
Re: Is this an exceptionally hard set of questions to answer?
Posted:
Nov 6, 2002 11:01 AM


Kevin Foltinek <foltinek@math.utexas.edu> said:
>Probability is not a measure of things past, it is a measure of things >future (or not currently known).
Probability is a measure on a metric space. Only belief can tie the definition of probability as a proportion to it being the measure of the likelihood of something to come. If I say that the probability of tossing a die and get 2 is 1/6, I'm not saying that I have 1/6 chance of getting a 2 in my next toss  I'm rather saying that, if I toss my die over and over again, the number of 2's I get will be roughly 1/6 of the total tosses.
>In any case, the reason I mentioned this was because you claimed that >"Unless we're dealing with infinite objects, intuition is seldom >wrong". If you now claim that "probability and intuition part ways >very early", then either intuition is often wrong, or we need >mathematics, rather than intuition, to understand probabilistic >situations (including every time we don't have perfect information).
Probability is an attempt to explain intuition, and a poor one at that. Intuition here is not only upstream from probability, but it seems to be kind of impervious to it. And here, please do make a sharp distinction between belief and intuition ! The fact that we believe that we have a 1/6 chance of getting that 2 when the die rolls doesn't necessarily mean that it's an intuitive thing. >That's the nice thing about probability  it lets us talk about the >probability that things will occur, without having to perform very >large numbers of experiments. In this particular case, we don't even >need an experiment, we just need a convincing argument that subsequent >tosses of even a biased coin are independent.
The probability that things "will" occur is only a 20/20 hindsight from the future. It can predict ratios, it cannot predict individual events. No matter how many arguments you get about that coin being unbiased, you cannot predict whether or not when you toss it it'll be heads. All you can say is, if you toss enough times, you're going to get half heads and half tails: in other words, hindsight, history. You can't say "it will be" with statistics, the only thing you can say is, "it will have been". There's a sharp difference between these two statements, and even if you ignore that difference, you can only use statistics to describe populations  not individual events. But our belief in chance entails the link between statistics and the chance of one event happening, and I don't think the mathematics can say that: only belief can. >This is not relevant to the context (which you conveniently deleted) >in which I made this statement, that adding is a function of two (or >more) things to be added.
I have now forgotten what the original context was, and in any case, go wash your mouth  I do not play games when I talk to people. My point is very simple: that our fingers are the only prop most of us will ever need to do arithmetic in early grades, and they're far superior to any other prop I can think of. >Have you read the literature?
Plenty. I have been teaching ever since I was a highschooler, and I'm in my late fifties now. >There are plenty of explanations of "the talent factor" other than >born intuition. For example, it could be related to an early ability >to learn the (relevant) intuition (or even to early environmental >conditions that create this learning ability).
There are plenty of ways to model ANYTHING. So ? But some models fit our problem space tighter than others. >Once again you deleted the context  the "evidence" and "conclusions" >are things like "hey, 2+5=7 and 3+4=7 and 4+3=7 and 2+2+3=7... maybe >there's something going on here". Your definition of "evidence" must >be almost as narrow as your definition of "mathematics".
Please give your own definitions of both, then. But again, I don't deal with definitions  we're not talking about mathematics here, we're talking about teaching and about handling real students. These spaces, unfortunately, aren't that amenable to precise models. >One cannot use a subject that one does not know.
I know enough of it to cover the need of a grad school level computer science or electrical engineering major. Which are two areas where one can say mathematics has a fair amount of application. I don't know much beyond it, but then, if even such strongly mathematical disciplines don't need the stuff, what's the point of throwing it at our K12 students ? Be realistic, man, get out of your shell and try to see the world as it really is. >You cannot drop things from a model until you have a model, and since >a model is (by definition) not the same thing as the reality, a model >is a way of relating something that is understood to something else. >The way of relating, and the understanding itself, can be and often >are rules.
A model is just what it is: a model. It may or may not apply rules. It may apply adhoc knowledge, fuzzy knowledge. It may be a trialanderror Monte Carlo or Las Vegas algorithm. It may be a lookforit Simulated Annealing or Tabu Search. Models can be based on rules, but they can be a lot looser than the garden variety mathematical formal model.
Incidentally, this is where a difference between math and computer stuff comes in. We can model things inside a computer without having to use a formal model, just by mimic nature and its interactions as is, in an adhoc way. Works wonders. >The ability to use without understanding what you are using is worse, >because you cannot know if you are using properly.
