructions, for example Poincare and Brower.
In the 20th and 21st centuries abstract formalism has won out in the market place of theoretical mathematics, but it was not without a fight. Cantor himself was a casualty of the philosophic wars concerning the nature of mathematics. The emnity he faced in academic circles aggravated his tendency to depression and he spent a good part of his adult life in mental institutions. Please see: http://en.wikipedia…ki/Georg_Cantor
Bob Kolker
I’m not sure how this suggests that we don’t have a firm grasp of what the real numbers are.
You guys are making this way too complicated. Mathematics is the science of measurement as I am sure most of you are aware. The purpose of mathematics is to be able to derive values for any measurable aspect of reality , using certain logically derived, objective methods.
Are these methods sometimes highly abstract and beyond an immediate connection to perceptual experience? Sure. Do some methods seem to be arbitrary? One might think that, but if they are logically derived from the basiic axioms of mathematics and allow one to acheive real-world values then they are not arbitrary at all. What matters here is that the mathematical methods used are not complicated beyond neccessity and that they can be used to acheive reasonably reliable and accurate results. Not only intuitive they may be, or how well they relate to every day experience.
Are these methods sometimes highly abstract and beyond an immediate connection to perceptual experience? Sure. Do some methods seem to be arbitrary? One might think that, but if they are logically derived from the basiic axioms of mathematics and allow one to acheive real-world values then they are not arbitrary at all.
So far, so good.
What matters here is that the mathematical methods used are not complicated beyond neccessity and that they can be used to acheive reasonably reliable and accurate results. Not only intuitive they may be, or how well they relate to every day experience.
And a little more confusing. Reasonably reliable? Reasonably accurate? From a science of measurement?
‘Not only intuitive they may be’ is really throwing me for a loop.
So far, so good.
And a little more confusing. Reasonably reliable? Reasonably accurate? From a science of measurement?
‘Not only intuitive they may be’ is really throwing me for a loop.
Oh forgive me, I typed up that post at like 1am this morning, and it seemed to make sense at the time. Alright…let me try to clear this up.
I mean that the results should be as accurate as possible. But that sometimes given the information available or the methods current known, that the results may be fairly approximate. Possibly due to technical limitations, or the the fact that the mathematical methods known are not quite refined enough to give results that are as precise as one might like. A good example of this is the method of finding the area under a curve by dividing the area in a series of rectangles. This gives an approximate result, however there is a better method, ie integration. But suppose integration had not been discovered, then well the rectangle method might be the best that one could do, even though it wont give exact answers.
The other thing seems to be a messed up sentence fragment…whoops. I beleive that I was attempting to point out some mathematical methods may not be simple to understand intuitively or by resorting to ones everyday experience. Such as imaginary numbers, which have no “physical” equivalent and is purely a method that allows certain types of mathematical problems to be solved. Niether are they a really intuitive concept or one which one can relate to the “physical world” as anything but a method used to measure things in the physical world.
So the imaginary numbers are utilized as a concept of method, similar to infinity, neither having a physical equivalent, but both providing a utilitarian purpose within math for resolving equations therein.
The modern “concept” of numbers ignores context and conflates distinct concepts. Each class of numbers in modern use is a product of a specific process of accounting. Different accounting context cannot be dropped except in very special cases, e.g., when one is only interested in the quantity and not its units.
Counting leads to counting numbers. The unit is the type of entity counted.
Relating counting numbers leads to rational numbers. The unit is the denominator.
Solving for quantities to fit equations leads to algebraic numbers. The unit is predetermined and then abstracted out of the process – but the domain of applicability of the equation determines the unit. Complex numbers are solutions of equations, no more, no less – and labeling the solution to x*x=-1 as “imaginary” is facetious. There is nothing imaginary about the solution to this equation, and physical examples of its reality abound, e.g., electro-magnetics in the context of conductors where phase frequency is complex; or in quantum mechanics where the operators are complex, but the observable eigenvalues are potentially directly measurable, i.e., “real”. Complex numbers cannot be directly measured or observed, duh, it would take at least two observations to pin down the real and imaginary components.
Taking the natural limits of sequences of rational numbers leads to the so-called real numbers. The unit is implicit in that the elements of the sequence must be commensurable. This allows one to model even the transcendentals. To my mind, the representation of numbers as limits of sequences, as used in constructing the real numbers, is the most general; but you can’t just blithely manipulate sequences as if they were unitary numerals, esp. the non-terminating sequences become bothersome. And you don’t get the complex numbers unless you use two sequences.
Thinking of the solution of x*x=-1 as the square root of -1 is just people mistaking formalism for fact. It is just a number defined by the fact that it solves the equation, and should have a disparate name such as “freddie” to make sure folk don’t conflate it with observable, measurable quantity.
The problem goes deeper. The notion that -1 or +1 are numbers is false. They have distance and direction. They are vectors. The integers are naturally represented as pairs (x,y) with the equivalence relation (x,y)==(s,t) iff x-y==s-t, with normal vector addition used to construct a vector space from the equivalence classes under this relation across the set of pairs of counting numbers.
So, if -1 is represented as (0,1), and +1 by (1,0), then what sense does it make to take second power roots of either of these vectors?
multiply(-1,-1) = (0,1) * (0,1) = (1,0)
How does that make sense?
