Generalizing mathematical concepts

Question

It's sometimes useful and interesting to generalize mathematical concepts - e.g., turning the familiar notion of number of things in a given set into the concept of cardinality of a set, etc. Lately I stumbled upon this Math.SE question about possible ways to generalize the concept of parity of integers to noninteger rationals and it got me thinking: why can't we just say that a rational is even iff its last digit represents an even number (of course we have to have some fixed base in mind, say the decimal one)? I.e., take a look at the following statement:

An integer X is even iff it can be represented as 2Y, where Y is an integer as well.

It is the definition of parity, and it makes conceptual sense. We can go further and say:

An integer is even iff its last digit represents an even integer.

(I.e. in decimal an integer is even iff its last digit is either '0', '2', '4', '6' or '8'. Also, to determine whether a given digit represents an even integer we have to use the definition, obviously.)

Now, the last statement makes technical sense - it's a perfectly valid mathematical theorem.

Here's the crux: to generalize the concept of parity, to me it seems that the only thing we should do is modify the last statement a bit, like so:

A rational number is even iff its last digit represents an even integer.

Here's what we just did: we came up with another definition under which the notion of parity for integers still holds and a new notion of parity for noninteger rationals emerges.

So we can say that, for example, the number 0.5 is odd, while 0.24 is even.

My question is: is this way of generalizing a mathematical concept valid at all?

P.S.It's also quite neat that if you replace "rational number" in the last definition by "real number" you get that any irrational number is both even and odd (vacuous truth, because no irrational has a last digit). So, we can say that any irrational is evodd (or maybe odden?) :)

EDIT: We're talking about the positive integer positional numeral systems here! Thanks to @Mauro ALLEGRANZA for making me state this explicitly.

EDIT 2: Never mind the infinite invisible zeros, e.g. even though 0.5=0.50=0.500=..., we still say that 0.5 is odd. Thanks to @Hunan Rostomyan for noticing this :)

Answer 1

The main philosophical objection to be levelled at your proposed conceptual analysis is that you're not matching your conditions on the left up with the range of concepts on the right, and this is introducing fuzzy boundaries for your concept. Not all rational numbers have "last digits". So if what you want is a concept that is determinate (that is, we get a yes or no answer as to the question of whether any given rational x is even), we should really qualify our definition on the right so as to make the assertion on the right something that comes out as true or false given the range of things on the left.

A simple existential qualifier usually does that particular trick - for example, we might re-construe your proposed analysis thus:

A (non-integer) rational number is even iff there is a minimal-length decimal numeral inscription of that number that has a final digit, such that the last digit itself inscribes an even integer.

So now that we have established a determinate analysis, there doesn't appear to be anything philosophically questionable about your proposed concept, as long as we're prepared to accept as more primitive the ideas of even integers and rational numbers, and to see this as a new concept extending or using those existing existing ones in some way.

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Some further thinking

Of course, you may find that this analysis of your concept lacks some of the force and utility of the concepts of evenness it extends. For instance, in the integer case, either an integer is even or it is not, and if it is not, we say that it is an odd integer. In your new rational case, however, while it is true that either a rational is even or it isn't, it doesn't seem natural to say that if it isn't it's odd. Why not? Because then the odd numbers would consist of all of the numbers whose last digits are odd, and also all of the numbers that don't have last digits. So, say, 8/9 (0.888...) would be odd on this suggestion. Why should rational odd be the property that gets all of the non-terminating ones - why wasn't rational even covering these instead? This throws up questions about whether our analysis of rational evens has missed something out.

Now maybe you might like to do some development work and add further refinements of your concept of even numbers. So let's say we add a "stability" qualifier to our definition of even numbers, such that in addition to the numbers we have above (where we have a last digit), we also add another disjoint condition:

... or the decimal numeral inscription of that number is such that for some integer i>0, the ith digit x of that number is even and nonzero, and every following digit is equal to x.

This would give us (0.888...) as even, though not, say, (0.6363...). We still wouldn't get a state where we'd be happy calling rational odd the negation of rational even, but importantly more things would be rational even in this definition that didn't work last time.

And of course you might note that this additional disjoint condition actually subsumes our original analysis, since every digit following the last digit of a number (i.e. 0 of 0) is equal to it. Maybe this is fits with what we want! Maybe it doesn't! Is 0.888... a rational even? What about 0.242424... - should we insist that this works too, and try to find an analysis that includes it? Do we want the concept of being rational even to stand in some relation to the concept of rational odd, and if so how do we think the two concepts relate to one another?

I tend to think that we only progress at this level through practical, rather than purely abstract theoretical, aspects of stress-testing. What are we interested in rational evenness for - what work is this new concept going to do for us? Maybe this sits in some particular abstract algebra context or maybe it's something interesting for computational analysis, but you'll really want a backdrop of some particular application in which to answer these kinds of questions of intuition. Otherwise the whole thing seems a bit like just spinning wheels for the sake of it! That's not to say there is no such backdrop, but rather that intuitions need to somehow be grounded in practice in order for there to be any sense of their verification - something very much needed in the lofty abstractions of mathematics!

Answer 2

Generalising mathematical concepts is a difficult task; it only looks easy in retrospect - some leading to blind alleys. It takes a certain amount of luck & intuition for it to work well.

