As I begin it, this snappy version takes into account two combox messages from him and me. If he answers, it will appear as "AlarmClock65 III" and if I answer that it will be known as "HGL IV", in each case appropriate parts, marked a, etc. His I is bold, so that the context can be seen by reading the bold and skimming over answers, if you want to before reading the rest.
- HGL II, my intro
- Sorry for delay, when I saw your work I was somewhat flustered on other grounds. I decided to save them to today. In return the answer has been worked through and is very detailed.
- Dive into previous O a
- AC65
- And, consider, as another example, Andrew Wiles' famous proof of Fermat's Last Theorem. In it, he had to use p-adic numbers, which are not even, in general, complex numbers, let alone natural numbers. Yet, using these numbers, he proved a statement solely about natural numbers: that there exists no quadruple of positive natural numbers (x, y, z, n) such that x^{n} + y^{n} = z^{n} for n>2. If p-adic numbers were not considered numbers, then such a proof, which undoubtedly relied on adding and multiplying them like normal numbers, would not have been possible.
- HGL
- Since in logic a proof may use the reduction ad absurdum, I cannot see why a proof about real numbers could not involve fictional ones. [LIKE:]
"Even if such and such were a number, such and such real numbers could not have such and such relations."
- AlarmClock65 I a
- This isn't reductio ad absurdum. Reductio ad absurdum involves applying a form of reasoning to a case in which it leads to a contradiction, demonstrating that it is not in general valid.
- HGL II a
- In logic it is about demonstrating that a proposition (stated with specification as to whether it is meant as applying generally or only sometimes) leads to a consequence which is absurd. Either self-contradictory or known to be untrue.
My point is that in order to make it, and I didn't know Mathematicians used one version of it as a technicality, you first have to assume what you wnat to disprove. Which is a step of FICTION - leading through logic back to fact.
Of course what I resumed it as would not be what you considered reductio ad absurdum to be from your experience of mathematics, but logically in a broader sense, yes, it would probably qualify as a reductio in absurdum.
- AlarmClock65 I b 1
- For example, one may think that if a real-valued function f is defined for all x>N for some natural number N, and it is monotonically increasing on that interval (i.e., if x>y, then f(x)>f(y)), then it must grow without bound, since it's always increasing, i.e., its limit at infinity must be infinity. This works for the square root function, for example. It is defined for all positive numbers, if you increase the radicand it increases the root, and, in fact, it does grow without bound: sqrt(10^{4})=10^{2}, sqrt(10^{40})=10^{20}, sqrt(10^{400})=10^{200}, etc.
- HGL II b 1
- OK
- AlarmClock65 I b 2
- But, this form of reasoning is invalid, as it does not always work. For example, the arctan function (inverse of the tan function) from trig is defined for all real numbers, and it is in fact always monotonically increasing. But, the conclusion does not follow in this case: it does NOT grow without bound; rather the arctangent of a very large slope approaches π/2 (assume for a moment that this exists...). So, this is an example of a reductio ad absurdum proof to show that such and such a form of reasoning does not always work and therefore cannot be used unless augmented by some qualification to show that it does work in a particular situation. Wiles, however, did not use p-adics only for reductio ad absurdum. He did not reason, "If I could find one of those quadruples, then p-adics would be numbers. They're not, so I can't."
[I did NOT give the reductio ad absurdum in such a form! See "dive into previous I a" above]
- HGL II b 2
- Is there really an arc tan function for ALL natural numbers? Or even ALL real numbers?
Btw, is arc tan rather than cotan really the inverse of tan?
As far as I knew, the standard model of calculating trigonometric functions (and since it is geometry, no problem, pi DOES exist in geometry, it is only in arithmetic it doesn't exist) involves a circle with 1 for radius. And dividing the circle in 360°, with 0/360 on top, 0 - 90 top right part of coordinate system, 90 - 180 bottom right part of it, 180 - 270 bottom left part and 270 - 360 top left part. THEN you compare lengths of triangle sides, which, being geometry, certainly involves more than what I call numbers and you call natural numbers.
- AlarmClock65 I b 3
- Rather, he calculated with them, treating them like numbers. So, if mathematicians didn't consider p-adics as legitimate numbers, then how would we have proved Fermat's Last Theorem?
- HGL II b 3
- Because sth fictionalised as a number may legitimately by being fictionally treated as a number give as much information about reality as a reductio in absurdum would with a fictional proposition.
- AlarmClock65 I b 4
- There might be another way to do it, but regardless, their use has certainly helped mathematics.
- HGL II b 4
- And considering p-adics as a useful fiction may get you searching for other ways to do it involving only what I would call "straight maths".
- AlarmClock65 I c 1
- I don't believe there's anything philosophically mistaken about the new terminology. Irrational seems to say exactly the same thing as incommensurable.
- HGL II c 1
- It is not the part "irrational" I objected to. It is the part "number" I objected to.
- AlarmClock65 I c 2
- To "commense" [commensurate, co-measure] things is to put them in a ratio.
- HGL II c 2
- Commensurable means having an ability to be measured together, i e by "same yardstick" i e having a common measure, and this as you say means putting them in a ratio and even more specifically a ratio of natural number to natural number.
- AlarmClock65 I d
- I do hate the terminology "imaginary" and "real," however, as this assumes that some numbers are legitimate numbers and others are just made up. In actuality, a number is anything we come up with / discover (can anyone really ever tell the difference?) with which we can add, multiply, divide, etc. and that can represent some quantity. But, "imaginary" and "real" are so widespread that there's no point in changing it now.
- HGL II d
- Number is properly speaking a thing to design multitude, just as length, area, volume, and similarily weight and a few more are things to design magnitude.
I have strictly speaking two stakes in this debate: a magnitude and a multitude are two different kinds of reality, even if some magnitude ratios (not π) parallel the multitude ratios. The other one is that "negative numbers" and "zero" are just as imaginary as i.