The conceptual flaw is that you can’t get somewhere by going in the opposite direction, so scaling +1 by multiplying it with -2 has no basis in reality.
Cheers.
The problem goes deeper. The notion that -1 or +1 are numbers is false. They have distance and direction. They are vectors. The integers are naturally represented as pairs (x,y) with the equivalence relation (x,y)==(s,t) iff x-y==s-t, with normal vector addition used to construct a vector space from the equivalence classes under this relation across the set of pairs of counting numbers.
So, if -1 is represented as (0,1), and +1 by (1,0), then what sense does it make to take second power roots of either of these vectors?
multiply(-1,-1) = (0,1) * (0,1) = (1,0)
How does that make sense?
The conceptual flaw is that you can’t get somewhere by going in the opposite direction, so scaling +1 by multiplying it with -2 has no basis in reality.
Cheers.
- ico
No, your issue is that you are mistaken in asserting that -1 and 1 have to be treated as two separate dimensions of a vector. You know that in fact, adding +1 and -1 gives you zero. Add your two proposed vectors and you get (1,1) which sure as heck ain’t zero. If you make a mistake like this nothing following it will make sense.
If you want to assert that 1 and -1 are vectors (as in some contexts they are) then they are the vectors (1,0) and (-1,0) since the negative sign would denote that -1 has the direction opposite of 1 (i.e., 180 degrees off). It certainly does not lie in the second dimension of the vector, since that would be orthogonal (basically, at a 90 degree angle) to the other dimension.
The second dimension of your vector could be filled by the solution of x*x=-1, which is generally denoted i. So i, not -1, is (0,1). That works well, and is used for the applications you listed above (basically electrical engineering type stuff) whilst complaining about the use of the word “imaginary” to denote such numbers. (The fact is, in mathematics the literal meaning of the word “imaginary” is simply ignored; the numbers are as factual as is π or 2.) Typically though electrical engineers denote such complex numbers with an angle and magnitude rather than using the vector–its equivalent and much more convenient in most contexts.
Thinking of the solution of x*x=-1 as the square root of -1 is just people mistaking formalism for fact. It is just a number defined by the fact that it solves the equation, and should have a disparate name such as “freddie” to make sure folk don’t conflate it with observable, measurable quantity.
Orthogonality is the fact captured by this formalism.
Mr. D’Ipolito, I respectfully ask that you check your premises.
Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other.
The premise I question is that opposing rays somehow coalesce into linear dependence, whilst rays at a tiny angular displacement from opposition are linearly independent.
The contradiction is due to context-dropping, and a bit of conflation, around the concept “linear”. Linear means something different than Euclidean rectitude in the more general context of so-called “linear” algebra, and unfortunately leads to the common mistake of assuming that lines are the basis of dimension in volumetric space.
Volumetric space is the conceptual context. What can be said with certainty about volumetric space? Well, it’s finite, bounded, enclosed. In other words, it has an inside, and an outside (editorial note: people traditionally have underestimated the importance of the inside/outside complementation that necessarily adheres to volumetric spaces). Now, the thing about inside and outside is that, like any truly complementary pair, they are not merely symmetric opposites. This becomes clear from a simple observation: you can only go inward so far before you pass through a volume and thenceforth move outward indefinitely (assuming a ray-like trajectory).
In volumetric space, in means in towards something, and directions are defined by reference to “inward-nesses”.
Mathematicians may play with more or less unrelated abstractions, but we humans live in volumetric space(s), and cannot assume things are where they are without seeing a signal indicating such facts … and the signal must traverse volumetric space to reach us … and two signals sent in different directions are clearly not gonna end up in the same place, clearly represent two distinct dimensions of information gathering. This is just another way of saying, as I originally stated: Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other.
Rays, not lines, are the proper basis of dimension when traversing motion is taken as the gold standard of proof of (potentially evolving) relative location. I challenge anyone to impeach this standard.
Ergo, lines are 2-dimensional.
And therefore, since a vectorial representation is required, I submit my representation of the integers as the most conceptually economic I could devise: pairs of counting numbers with equality defined by reference to equivalent net displacement when one imagines finite, constant-length steps taken to the left, or the right.
I can keep trying to make it clear if you choose to pursue it – it is worth the effort, I promise.
Cheers.
Mr. D’Ipolito, I respectfully ask that you check your premises.
Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other.
Two distinct rays that share the same origin are not actually distinct.
On reading my prior post, I think it appropriate to give a simple visualization for my model.
Consider a discrete random walk where each step is of the same length and can be taken either left or right.
Let (x,y) represent all paths that subtend x steps left and y steps right.
Notice that (x,y) and (x+q,y+q) refer to the same endpoint in a walk starting at (0,0)
(The number of steps is ignored for this purpose … but should not be in general, e.g., when planning lunch dates. But that’s another, more involved, story of mine; I’ll fire it up on a new thread at some point.)
So, I simply take (x,y) to represent the integer y-x, and accept the fact that there are multiple ways to refer to this integer, kinda like the case with rationals where both numerator and denominator can be scaled without changing the value. (Actually, I end up exploiting these multiple ways to model translational momentum, but that is part of the other story).
Cheers.
Numbers are not rays and should not be modeled as such. By doing so you assume what you are trying to assert.