For example, in calculus one can apply differentiation any integral number of times; and one can then ask can one make sense of it when it is 'applied' a rational number of times; obviously one cannot literally use the justification of application as this is literally impossible; one must look for a different justification - historically this has been done; but the resulting theory hasn't had a deep nor wide impact.

The generalisation that has worked is complex calculus which has generated a large number of deep ideas with a wide impact. This rather than looking at the nature of the exponent in differentiation looks at what is being differentiated ie complex functions.

Similarly given the huge expansion of alternative number systems - monoids, groups & rings; the important question is to generalise it to this context.

Taking the example of a ring (think of this as the ring of rational numbers); the usual manoeuvre is to find its ring of integral elements (think of this as the ring of integers); then its easy to supply a natural definition of parity for this ring.

What this shows that generalisations usually don't follow simple-patterns but sometimes they do.

Answer 3

The way I see it, your question has two implicit parts:

is this way of generalizing a mathematical concept valid at all?

The first part of the question concerns validity. Is the definition "a rational number is even iff its last digit represents an even integer" complete and consistent mathematically? This isn't really a philosophy question, it's a mathematical one (though see Paul Ross' excellent answer for some analysis), so I won't go into it further.

The second part of your question concerns generalizability. Assuming the definition is mathematically valid, does it make sense? Does your idea of rational even-ness mirror the usual concept of even-ness in a useful and intuitive way? In other words, is it conceptually homologous, or is the resemblance only skin-deep? This seems more like a philosophy question, because now we're getting into issues of aesthetics and ontologies, rather than simply validity.

Aesthetically speaking, then, is your concept of even for the rationals similar to the concept of even for the integers? Well, there's at least one big difference: not every rational number can be easily categorized. No matter how you address the problem of rationals that don't have a "last digit" you are inevitably going to end up with some numbers that seem like they should be even, but aren't (0.2424...), as well as some numbers that don't seem to be either even or odd (0.5656...), both of which are departures from the simple duality of odd vs. even for the integers.

Another huge difference is that, whereas a number's relationship to 2 is essential to the concept of "even" for the integers, it has no bearing at all on your definition: 0.2 is even and 0.5 is odd even though the first is 1/5 and the second is 1/2.

The biggest gap, however, is that while odd and even for the integers describe properties of the numbers themselves, your definition describes a property of the number's representation. This could still be useful (it's often important in programming, for instance, to know how numbers are physically stored in a computer), but it's a huge aesthetic difference from the original concept of odd and even. If an integer is even, it's even no matter what--even-ness is absolute, like primality or being a perfect square--but this isn't the case when using your definition for the rationals. 1/5, for instance, is even when written in decimal (0.2), but when written in binary it's 0.00110011..., which is not even. In fact, no rational is even in binary, because every number either has one as its last digit, or has no last digit at all! The idea that you could "switch" a number from even to odd just by writing it out a certain way clashes strongly with our existing intuitions.

This leads me to conclude that, although your definition might be made mathematically valid, it is not a generalization of the concept of even and odd--it's some other concept. As such, it might still have useful applications or interesting properties, but that's getting back into mathematics and not philosophy.

Answer 4

You gave three potential definitions for some kind of even number.

The first one is the one that is used in mathematics.

The second one "doesn't make sense". An integer doesn't have "a last digit". The digits of an integer are a concept that follows from representing integers using digits. We usually represent integers as decimal numbers, but we could use different bases, for example base 16 is quite commonly used, where the last digit could be 'A', 'B', 'C', 'D', 'E', or 'F' and isn't a digit at all (you might say that twenty-six = 1A in hexadecimal, so the last digit is 1). Or base 3 could be used. The integer four is written in base 3 as 11. That's the first thing in mathematics when you write down a definition: You check whether that definition makes sense at all.

In the third definition, what is the last digit of a fraction supposed to be? That doesn't make any sense at all. Before you use this definition, you have to state what you actually mean by the terms in the definition. (The difference to the second definition is just gradual. In the second definition, one might think that it makes sense, but it doesn't when you look closer. In the third definition, it doesn't make any sense at all).

That said, you can define whatever you want. The question is whether the result is useful in any way. For example, extending the definition of "even number" to fractions should be consistent with the original definition. That means 4 or 5 should have the same evenness or oddness independent of which definition you use. If your definition for rationals tells me that 4 = 4/1 and the last digit 1 in that fraction is odd, then the definition is inconsistent with the original definition. That is very likely to make it less useful.

Another requirement for usefulness is that the definition helps you in any way. For example, for integers you can say "the sum of two integers is even if the integers are both even or both odd, and odd if one is even and one is odd". Or "the sum of four consecutive integers is always even". Expressing these without the definition would take a lot longer. If all your definition does is to allow you calling some rational numbers even and others odd, without any situations where this allows you to express some statement in an easier way, then the definition is useless.

Paul Ross gave some excellent examples how you could make a definition for rational numbers that makes sense. However, even his examples only allow you to classify rational numbers into even and odd; you don't get anything useful out of it. For example stated above was a rule for the sum of odd and even integers: 0.14 and 0.26 are "even" and their sum 0.4 is "even", while 0.14 + 0.16 = 0.3 has a sum that is "odd" according to that definition.