Oh, cool. So I shouldn't use my car because I don't understand how it works. I shouldn't take a plane because I don't understand how it works. I shouldn't use a doctor because I don't know medicine. I shouldn't take medicines because I don't know how they work.
Hello ? >Systems of rules can be anything you want them to be. They don't have >to model anything.
Systems of rules, by definition, are models. Actually, a rule is already a model ! Reality has no rules, and if it isn't reality, it's a model. >I don't disagree that we need to teach students things other than >mathematics, but I think we should be teaching them *some* mathematics >(which often we are not). I have already stated my reasons.
Now, we agree superficially, but let's see. I have gone through two professions that use a lot of math: computer science and electrical engineering. I look at them, and at my students' needs, and I cannot reconcile what I call "mathematics" in that context to the subset of mathematics you advocate we should be teaching them. While I do see a need to teach rulebased axiomgenerated modeling, I put that need at a relatively low priority when compared to the things people like me need every day  and those involve number sense and an ability to do onthespot mental computations and visualization.
So, to put it specifically, should we teach them formal logic, rulebased inference, set theory, abstract algebra, metric spaces, topology, analysis ? Maybe. From the start ? I don't think so. As I keep pointing out, those definitions and properties may be mathematics, but as such they're already a model of a model  and before I teach my students that model^2, I must get them to first understand the model^1. Hence, being even more specific, do I want to teach them about the commutativity of addition ? I don't think so, because in a few years, if I do my job right, that commutativity will be so evident that I don't even have to formalize it as a rule. But if I formalize it as a rule without anchoring it in the behavior of the operation itself, I'm doing things way out of order, and I'm creating a gap in their knowledge that's going to be hard to fill up. And again, maybe that's the way mathematicians learn the stuff, but that's not ok to the rest of us. >If you look closely at your computations and algorithms you will see >that the rules or axioms are used extensively. The algorithms are >consequences of the axioms.
No, Kevin, it's the other way around. The axioms and the rules model the algorithms. The algorithms come out of number sense. Number sense comes out of familiarity with the basic behavior of counting. >So why did you object to the "can you find a general rule" question >about discovering commutativitytype properties?
Because it's not about finding rules, it's about GENERATING them. It's not a discovery, it's CREATION. It's not about analysis, it's about SYNTHESIS. Does addition entail commutativity ? Fine but who cares, I'll deal with it if I ever have to model addition at that level. But I will need to create a lot of models for things that don't even remotely behave like addition, and chances are I may end up overloading that "+" sign to mean something well beyond what mathematics required it to. >Most call it "modelling" and recognize that it is quite distinct from >"computer programming".
Modeling without a computer can be pretty useless today, eh ? This is the twenty first century. >The sequence 1 2 / is not defined in the integers. Your computer >implementation of ((int)1)/((int)2) is not an implementation of the >integers.
If I define my "/" operation to be from a pair of integers to an integer, the sequence 1 2 / is now defined in the integers. Because I happen to define it so within the model I'm building in my computer, and I don't care if the math book says otherwise. See, here again, you reject the idea that I can create my own models, and that models may depart from the party line. Worse, 4/3 can be defined to be 1 or 2 depending on what kind of division I use.
You see, "/" is no longer that holy untouchable; it can be redefined at the whim of the modeler, and it often is. Now, THAT I find important about rules: to teach our students that NO RULE IS TOO HOLY AS NOT TO BE TOUCHABLE BY A MODELER. Let's not teach them "the rules", let's rather teach them how to create their own rules.
>The fraction 1/2 is something called a "rational number"  I know >you've heard of them because you have mentioned them. The rational >numbers form a field; algebraists call it the fraction field formed >from the integers. (Fraction fields can be formed from many other >objects.) The fraction 1/2 is not a promise of a future division; it >is an object all on its own.
The "fraction" 1/2 is not a number, it's a notational sequence. The number is the RESULT of dividing one by two. And numbers are not elements in a field, numbers are MODELED as elements in a field. The distinction is fundamental, numbers exist in nature independent of whatever model you throw at them.
And precisely which numbers ? Do I need to resort to a mathematical definition of number before I can say that rationals are elements of a field ? And then, do these numbers really exist in real life ? I can easily take the road that a number is not an entity but an adjective, so I say "one rod" the same way I say "blue rod"  onenness is an attribute of a collection and hence it may not be easily modeled as an element in a set.