Where ever I have seen them occur, zero has not been a mere absence, it has been a value conventionally designed as zero.
- AlarmClock65 I e 1
- Finally, you're admission that you've never heard of any of the theorems I mentioned shows that you're really not in a position to judge whether irrational numbers should be treated as numbers.
- HGL II e 1
- Not so, since I consider that common sense on some very clear level has to be able to judge between specialised knowledge presented as such and gobbledeegook presented as specialised knowledge.
- AlarmClock65 I e 2
- It's like you're opposing reelection of a candidate with little to no knowledge of her actions while in office.
- HGL II e 2
- Not so. Number is not an elective office, and if it were, magnitude relations as such would not be eligible.
- AlarmClock65 I e 3
- You don't have any idea what irrational numbers have accomplished, but you want them out.
- HGL II e 3
- Only out of the category NUMBER, not out in general.
- AlarmClock65 I f 1
- As long as there are no logical contradictions, the ends justify the means in math. If such and such a construct gives good results, and it is perfectly consistent, then regardless of how intuitive it is, math keeps it.
- HGL II f 1
- That is like arguing that fiction is perfectly good fact, because it can be put to consistent use.
- AlarmClock65 I f 2
- Hence non-Euclidean geometries, ...
- HGL II f 2
- Misnomer, the things so referred to do not contradict Euclidean geometry. A "non-Euclidean triangle" is not a triangle. Because it is not in a plane. Because the lines joining the angles are not straight but only "as straight as possible on that surface" which for non-flat surfaces means = not straight.
So a non-Euclidean triangle not being a triangle, there is nothing in non-Euclidean geometry which contradicts Euclidean geometry.
- AlarmClock65 I f 3
- ... infinite sets with the same cardinality as a proper subset thereof, ...
- HGL II f 3
- o ... k ... you have me kind of lost here, a subset of non-Euclidean geometry, but what about "infinite sets" or "same cardinality"?
Cardinality sounds a bit like dimension in the three dimensions, right?
- AlarmClock65 I g 1
- ... different kinds of infinities, ...
- HGL II g 1
- How about admitting right from the start mathematics never deals with infinity properly speaking?
I am not against potential infinity of natural numbers (never realised) or potential infinity of magnitudes (which in actuality are limited) and I am very much FOR considering these two "infinities" as different from each other as well as from infinitesimals and as from ACTUAL infinity which only belongs to God.
I have never yet seen a mathematician actualise the potential infinity he considers as infinite. And will never.
- AlarmClock65 I h
- ... arithmetic with infinities wherein some operations are defined and others are not, etc.
- HGL II h
- Sounds like a fun game, a nice pastime for mathematicians, I don't want to take it away from you, but also sounds like fiction rather than as the study of mathematical realities.
- AlarmClock65 I i
- If the most efficient way to do something is unintuitive for some, then math will keep using it, because the results are all that matter in the end.
- HGL II i
- Keep using them. But don't keep calling them as real as 1, 2, 3, 4 or body, surface, line, point. Nor these as madeup as the enumeration you did.
- AlarmClock65 I g 2
- Also, I fail to see how your "method" of taking logs helps any. You will never find any integers x and y so that 2^{x}=10^{y}, so there's really no way to avoid irrationals with logs.
- HGL II g 2
- Do you have an 80-year old math professor who got mad at me when he was 70 as a backseat driver in this debate?
It seems some kind of backseat driver is forcing you to ignore what you otherwiose could have understood of what I actually meant and is forcing you to misunderstand it over and over and over again!
I am, ONCE AGAIN, not trying to keep irrationals out of mathematics, I am refusing to call them numbers.
- AlarmClock65 I g 3
- You seem to be set on avoiding them at all costs, even when they make everything simpler and mathematicians have been using them since Euclid.
- HGL II g 3
- But Euclid did not call them NUMBERS. That is my one quarrel in this respect of modern terminology.
[For that matter, neither did Euler, he called the most general mathematic concept "quantitates" rather than "numeri"]
He on his part thought Geometry and Arithmetic was a study of reality on one of its most general aspects, not just about getting useful results.
That is why he called magnitudes magnitudes and multitudes multitudes without confusing the two, despite their obvious parallelism.
- AlarmClock65 I j 1
- What do you think is better about your system?
- HGL II j 1
- It's not mine, I am just an amateur bringing geometry back to the basics set by Euclid and arithmetic back to the basics set by Pythagoras and both as developed and understood by Boëthius and by St Thomas Aquinas.
And I am up against people who are professional in math and on one day will tell me how much good irrationals have done since AFTER Thomas aquinas since they were called irrational numbers and on the other hand want to tell me they've been around SINCE Euclid.
St Thomas Aquinas had access to Euclid or summaries thereof. And he understood very well the difference between magnitude and multitude - the one which has been blurred (at least as far as Mathematicians communicate with the rest of the world!) since irrationals began to be called "real numbers" when they are really NOT numbers.
And no, this terminological move has allowed you no useful thing which you cannot have by [or at least while] keeping or going back to the old system. As I have tried to make clear on occasions.
- AlarmClock65 I j 2
- Would it generate more useful results? No. It would have to throw out all of calculus and a lot of other stuff, too.
- HGL II j 2
- I have demonstrated the opposite by one example of logarithms.
As said, 2=10^{x/y} MEANS 2^{y}=10^{x}. If this is wrong, of course my understanding of logarithms is wrong and I won't get any results approaching the real values.
If it is a good hunch, I might already have done such a thing.
Ready for a little disclosure? Text in French, but I do think most is understandable.
Since the real use of logarithms involves slide rules, I used feet (for units), inches (duodecimal fractions so that I count per 1/12 instead of first decimal), lines (1/144 instead of second decimal), points (1/1728 brings me nearly twice as fine shades as third decimal and they are about five times smaller than a millimetre - that is as precise as you can want any slide rule).
deretour : exploration des logarithmes sans le flou définitoir des maths modernes
http://hglundahlsblog.blogspot.com/2009/05/exploration-des-logarithmes-sans-le.html
deretour : Mise à jour, logarithmes
http://hglundahlsblog.blogspot.com/2009/06/mise-jour-logarithmes.html
- AlarmClock65 I j 3
- Would it be easier to understand? No.
- HGL II j 3
- Mathematics these days counts as a special talent. Part of the limited appeal of the discipline is precisely that the modern terminology which doesn't distinguish between what is real (though abstract) and what is fiction (i e more "abstracted and arbitrarily recombined" than just "abstracted").
- AlarmClock65 I j 4
- Imagine prefacing any discussion of logs, most limits, etc., with the statement that they can never be found exactly because irrational numbers don't exist.
- HGL II j 4
- A logarithm with exact value can never be found IN ARETHMETIC, but only IN GEOMETRY because irrationals are magnitudes not numbers, thank you!
As to arithmetic, no one has given any of these with the infinite series of decimals, so any actual logarithm is indeed inexact. Logarithm for 2 as 0.301 is a rational number, unlike the logarithm itself, which is unreachable. Mine are even more so, but sufficiently close to the given ones (once you transform the units) to validate that they are based on some correct understanding on some level.
TRANSLATION of DEFINITIONS:
[Could first have given the geometric definition : a logarithm is a graph plot along one axis growing as added numbers while other axis grows neither as added numbers, nor as multiples, but as arithmetic logarithms. Note that the geometric description of a logarithm refers to an arithmetic one, which I gave:]
Logarithm for 10 is 1 (or one foot). Logarithm for 100 is 2 (or two feet) since exponent of ten that gives 100 is two. Logarithm for 1000 is for same reason three And So On - and these are the real logarithms, with real numerical values.
Conventionally assimilated to them are near-logarithms (remember that 0.301 is also a near-logarithm and not an exact logarithm), which are defined as fractional exponents (0.301 is also such - it is 301/1000 written in another way).
What is a fractional exponent per se? If 100 were the basis of a logarithm, the logarithm for 1000 would be 3/2 because 100^{3}=1000^{2}.
When a fraction is only posited as approximation for a logarithm (like 301/1000, remember), what holds true is that
IF 10^{3/10} < 2 THEN 10^{3} < 2^{10} (1000 < 1024) [approximative value 0.3 is lower than real value of irrational geometric logarithm if it WERE expressable as arithmetic]
IF 10^{10/33} > 2 THEN 10^{10} > 2^{33} [approximative value 0.3030303 ... is higher than real value of irrational geometric logarithm if it WERE expressable as arithmetic]
In other words fractional exponents are not accepted as per se real but as algebraic shortcuts for sth more real. Which works.
And if you want a good pedagogy to bring home what logarithms are, feel free to use this!
Most people are aware, more keenly than mathematicians, of the difference between per se a good and intuitive description and per se counterintuitive as a description, but correct as a shorthand for a more intuitive but less useful description.
Break a leg is not a correct description of what you wish someone, usually, but people understand it once you explain why you don't use the more straightforward good luck.
- AlarmClock65 I g 4
- You haven't given any actual reason not to use irrational numbers other than that you just don't like them.
- HGL II g 4
- What I have said is, use them but keep in mind they are not numbers. And that numbers you use for them (like 0.301 for ten-log of two) are approximations.
- From here
- snappy version becomes a bit superfluous, since debates are not as interlocked in several layers over each message.
- AlarmClock65
- Okay, sorry for the confusion. For some reason, I couldn't read the "Glad you agree Roggeveen and James Cook..." comment of yours before on YouTube. I also couldn't get the full content of your latest one until just now when I realized that, for some reason, both of them were listed only on Google+, not on YouTube. So, now that I've been able to read both of them, I think I understand your view much better.
It seems you're concerned much more about how we think about irrationals, complexes, etc. than about how we manipulate them. In your view, we can add, subtract, multiply, and divide them, but they're not really numbers. And for you, what one terms a "number" is dependent not only on an arbitrary choice to include or exclude certain structures from the category but also on what is "real" and what is "fictional." You also speak of an overarching distinction between geometry and arithmetic. The problem is that it is impossible to formalize this distinction mathematically.
For example, consider a two-dimensional Euclidean space equipped with a Cartesian coordinate system. Then, each point corresponds to an ordered pair of reals, like (2,1) or (0,0). But what is the difference between (a) the set of all the points on the plane and (b) the set of all ordered pairs of reals, which is called R^2? Obviously, the former is a geometric set, and the latter is an algebraic set. But what IS the definition of a point anyway? In turns out that in most modern treatments of geometry, it is taken as a primitive notion. This means that it is not defined. After all, not everything can be defined. You have to start with some undefined concepts in order to avoid infinite regress or circular reasoning. So, from a mathematical perspective, those two sets are identical. From a philosophical perspective, though, there is an obvious difference between pairs of numbers and points. But I think what it comes down to is that in math, there is no place for concepts that cannot be stated in purely logical terms. The intuitive notions we have about points being zero-dimensional and lines being straight have no logical content: they are merely intuitive perceptions of the world. Think of it this way: can a computer understand what it means for two lines on a plane to be parallel? Sure it can. It means that they have the same slope and are not the same object. These statements can easily be translated into terms that a computer can understand. But can a computer understand what a point is? No. Such a concept of infinitesimal location is graspable only by direct experience, and computers are not capable of direct experience. Really, I guess, math could be understood as the study of that which a computer can understand. Anything else, anything that requires human intuition, that is, can only be approximated by math. What is truth? What is beauty? Are numbers real? Such questions go to the philosopher, not to the mathematician. My point is that it's fine if you want to say that irrational numbers aren't really numbers; just know that your claim is philosophical, not mathematical, in nature, as "number" is really, like "point," a primitive notion in math.