And of course you can model 1/2 as an object, but then, you are, again, creating a model. It's a valid model, but it isn't that a fraction "is" anything like that model, it is that you MODEL a fraction that way. There are other ways to model it, if what what we want to model is the concept of deferred division; and in elementary levels, we may care more for the actualy concept than about the modeling of it. >You would do well to educate yourself. I'd suggest learning some >algebra.
And I suggest you jump into the twenty first century and start looking at things from the computer side for a change. Fractions are deferred division, nothing more. The rest of it is models built on top of the concept. >Probably the standard example of "syntactic sugar" is a[i] used in >place of *(a+i). The analog of this is the different notations that >we have been discussing: 2/3, 2 3 /, [2;3], etc. The fundamental >object, analogous to *(a+i), is [(2,3)], the equivalence class of the >pair (2,3).
Actually, a(i) is not necessarily the same thing as *(a+i). For example, "i" can be a string, or a structure. And then, I can redefine "*" and "+", and even "( )", to mean whatever I want to. It's called "object orientation", and it's one of the cornerstone of contemporary programming. Now, coming back to 2/3, however you decide to model it, it's all syntax: if the semantics you get is one where 2/3 doesn't yield 0.666..., your model does not match what reality needs it to be. So, back to the drawing board: before we study those models, we must study the concept those models are trying to address. > >This equivalence class is far from syntactic sugar; on the contrary, >the notion of a fraction is analogous to a class in objectoriented >programming.
YOUR notion of a fraction, as the model of a more basic concept, is what I can call an object in an OO program. But that's the whole point: once we look at these things as OO classes, they become just a model, and no longer "the" model. So, I can easily write
class fraction { int a, b; float result( ) = (float) a / (float) b; float operator + ( fraction f) { return new fraction ((a*f.b +b*f.a)/(b*f.b) ; } ... }
and so on, and presto, I have my own fraction object. But then, I can define anything in that way, so, the emphasis become on not what your model "is", but what your model CAN BE  and the emphasis is not in learning about fractions, BUT IN LEARNING HOW TO GET TO THE MODEL  and here, the model is obviously intended to address the more basic concept of deferred division. >Very few people understand what a real number actually is, and it is >extremely difficult if not impossible to talk about real numbers >without first talking about rationals (fractions).
Again, you confuse reality with model. Assuming for example that I rule out real numbers with an infinite number of digits, I'm back into the realm of rationals, and, what do you know, THAT'S ENOUGH FOR JUST ABOUT ANYTHING APPLIED I CAN THINK OF. So, here again, we come back to that point that some of the things mathematicians love may not be that relevant to the rest of us ! I don't care for Dedekind cuts or anything like that in my applied real life, so, I am happy to stay in my side of the fence where there's no such a thing as infinite and where every number has a finite number of digits. Your view of real numbers are ok for math majors, but maybe the rest of us don't really need it. >I could just as easily state that the sky is green and the sun rises >in the west.
Just turn on any computer game and you will find all sort of assumptions that don't match objective reality  and that is because what the game is trying to address is NOT objective reality. Your very math is full of objects that I find it very debatable whether they exist at all, that is, outside the math model itself. A lot of science is built on preposterous assumptions that just do not match reality, yet it works fine within the level of precision we are able to use. >There is no excuse for this level of ignorance. Equality is not the >same as approximation.
You are assuming something that may not exist in the real world: a number with an infinite number of digits. Equality in real terms is very simple, two things are equal if I can't tell them apart by measurement or observation.
>(1/2)+(2/3) in the realm of reals, is exactly the same(*) as >(1/2)+(2/3) in the realm of the rationals, neither of which is the >same as 0.5+0.6. You can approximate it in both the reals and the >rationals by 0.5+0.6, or 0.50+0.66, or 0.50+0.67, or (etc.). >(*) By "exactly the same", I mean that I am identifying the usual >subset of the reals with the rationals.
If I'm in the realm of integers, 1/2 = 2/3 = 0. If I'm within the set of 5digit rational numbers, 2/3 = 0.666666, period. The "exactly the same" you are using assumes an entity that I may never bump into in my real life: a number with an infinite number of digits. So, fine, they exist within your mathematical model, and they're of great interest to mathematicians. So ? To the rest of us, every number eventually ends, there's no such a thing in reality as a number that's not rational, and consequently an integer up to a scale factor. You see, whenever the axiom of infinity creeps in, it becomes of interest to mathematicians, but not necessarily to the rest of us. >No, the parentheses make it clear which to do first, you don't have to >use this rule. In any case, the only thing the notation does is tell >you that we need to add two fractions.