With that in mind, I do not understand how you claim not to be excluding irrational and complex numbers (instead claiming only to be objecting to the term "number") yet you try to recraft logs and 2D vectors as creatures based on rational numbers. Maybe natural numbers are the only true numbers; I don't know, but regardless of whether irrationals are numbers, there is no reason why they can't be used as the value of logs. This has worked for hundreds of years, never leading to a contradiction or a major issue. Your idea of truncating them after three decimal places using base 12 is nice, but what if someone desires a greater degree of precision (i.e., if they are doing nuclear physics or something)? Then what will she do? The beautiful thing about irrational numbers (regardless of whether they're real "numbers") is that they have infinitely many digts, so you can choose however many are necessary to the task at hand, without ever running the risk of being "wrong." It's just like with regular numbers as used for labeling. If you have 1000 customers, then you can give each a number between 1 and 1000. If you have 1,000,000, then you can just take more numbers out of the infinity of them that exist. No matter how many people you would like to label, you can find enough numbers. Similarly, no matter how much precision you would like with an irrational/incommensurable quantity, you can have that precision. So there's no reason to change the log function to make it more rational.
Thus, whether or not the terminology's actually correct, it's here, and it's not substantively changing math. You can also add, subtract, multiply, and negate vectors, but are they numbers? I don't know. I just know what I can do with them. Matrices certainly aren't numbers, right? Are imaginary numbers imaginary? Are all reals real? I don't know. I think it's more of a faith-based issue than a mathematical one. So, in regard to whether irrationals are numbers, I suppose they can be whatever you want them to be.
A note in regard to reductio ad absurdum: this is how it works. It's a special case of proof by contradiction. With that, you show that P -> Q and that ~Q (Q is false). Since P -> Q is equivalent to ~Q -> ~P, by the law of contraposition, it follows that ~P. So, if Wiles were proving Fermat's Last Theorem with a reduction ad absurdum, then he would be aiming to get ~P, where P is the negation of Fermat's Last Theorem. Thus, he'd have to start out assuming the negation of Fermat's Last Theorem, then show that this led to a contradiction (a falsehood). Nowhere is he allowed to randomly throw in "fictional" numbers. They would come about only if he could derive them from the negation of Fermat's Last Theorem, and that's just not how it works. Rather, what he actually did is he used them as manipulable objects. He basically converted a problem about natural numbers into a problem about p-adic numbers and objects in hyperbolic geometry. You may say that hyperbolic geometry is "fictional," but the universe is not Euclidean either. The theorems of Euclidean geometry do not always hold in our universe. The "real" geometry is so complicated we don't completely understand it yet. Rather, we use Euclidean and other models to deal with it.
Finally, it really seems to me that you object to most of the modern uses of irrational numbers. You object, for example, to having them be the values of logarithms. Why? So what if they're not numbers? I don't care. Call them something else. Just don't try to change the way we actually use them, since there isn't any problem with that. In modern mathematics, the only way to understand a length is as a number. Mathematics is basically all about removing the intuitive impurities of things to turn them into completely symbolic concepts. Math doesn't study the real world. That's for science to do. Math takes real world phenomena and creates watertight, logical models to support them. For example, nobody has one perfectly well-defined height due to the daily fluctuations in her hair and scalp, but she can be assigned a number as an approximate height. The irreducible impressions we have (fast vs. slow, big vs. small, 2D vs. 3D) are only partially expressible in math. So I guess what I'm trying to say is don't expect math to perfectly explain your intuition, because it can't ever. Irrational numbers, whatever they are, give us a great model for a great many things, even though the world is so much more than mere numbers. Mathematics is like panning for gold: you fill a pan with water, taking in some gold in the process; then you skim off the water, leaving only the gold. Similarly, you need intuition to come up with any useful ideas in math, but once you have a pan full of intuition and rigorous ideas, the next step is to separate the intuition (water) from the purely logical mathematics (gold).
- Hans-Georg Lundahl
- "You also speak of an overarching distinction between geometry and arithmetic. The problem is that it is impossible to formalize this distinction mathematically."
Well, as I said to someone before, my issue is not how to formalise things as in expressing them using only symbols.
My issue is to define philosophically.
In common sense I hope you have no problem with the distinction, whatever your issues about formalising it.
I write this after reading your first paragraph, and I will read on after posting it.
"For example, consider a two-dimensional Euclidean space equipped with a Cartesian coordinate system. Then, each point corresponds to an ordered pair of reals, like (2,1) or (0,0). But what is the difference between (a) the set of all the points on the plane and (b) the set of all ordered pairs of reals, which is called R^2? Obviously, the former is a geometric set, and the latter is an algebraic set."
Sure. But in this case the algebraic set is a shorthand for geometry rather than strictly speaking for arithmetic.
As you mentioned the Cartesian coordinate system, this is the main practical use of considering zero and negatives as numbers, along with thermometers.
In both cases the zero is not really an absense but an arbitrary midpoint.
In arithmetic 3 - 3 = 0 is to my mind not a real subtraction.
Have three apples, eat three apples, the main thing is you have eaten all you had and so there is no subtraction, since there is no remnant. There is only a transfer. Which is why zero doesn't qualify as a number.
"But what IS the definition of a point anyway? In turns out that in most modern treatments of geometry, it is taken as a primitive notion. This means that it is not defined. After all, not everything can be defined."
The latter point (excuse pun!) is indeed true.
Usually obvious things are primitives and so definitions start from primitives that are obvious things.
To earlier geometry point was NOT the starting point of definitions. Rather:
- surface is the limit of a body
- line is the limit of a surface
- point is the limit of a line.
This means for instance that each straight line has only two actual points until you start off marking out more of them on it : the point beginning it and the point ending it.
"You have to start with some undefined concepts in order to avoid infinite regress or circular reasoning."
Indeed. Classical geometry starts off with body, the definition of which is left to some higher science. It studies specifically its property extension.
"From a philosophical perspective, though, there is an obvious difference between pairs of numbers and points. But I think what it comes down to is that in math, there is no place for concepts that cannot be stated in purely logical terms. The intuitive notions we have about points being zero-dimensional and lines being straight have no logical content: they are merely intuitive perceptions of the world."
Intuitive and logical are not opposites.
Logical procedures are there to connect intuition with intuition. Until a true but non-obvious conclusion by the linking becomes as intuitive as the starting points.
"Think of it this way: can a computer understand what it means for two lines on a plane to be parallel? Sure it can. It means that they have the same slope and are not the same object. These statements can easily be translated into terms that a computer can understand. But can a computer understand what a point is? No. Such a concept of infinitesimal location is graspable only by direct experience, and computers are not capable of direct experience."
A computer can understand nothing.
Saying a computer UNDERSTANDS what paralle lines are (by the means suggested) is a "pathetic fallacy" or rather worse, animals do have some direct experience with which to commence some rudiments of some kind of understanding, even if it is not the real understanding as we know it.
Therefore, a computer can be no ideal model for showing what logic is.
"Really, I guess, math could be understood as the study of that which a computer can understand. Anything else, anything that requires human intuition, that is, can only be approximated by math."
Wrong.
Math in both main branches starts off as human intuition of what one means.
Arithmetic studies "one as opposed to many" and geometry "whole as opposed to its parts".
Saying what a computer can't handle isn't maths is like saying what an abacus can't handle isn't arithmetic.
"My point is that it's fine if you want to say that irrational numbers aren't really numbers; just know that your claim is philosophical, not mathematical, in nature, as "number" is really, like "point," a primitive notion in math."
Boethius defines "number" as "one added to itself".
Thing is, back in those days, philosophy was what math started out from.
And I hanker back to them.
"With that in mind, I do not understand how you claim not to be excluding irrational and complex numbers (instead claiming only to be objecting to the term "number") yet you try to recraft logs and 2D vectors as creatures based on rational numbers. Maybe natural numbers are the only true numbers; I don't know, but regardless of whether irrationals are numbers, there is no reason why they can't be used as the value of logs. This has worked for hundreds of years, never leading to a contradiction or a major issue."
My point is that a three decimal approximation is not the irrational itself, it is an approximation which is in and of itself rational.
A six decimal approximation of a log is not the irrational itself, it is in and of itself an approximation which is rational.
A twelve decimal approximation etc .
"Your idea of truncating them after three decimal places using base 12 is nice, but what if someone desires a greater degree of precision (i.e., if they are doing nuclear physics or something)? Then what will she do?"
The base twelve fractions I used were only used because I was imagining a concrete geometric object, a slide rule, on which very certainly the logarithm is realised. I was only concerned with giving as much precision as would be adequate on placing the 2 pooint on a slide rule, the 5 point on a slide rule.
No problem at all with pursuing it further, do for as many places as you like. But whatever kind of place, decimal or duodecimal fractions, and how many of them you use, you very certainly will be using a rational approximation. And not the irrational itself.
"The beautiful thing about irrational numbers (regardless of whether they're real "numbers") is that they have infinitely many digts, so you can choose however many are necessary to the task at hand, without ever running the risk of being "wrong.""
Which means that whatever choice you make, you will be using a limited number of decimal places and therefore in reality not the irrational as such, but a rational approximation.
Irrationals are real geometric ratios of the irrational or non-numerical-rational kind.
As they are geometrical, they are not in and of themselves numbers, so any numericalisation written out is an approximation - precisely because the numericalisation is dealing with it after the manner of a number ratio.
Pursue π to 3.14 or to 3.1416 or to whatever precision you like, but such figures in arabic numerals in the end are in and of themselves rational numbers, and therefore not π itself as a geometric ratio value.
"It's just like with regular numbers as used for labeling. If you have 1000 customers, then you can give each a number between 1 and 1000. If you have 1,000,000, then you can just take more numbers out of the infinity of them that exist. No matter how many people you would like to label, you can find enough numbers."
However many people I'd like to label, we would be dealing with a real number - a number actually occurring, while counting these people.
Infinity of numbers is only potential. Imagine the highest number of people one can have - like after Judgement Day the populations of Heaven and Hell taken together. Both angelical beings and human ones. That would be all people who have ever lived since angels were first created and all people who have ever lived since Adam and Eve.
The number is finite.
So, number of people not being enough, let's take number of subjectively distinct temporal moments in the lives of all - a vastly greater number - but still finite.
And if we take the number of moments the population of Heaven experiences from Judgement Day on, we will have a number of moments which is growing TOWARDS infinity, but which in every actual moment still even so remains finite.
That is the philosophical difference between sempiternity of souls and angelic beings and eternity (with its totum simul) of God.
As you may guess, this kind of question is more my concern than the usual mathematical stuff of "how do we calculate this kind of problem".
But even for that, it may be a good idea to recall that mathematics is only dealing with potential infinities.
"A note in regard to reductio ad absurdum: this is how it works. It's a special case of proof by contradiction. With that, you show that P -> Q and that ~Q (Q is false). Since P -> Q is equivalent to ~Q -> ~P, by the law of contraposition, it follows that ~P. So, if Wiles were proving Fermat's Last Theorem with a reduction ad absurdum, then he would be aiming to get ~P, where P is the negation of Fermat's Last Theorem. Thus, he'd have to start out assuming the negation of Fermat's Last Theorem, then show that this led to a contradiction (a falsehood)."
Negation of a true theorem - say Pythagoras - is fictional.
There are more ways than one in logic to start from there and get back to real stuff by seeing how it leads to a contradiction.
"Nowhere is he allowed to randomly throw in "fictional" numbers."