The parentheses STATE the order of operations. (1/2)+(2/3) ain't the same as 1/(2+2)/3, and that because the parentheses say so. But if I write 1/2+2/3, parsing that sequence either uses precedence rules or it doesn't. It's not that the notation is making a concept clear, the notation just is: it tells you what to do. >It's not just intuitively simpler, it *is* simpler, because 6/2 and >8/4 are both defined in the integers.
And 2/3 is defined in the set of finiteprecision approximations of rationals  which is by far the only thing I'll ever handle in my professional life. So, if I have two digits of precision, 8/4 is 2.00 while 2/3 is 0.66 which is the same as 66/100: my model is no longer a pure math model, but a real life one. >See, you know that rational numbers exist, but for some reason you >don't recognize that a fraction is a rational number.
For the umpteenth time, man, a fraction is just a notation. You can define it as an object, OO style, but then, it boils down to a division. The RESULT of that division is NOT the same as the division, so, NO, A FRACTION IS NOT A RATIONAL NUMBER, what happens is that the RESULT of performing the division deferred by the fraction yields a rational number  and even then, only if I'm operating with rationals, because if I'm operating with integers only, the result of doing the deferred division is an integer.
>The notation does not matter  the concept of fractions or rationals >is what matters.
No, the concept of DIVISION is what matters. The rest is downstream. >When you divide first, you get the rational numbers [(1,2)] and >[(2,3)] (usually denoted 1/2 and 2/3, respectively).
You get the numbers 0.500... and 0.666..., up to whatever level of precision. Now, you MODEL these with your rationals and your fraction objects. But there's a problem space concept upstream, and it's not a good idea ot model something we don't know what it is, and it's even a worse idea to replace model for reality.
>By the way  what does the notation "0.5" mean?
It means the result of multiplying 5 by the result of dividing 1 by 10. The same as 5 1 10 / *. >You seem to be completely oblivious to what I have been saying for >some time now  "do the operations with rationals" is where the >students have problems.
No, sir, that's not where the students have problems. The students have problems when we throw models at them that don't match their own notions of the basic concepts behind operating on rationals. They are not taught that the decimal point is merely a question of scale, and that if I want to add 0.5 to 14.99, the easiest thing is to (1) append zeros to get the numbers to have the same number of digits to the right of the dot, (2) throw away the dot, (3) add as if they're integers, and (4) put the dot back. This is equivalent to saying that 0.5+14.99 is the same as 50 /100 + 1499/100, so we abstract the scale factor 100, then we add 50 + 1499 = 1549, then we put the scale factor back and get 15.49. Like I keep saying, the easiest model to teach is one that reduces everything to integers. >A fraction (with integer numerator and denominator) is a rational >number. A rational number is a fraction of integers. It's a very big >concept. The set of rational numbers *is* the fraction field over the >integers.
A fraction is a piece of notation. The RESULT of doing the operation expressed by the fraction is a fractional number. Fractional numbers from objective reality are MODELED as rational numbers, and division is MODELED by your "fraction objects", and there's a onetoone correspondence between your fraction objects and your rational numbers within your own model  but still, the fact that you model fractional numbers from our objective reality that way doesn't mean that they "are" that way, it only means that you model them that way. Now, I can model it differently, and I often do when it suits me ! >Nowhere do we need the notion of a real number to compute >(1/2)+(2/3). We do need the concept of fraction or rational number.
A rational number is a real number, no ? I can model Q as a subset of R, we do that inside our computers all the time. If that's not the way you pure math guys see it, well, fine, but I'm not sure we can use your model to get on with our stuff. >Fraction = Rational.
That's one way of looking at it, but a jolly ineffective way from my applied world point of view. To me, a fraction is just a piece of notation, nothing else. I can build it as an OO class, but I don't think I need to bother with it. >On the contrary, that is *exactly* how it is defined.
If that's the way YOU define it, fine  but that definition may not be very useful in teaching my students. I'd rather tell them that a/b is a shorthand for "a divided by b", end of story and end of mental masturbation. The rest, including your definition, is merely a consequence of this starting point and of the commonsense rules of counting. >And we add rational numbers how?
Like I said before: append zeros to one of them to give them the same scale factor, get rid of the dot, add them as integers, and put the dot back. In other words, every rational number can be written in the form n*10^s, where s is the scale factor and n is an integer. O add m*10^p to n*10^q, first rewrite them both as a*10^s and b*10^s, and the result is (a+b)^10^s. Actually, you can replace the "10" by anything, 2, for example, and express it in binary, or 16 and do it in hex. >Again, how do we add the result of doing those two divisions, and how >do we "do" those two divisions? (Make sure you've told me what "0.5" >means.)
See above. 0.5 is the same as 5 x 10^1, or 50 x 10^2, or 500*10^3, or 32*16^2, and so on. >Directly presumably meaning by "doing those divisions"?
Indeed. Remember, a/b is nothing but "divide a by b". >Often one writes it down "in place"  that's what's involved in >"showing your work". For example, > (2+3)*(4+5) = (5)*(9) = 45 . >The (5) and the (9) are "in place".
Sure, that's two places: two memory locations, two stack locations, whatever you call it. Two placeholders. If you don't have at least two pieces of permanent memory, you cannot compute (2+3)*(4+5).
>The use of "registers" or pieces of paper or whatever is a >complication.
They're a consequence of the operation. Your "in place" abstraction is just another name for a register.
>The same applies to postfix stack operations: > 2 3 + 4 5 + * = (2 3 +) (4 5 +) * = (5) (9) * = 45 >(I introduced the parentheses only for easier direct comparison with >the infix notation).
You still need those two locations, call them whatever you want: "in place", registers, memory, pieces of paper, placeholders. There's no way around it. >I would strongly disagree that these steps are easier to memorize than >the traditional formula.
Have you tried ? I have learned a fair amount of applied math over time, and I seldom if ever bothered to memorize a formula  because I could derive any formula I needed in a couple of seconds, based on number sense, algebraic manipulation skills, and understanding of the process that leads to the formula. Every time I throw this kind of thing at a student, I see a marked improvement, often pretty fast.
>You'll also run into problems when you start talking about complex >numbers  your algorithm rolls over and dies at step 2, whereas the >traditional formula is (with a only small amount of interpretation >about what "sqrt" means) perfectly applicable.
A complex number, now is one of your objects, pretty like a fraction. Now, if I define a complex number such that negative numbers have square roots, I will need a different sequencing. But that's much later, typically at senior high, and if the students know the stuff well, extending it is not a big deal. I find that generalizing is seldom hard, provided the students know the original material well. >The *reasons* for the one or two equations are not at all obvious. >They're just sitting there in the mysterious algorithm (which is just >as mysterious as the traditional formula, until you've seen a >derivation of it).
There's a multitude of ways to shore the sequencing in more traditional algebra in a way that is more easily digested. The b^24ac part has obviously to be justified, and proof will be needed, but here again, maybe we don't want to dig too much into the actual formula: we stop when we bump into the expression, and we can talk about solutions or the absense thereof even if we didn't as yet compute them. >Perhaps because of a lack of familiarity with rules and their >consequences, or perhaps because the "factfinding" is better done in >the derivation itself.
Perhaps because of a lack of connection with the students' mental reality ? Out of all the things the derivation brings, the formula is the least important, and I have my doubts whether deriving the formula itself has any merits  while, as you rightly point out, investigating the main factors behind the process is important. >Neither does the object I represent by 1.
Not true. If I move right 1 unit I call it +1, if I move left 1 unit I call it 1. Now, reality has another kind of numbers, the unsigned kind: unsigned 1 is neither +1 nor 1. >What does "0.6" mean, and why does it exist but not 2/3?
Again, 0.6 is the result of applying a scale factor of 10 to the number 6: the result of six divided by ten. >Again, 2/3 is rational, and division of 2 by 3 in the integers is not >defined.
In your model, the division of 2 by 3 in the integers is not defined. In my model, 2 divided by 3 in the realm of integers is zero. >2/3 is a perfectly acceptable representation.
COOL ! He said the word.
REPRESENTATION, you said. YES ! Representation of a division. In other words, a notation. A model, nothing more. >Probably because it's not a fact, it's a lie.
It's not a lie, it's reality. In real life, the notation 2/3 can stand for 0, for 0.6, for 0.66, for 0.666, and for a wide range of other values, depending on what precision I'm working with. >We gain the rational numbers, from which we can build the real >numbers, the complex numbers, the quaternions... Did you know that the >quaternions can be used to model rotation in space?