Anything which is geometric in itself but is expressed as numbers is fictionally a number.
Zero is a fictional number, negatives are fictional numbers, things algebraic are if not numbers then fictional numbers.
My point is that fiction does not mean randomness and that fictional numbers may be as useful for elucidating of truth as that other fiction used in any reductio ad absurdum which involves starting with the fiction of denying the truth one is really trying to prove.
No closer parallel than that was intended with reductio in absurdum.
"Rather, what he actually did is he used them as manipulable objects. He basically converted a problem about natural numbers into a problem about p-adic numbers and objects in hyperbolic geometry."
I still have no idea what "p-adic" means, but sounds exotic.
Expressing numbers in geometry is nearly as fictional as expressing geometry in numbers.
Am I one-eighty-six or am I six-four? Depends on whether you measure my length in metres and centimeters or in feet and inches. One numeral is as correct as the other. I e, my length is per se neither numeral.
With numbers expressed as geometry, there is another matter, namely that each ratio in arithmetic actually has a corresponding ratio in geometry. So, expressing numbers as geometry is fictional, but less so than the reverse.
"You may say that hyperbolic geometry is "fictional," but the universe is not Euclidean either. The theorems of Euclidean geometry do not always hold in our universe. The "real" geometry is so complicated we don't completely understand it yet. Rather, we use Euclidean and other models to deal with it."
I am neither saying hyperbolic geometry is fictional, nor that it is non-Euclidean.
I am not stating as a fact that the universe is as such non-Euclidean, and your stating the real geometry is really such (not just as in globe surface geometry, hyperbolic geometry and a few more which simply deal with problems too complex for Euclid's tools to deal with, but really non-Euclidean as in contradicting Euclid on what he did show).
Euclidean cases may be simpler than the real geometrical shapes of the universe, and these could still be obeying the laws which Euclid applied to the shapes he did study.
"Finally, it really seems to me that you object to most of the modern uses of irrational numbers. You object, for example, to having them be the values of logarithms. Why? So what if they're not numbers?"
Not really, no.
I do not really object to having irrationals as exact values of logarithms, but I do object to having numericalisations of approximate values logically considered as identical to the irrational and purely geometric values.
"0.301" is not the logarithm of 2.
"0.3010300" (took this from a logarithmic table) is not the logarith of 2 either.
Both are expressed as numeric fractions and both are therefore rational fractions.
301/1000 is rational.
3,010,300/10,000,000 is rational.
ANY value you chose to use for actual calculations by using the numbers will still be rational.
Let us take the two approximations I had found for log of 2, 3/10 is higher than the log of 1.99 and 10/33 is lower than the log of 2.01, so my values were pretty accurate.
But even if they had been ten times as accurate, they would still have been rational numbers that only approximated the ten log of 2, just as any pair of Fibonacci numbers is a ratio which only approximates phi.
Ten log of 2 as such is a geometric entity (or a number of entities - a logarithmic curve would have the same proportions between numeral and logarithmic dimension whereever you arbitrarily place the logarithmic entity one for the numeral entity ten). π is a geometric entity. φ is a geometric entity.
ALL of these are irrationals, which means NONE of these is exactly expressable in numbers.
"Mathematics is basically all about removing the intuitive impurities of things to turn them into completely symbolic concepts. Math doesn't study the real world. That's for science to do. Math takes real world phenomena and creates watertight, logical models to support them."
Math studies traditionally the real world insofar as it is expressible either in one and its multiples or in body, surface, line and point.
"For example, nobody has one perfectly well-defined height due to the daily fluctuations in her hair and scalp, but she can be assigned a number as an approximate height."
Nonetheless, whether the height at any given moment is knowable or not, at any given moment there is such a thing as the actual height.
And no, the approximation as in sample measuring does not per se assign a number. When it comes to numbers, I am as much 1-86 as I am 6-4. I would be yet another number if you used "pieds du roi" (the measures that have the third level of duodecimal fractions, abolished during French Revolution). And yet another number if you used Egyptian Cubits.
On the other hand, the sample measure could be made by a notch and therefore without any number.
Whether your actual height fluctuates around or below or above the notch, the length given by the notch measure is a real and geometrically defined length.
And it is a different thing from its numericalisation.
"The irreducible impressions we have (fast vs. slow, big vs. small, 2D vs. 3D) are only partially expressible in math."
For fast vs slow, big vs small, agreed. Homo mensura.
[x faster or bigger than y - empirically given unless illusions of very close values or of camouflaging nature of one value is involved, but x fast enough to be fast or y big enough to be big - homo mensura.]
For 2D vs 3D, not agreed.
For NUMBER (as in "how many apples"), not agreed at all.
"So I guess what I'm trying to say is don't expect math to perfectly explain your intuition, because it can't ever. Irrational numbers, whatever they are, give us a great model for a great many things, even though the world is so much more than mere numbers. Mathematics is like panning for gold: you fill a pan with water, taking in some gold in the process; then you skim off the water, leaving only the gold. Similarly, you need intuition to come up with any useful ideas in math, but once you have a pan full of intuition and rigorous ideas, the next step is to separate the intuition (water) from the purely logical mathematics (gold)."
Very disagreed.
There is no such opposition between intuition and logic.
Logic is about working out the intuitions.
Therefore it cannot be about discarding them.
As to "irrational numbers", my point is once again, that irrational ratios of geometry are never numerical ratios of arimthmetic. Even if they are approaching sufficiently for such or such a practical purpose.
Finally:
"Okay, sorry for the confusion."
No problem. Happens.