Sure, let's teach quaternions to first graders. And dude, I did computer graphics for a living for many years, and I still teach it. >Nothing except for a whole realm of abstract stuff beyond >fingercounting.
Which, again, may not add anything worth adding to people who are not going to be math majors. >Physics would be nothing without calculus, which (by definition) >requires the real numbers  which as I've said, requires the >rationals.
No, Kevin, no. Real life calculus does not need reals. An integral is a sum of products. A derivative is a ratio of two values. We do calculus inside our computers every day and we don't need reals. >You do know that the IEEE floating point standard uses a subset of the >rational numbers extensively, don't you?
A SUBSET. And the reason is, we have limited precision. No matter how you hack it, if I have 32 bits I cannot represent more than 2^32 numbers with it. >What is 0.4? What is 0.6....?
0.4 = 4 with a 10 scale factor. 0.6 = 6 with a 10 scale factor. If you abstract scale factors, every rational reduces to an integer. That is, in fact, the concept behind computer floating point, and that originated from engineers' slide rules, where, again, rationals get abstracted from scale factors. Get a slide rule, divide 2 by 3, and you get precisely to the same spot as if you divided 20 by 3 or 200 by 3. >But in music, twothenthree and threethentwo are not really the >same thing (5/4 time may have emphasis on the 1 and 3 or the 1 and 4).
Of course not, which leads to the concept of asymetric counting. In music, 10 is perfectly divisible by four: 12, 345, 67, 8910. That is, 10/4=4.
So much for your rationals and for your fractions. >One of my points is that mathematics is far broader than you think it >is, and often people do mathematics without realizing that they are. >Another of my points is that learning this mathematics helps in areas >other than mathematics. I am not suggesting that all other subject be >abandoned; I am suggesting that math class should be about broader >mathematics, not about number crunching.
Dude, I'm an electrical engineer and a computer man. Both professions use a fair amount of math, substantially more than many other professions. If I cannot conjure the need for mathmajor math in my reputedly mathintensive profession, how can I justify it for other, less mathintense fields ? >This explains why so much of visualization involves plotting, the use >of colours and shapes, animation, and various other approximations of >analog things.
Approximations of analog things are digital. >If there's no essential difference between those two, then there is no >essential difference between 1 and 2, between 10 and 20; really there >are only three things, negative, zero, and positive.
If you want to model it that way, I bet there's going to be something that can be successfully modeled by that simple scheme. It's not that things "are", it's how we want them to be inside our model ! >Arithmetic probably doesn't help, but more general mathematics might >(probabilistic models, for example). In any case, the point was to >dispute your claim that our intuition is digital.
Our intuition IS digital, and it ends at the limits of the precision of our perception. I can set a computer graphics experiment and easily show that most of us can only distinguish variations of color on a digital scale, for example. >Lack of precision does not imply discrete transitions. Lack of >precision might mean that a value is 5.3 plus or minus 0.5; the plus >or minus should often be interpreted as standard deviations of a >Gaussian density.
Lack of precision necessarily entails discretization. If a value is 5.3 plus or minus 0.5, you've already discretized it to two digits of precision: it means that you're going from 4.8 to 4.9 but not from 4.8 to 4.81. Now, you can measure it as 5.30 plus or minus 0.50, but that merely raises it to three digits of precision  still digital. >I guessed that's what you were talking about. I understood the >scaling given by the decimal point before I learned to use a slide >rule; it's something I learned when I learned about "scientific >notation" and computing with it by hand. Of course, one need merely >change base to see how stupid it is to say that 3.14 is essentially >the same as 314: 3.14[10]=3.16[11] would be essentially the same as >316[11]=380[10] (that's why I said above that all positive numbers >would be essentially the same).
The idea that we can treat 3.14 as 314 by abstracting the scale factor is pervasive in real life application: when I look at a ruler I don't just say "one", I say "one inch"  and what I'm doing is, I'm abstracting from that "one" the scale factor, which is "inch".
>This decimal point business is extremely dependent on the base we use, >and has nothing to do with either the numbers themselves or any >reality we might be modelling with them.
Bases and decimal points are issues of representation, and hence notation. Here, again, let's not confuse notation with concept. But sure, we must use the same base to abstract scale factors, yet, that's kind of intuitively obvious, no ?