- AlarmClock65
- All right, as long as we both support the main branches of modern mathematics, then we're essentially in agreement. When you said that you weren't sure whether the arctan function was defined for all reals, I thought you weren't sure about functions of a real variable, but I think now you just probably hadn't done that kind of trig in a while (the tangent of an angle can be defined as the slope of the ray starting at the origin that forms the angle in question with the positive x-axis; you can get any slope by choosing an appropriate angle, so you can take the arctan of any slope, i.e., any real number). Really, I can actually say now I understand and agree with what you mean about irrational numbers being fictional. They exist in geometry, but since we can never fully write them out as decimal numerals, then they don't really exist in arithmetic. I think most of the confusion was between arithmetic and algebra.
I agree that you can't ever work with them directly (other than in simple cases like sqrt(2) - sqrt(2) = 0). But in algebra, then you can work with them, albeit indirectly! What are the solutions to x^2 - 8 = 0? Why, 2sqrt(2) and -2sqrt(2), which are the same as sqrt(8) and -sqrt(8). Notice that we couldn't ever really write them out. We had to use shorthand. So in algebra, they are numbers, just not in arithmetic. But then, the question arises, is algebra really studying geometry or arithmetic? For this kind of algebra, the answer's geometry. So, a "number" in algebra is really a geometrical quantity, whereas a "number" in arithmetic is a counting number, or perhaps zero. And what I think you're getting at is that these numbers do not really come from a geometric background. They're sort of more inherent, more "natural," which is why they're called "natural numbers."
I suppose it does make sense to call these the "true" numbers at some level, for every other numerical construct (zero, negative integers, infinity, rationals, reals, complex numbers, vectors, matrices, functions, etc.) sprang from them. Yet some of those constructs are called numbers (zero, negative integers, rationals, reals, complex numbers) whereas others (vectors, matrices, functions) are not. What is the difference between the two groups? A rational number and a 2D vector are both really ordered pairs of numbers under some interpretation. So, really, I suppose the only "true" numbers are the natural ones, though all others are perfectly valid mathematical constructs, regardless of whether they're numbers. So, with this, I guess I agree.
To a mathematician, I guess, the most important thing is how constructions can be used and connected to other constructions. So a mathematician would not really care whether vectors are considered numbers, so long as it is clear that they are made up of real numbers, that they can be scaled by reals but multiplied with each other only in a special process, etc.
Now to zero, which we still kind of disagree on. Everyone in the world right now either has some apples with him or none. Considering no apples to be zero apples opens up the door for calculation. Now, if someone with no apples and someone with five apples combine their resources, how many apples are there altogether? 0 + 5 = 5, so five apples. If the 5-appled person had instead lost five apples, she would have been left with no apples because 5 -5 = 0. So, this is how zero is useful. Is it as real as the positive integers? It certainly came about initially as an arithmetic rather than geometric quantity, so I would say it is.
As for computers and the panning for gold metaphor, what I mean is that math cannot be defined in terms of subjective experience, so in math itself, everything needs to be defined perfectly in terms of primitive notions, and every theorem needs to rest firmly upon axioms and/or upon other theorems. In philosophy, however, this is not the case. For example, I take it to be self-evident that the adjectives "good" and "bad" as applied to courses of action are irreducible. It can be debatable whether a particular course of action is good, but "good" itself cannot be defined. I say the same about "subjective experience." I laugh at those who say that we shouldn't trust our minds, that maybe we don't have subjective experience. We can't be fooled unless we exist. Anyway, in math, it's not that cool. Things are whatever they're defined to be.
So, I agree with what you said about the "potential infinity." We can't ever actually have infinity. Nor can we ever have any geometric quantities be in a perfect 2:1 ratio. But we can have exactly two apples, or exactly no (zero) apples, so in this sense natural numbers are the most real.
A quick note about non-Euclidean geometry: it does sort of contradict Euclidean geometry. For example, Euclidean geometry says that the shortest segment between any two points is a straight line segment, but in other geometries this need not hold. In the universe, on a large scale, it need not hold. It's like the 3D space in which we live is curved through other dimensions in such a way that points are closer together or farther apart than one using a Euclidean model would expect. I don't know exactly how this works, but it happens.
Also, you can square the circle in some other kind of geometry, I think, which is impossible in Euclidean geometry.
[The shortest - and really geometrically so - connexion between two points is a straight line. A phrase like "the shortest way ON A GLOBE" negates the shortest and therefore gives no contradictory information about it. Not only case where shortest way available is not shortest simpliciter. To pedestrians the shortest way on avilable roads may even be longer than "the shortest way on a globe" or "the shortest way as the bird flies".]
- Hans-Georg Lundahl
- "Now, if someone with no apples and someone with five apples combine their resources, how many apples are there altogether? 0 + 5 = 5, so five apples."
My point is that just as 3 - 3 = 0 is too obvious to count as a subtraction as well as there being no remainder, 0 + 5 = 5 is too obvious to count as an addition as well as not really augmenting the 5.
"So, this is how zero is useful. Is it as real as the positive integers? It certainly came about initially as an arithmetic rather than geometric quantity, so I would say it is."
Zero first off has quite another use: in Arabic numerals it is a placeholder.
If 1 comes as only or right-most numeral it means 1. If one comes as second from right numeral it means ten and so on.
I want to write XI - no problem: 11.
But I want to write X? If I only write a 1, how do I tell it is not right-most but second from right? Zero means a position is empty. We all know it becomes 10, with 0=zero, from an Arabic word like tsefrah or thefrah meaning empty.
"But then, the question arises, is algebra really studying geometry or arithmetic? For this kind of algebra, the answer's geometry. So, a "number" in algebra is really a geometrical quantity, whereas a "number" in arithmetic is a counting number, or perhaps zero."
I agree. Or nearly. Some years ago, I would rather have said algebra is a useful fiction denoting quantities without some of the constraints of real ones, and one of the ditched ones would be the distinction between geometry and arithmetic.
Or the distinction between operations and cancelling polarities counted. In chemistry you can get ions with load -1 or or -2 sth, not because we deal with "less than zero", but because the polarity assigned "left hand status" is outnumbering the polarity assigned right-hand status by one particle or by two particles or sth.
Since then I have come across the notion that earliest examples of algebra would perhaps be found somehwere in Euclid, so algebra is another version of geometry.
Just as you can use origami instead of straight edge plus compass, you can also use letters and apart from that a quasi-arithmetic notation to do geometry. Does not mean algebra can't be sused to solve real arithmetic problems too, since geometry done by Euclid's means can also be used that way.
Have you come across Japanese multiplication?
12 * 25
One line and a notable distance and two lines with a / slant, two lines and a notable distance and five lines with a \ slant. Where one line cross two lines, you count hundreds, where there are two crossings in the middle you count tens, and where two lines cross five lines you count units. Btw, that example would arguably be threehundred, without my doing it. However, that, or using fingers or using an abacus is really doing arithmetic with geometrical means.
But since algebra is the art par excellence of short cuts and pragmatic restatements, guess why I called restating 2^{y}=10^{x} as 2=10^{x/y} a precisely algebraic reshuffling.
While we are talking fiction, it doesn't mean arbitrary. For any piece of fiction (as in non-documentary narrative writing) the starting point is arbitrary, but the closer you come to writing the story out in full, the less arbitrary the fiction is.
Even the arbitrary starting point must have SOME intelligent relation to reality.
So, 1000 = 1024 is literally false but a fiction useful to illustrate what 10^{x} = 2^{y} means.
And of course whatever actual number of decimals you use on a logarithm, the number you actually count with (except 1 for 10, 2 for hundred etc) will also be only fictionally, i e approximately equivalent to the actual geometrical logarithm.
- AlarmClock65
- With respect to zero, admittedly, this use is trivial. But math always has to start with the trivial in order to get anywhere. Consider multiplication by zero, for example. 5 * 0 can be viewed as 0 + 0 + 0 + 0 + 0, iterated addition of zero with multiplicity five. Now, even though we seldom multiply things by zero in everyday life, this reasoning does lead to the fact that a product is zero iff at least one of the factors is zero. This is called the Zero Product Property, and it is what allows us to use the factored form of a polynomial expression to solve the corresponding equation: for what x is x^{2} - 50x + 600 = 0? Suppose that expression gives profit as a function of tickets sold (x). Then the solutions will give the ticket-sale numbers for which we break-even, surely a non-trivial result. In fact, the quadratic factors as (x - 20)(x - 30), so the equation we have to solve is equivalent to either (x - 20) being zero or (x - 30) being zero, by the ZPP. Thus, the break-even points are at 20 and 30 tickets, and zero is useful and real after all, even though its initial use was trivial.
With place-holding, zero once again represents the absence of a positive value in a particular place, and absence is quite a real phenomenon, so this only goes to show that zero is as real as the other natural numbers.
And I agree that algebra is more closely tied to geometry than to arithmetic, since most algebraic quantities can be represented exactly in geometry but not in arithmetic.
- Hans-Georg Lundahl
- Gain and loss are not plus numbers and mjnus numbers.
Gain and loss are account balances.
Now, the equation (which starts out treating gain and loss as sth they are not) says that tickets sold will break even - i e neither gain nor loose - at EITHER 20 OR 30 tickets. But surely one of the solutions must be wrong?
If you break even at 20, you are already gaining at 30. If you break even at 30, you were still loosing at 20.
So, the equation was somehow wrong.
If instead you had taken a table method, or a graph, starting off with brut gain per ticket and deduction for occasion to see when you get a net gain overall, you would have known.
Absence is a real phenomenon - as in absense of what is expected from such and such a comparison with something else. But it is not a real number.
But TOTAL absense is not a real phenomenon, and Krauss is wrong to say that is where Universe started off by "quantum fluctuations between plus and minus".
Now 5 * 0 can be defined as 0+0+0+0+0 - but if even 5 + 0 is no real addition, how can 0 + 0 be one, or for that matter a multiplication?
- AlarmClock65
- +Hans-Georg Lundahl
With respect to the profit equation, you're right, that equation doesn't make any sense! What I suppose I meant to say was that x is the ticket price, in $. In that case, it makes sense that profit can go up and down with ticket price, but it doesn't make any sense for more sales to lead to less profit.
And "total absence" is indeed a real phenomenon. Breaking even is the total absence of profit or loss.
- Hans-Georg Lundahl
- Breaking even is exactly absense of profit and loss, but it is not absence of activity, of life, of material being etc.
That is what I meant with TOTAL absence not being a real phenomenon. I wasn't using the phrase as usually "absence of any x at all as opposed to absence of nearly all x", but as absence of any category at all of presence or being.
As far as breaking even is absence of profit and loss, it is not counting one of them more than the other - and thus not counting anything. Hence, not a real number.
"What I suppose I meant to say was that x is the ticket price, in $."
(x - 20)(x - 30) = 0?
Either way, a few hours in advance (my time zone):
Merry Christmas!
- AlarmClock65
- +Hans-Georg Lundahl
Merry Christmas to you, too!
Maybe zero is better thought of as an origin, a base state. Having zero apples can be thought of as the "reference" state, as can breaking even: everything is balanced.
- Hans-Georg Lundahl
- Well, but balanced is not a mathematical term as much as a physical one.
Besides zero in any such case would not mean everything is balanced but only that relevant parameter is so between its polarities.
- AlarmClock65
- +Hans-Georg Lundahl Well, whatever. Zero is what it is.
- Hans-Georg Lundahl
- What it is is precisely a debate.
1500 almost no one called it a number, 1600 almost every mathematician did.
Zero as a midpoint between polarity-opposed directions labelled + and - is about quality, not quantity, since quantity as such has no polarity. Therefore, in such cases, mathematics is trespassing into physics, chemistry, bookkeeping or whatever.
If magnetic polarity could be called N and S, presumably instead of positive and negative electricty you could have E and W electricity. Or [E/W] ions, since these have electric charges.