Numbers: Large and small; Real and fake
leonstreet
NUMBERS: LARGE AND SMALL; REAL AND FAKE
This is rather long for a newsgroup post, for newsgroups like this
have something of the car-boot sale about them (perhaps garage sale is the
nearest american equivalent): things easily picked up and put down again,
tat everywhere, but always the chance of finding something interesting, or
even valuable. Since it's long anyway, it might as well have this preamble,
in which I apologize in advance for speaking from mere pockets of
knowledge, for spending too much time thinking airiy about foundational
issues and not enough time mastering the relevant mathematics. Yet I have
struggled, in no doubt a very amateur way, with the mathematics and the
concept of number, and I have tried to be constructive as well as critical,
and though this post may prove to be more piffle than pedigree, perhaps
some passer-by or stall-holder, with a little more time on his hands than
usual, will turn over this unwieldy object, and manage, between gasps of
effort, to make some knowledgeable observations.
The counting process, as I shall call it, is the progressive,
stepwise transfer from the to-be-counted to the
has-been-counted. It could be depicted as follows:
To-be-counted -> Has-been-counted
/ / /
/ / / / empty
/ / /
/ / /
/ / / -> /
/ / /
/ / / -> / /
/ /
/ / /
: :
: -> :
: :
/ / //
empty / /
/ / / /
We do not yet have number, but merely the concept of aggregatable
units.
Given the basic counting process, one can go on to develop the
notion of a model counting process, which is itself a counting process but
serves as a standard or yardstick for all counting processes. For
example,the counting process performed with the ten fingers. Since the
finger gestures become representative of all counting processes, they
acquire a symbolic character, which will acrue oral and literary
equivalents, be subject to abbreviation and replacement by more
convenient symbols, be subject, in short, to abstraction.
From a purely conceptual viewpoint, however, the development of a
standard counting process is unnecessary. All that is required is the
acquisition of names for the stages of the counting process. After all,
when I idly 'count' ("One, two, three...."), there is not anything I am
counting, but I am reciting the sequence of names for the stages of the
counting process: like an engine waiting to be put into gear. But when I do
count something, my counting is like a lyric overlay to the rythmic beat of
the counting process. The counting process is counting in all but name.
It may be noticed that I have not made reference to the notion of a
one-to-one correspondence. It should be fairly obvious that the act of
putting the elements from two collections in one-to-one correspondence with
each other is the performance of two counting processes in parallel (or
equivalently, the counting process performed on pairs of objects). It
would be silly to suppose that the parallel counting process has to be
included in an account of the single counting process. Even in the case of
finger-counting, where it could be said there is a one-to-one
correspondence between object and finger, it is not this one-to-one
correspondence which is telling me that the number of objects and fingers
are the same, but that this whole process defines for us what number is.
There are not two concepts of number here, but only one.
It may be said that number represents not the name of the stage of
the counting process, but the result. in the has-been-counted column, after
that stage of the counting process. That is correct, but that result cannot
be thought of as having some numerical status apart from the counting
process. Names for stages of the counting process already carries this
information about the result. Whilst it is true that in the number namesthe
focus is on the result (what is in the has-been-counted column), this
'result' should not be wrenched from the context of the counting process.
It is not something apart from this process, like a knife or fork exists
apart from the machine that manufactured it. In this output, this count,
it is not that the dumb, blank-eyed four-ness (say) of some collection
means something in and of itself, but rather meaning lies in the counting
process in which it is embedded.
What then is the significance of the multiplicity of these fingers
when four fingers are raised in a count? It is merely the explicitness of a
fundamental analogy which enables four fingers, or four sheep or four
spears to be seen as equivalent. But this is already implicit in ordinary,
abstract counting, and is expressed through the fact that any kind of
thing (or indeed mixture of different kinds of things) can be subject to
the counting process. Conceptually, abstract counting is fundamental, and
in it there is no one-to-one sorting, merely the application of the
counting process with the names for its stages. Abstract counting is not a
disguised form of analogical counting. On the contrary, analogical counting
is merely a crutch, a half-way house, like a sign language I used before I
learned properly to speak. If it is said that there is after all a
one-to-one correspondence, in counting, between the objects and the
counting process stages or names for stages, still, the one-to-one
correspondence does not function as input to the concept of number. When I
have the names, I already have the numbers. Our standard numbers are at
once and inseparably ordinal and cardinal.
Of course we know that if the elements of two collections can be
put in one-to-one correspondence with each other then there are the same
number of elements in each collection. But then we know that the number of
elements in either collection is 1 or 2 or 3 or......... It is sufficient
to have the concept of number to deduce that two collections in
one-to-one correspondence are equinumerous. The question is: Does
one-to-one correspondence in and of itself give us or engender a purely
cardinal notion of number? Let us ignore, so far as we can, how exactly
such a purely cardinal notion of number is to be integrated into an account
of the ordinary counting numbers, and try to explore the plausibility of
this notion in its own right. It is not that the 'short' answer to the
supposedly elemental status of the one-to-one correspondence (that a single
counting process is more elemental than two counting processes in parallel)
is inadequate, but we want to see what lies behind this idea. For it can
seem that we can just see directly somehow, that two collections are
equinumerous when their elements are in one-to-one cross correspondence.
Dantzig speculates that there must have been model cardinals in the
early development of number. For example, the wings of a bird for 2, clover
leaves for 3, animal legs for 4, fingers on a hand for 5, and etc..Counting
processes apart, the problem is that faced with a collection of things, how
does one even begin to estimate which model cardinal is to be
attached to it, unless the model cardinals are understood to be in some
sort of sequence. Later he candidly observes that though (on his, the
conventional view) cardinal number is conceptually more primitive than
ordinal number, there is no anthropological or philological evidence for
this priority.
Lines of correspondence between two collections seem like skewers,
fastening each object to its opposite in the pair. Skewers would be a great
visual aid. We would have to suppose these are not strewn about, like so
many pick-up sticks, but straightened and sorted. We could then see in one
glance perhap 5 or 6 skewered pairs. A few more? But of course if there are
hundreds we resort to the counting process.
Imagine you are sitting on a train, and another train is travelling
alongside in the same direction on an adjacent track. You see on your left
fields, buildings slipping by, but suppose you have no concept of speed.
Are you able to by looking out of the window on the right to see directly
that the other train is moving faster, slower or just as fast as
yours? Or will you just see a carriage either slipping away to the left or
right or just stationary?
I have no concept of number. I have no use for it. I do not herd
animals. I do not measure time. If one of my children is missing at a
family gathering this is not a numerical event: I know he or she is missing
because I know each child by name. If an object is missing I know it
because it thwarts an action I was going to perfrom with it. You perform in
front of me the act of placing objects from two collections in one-to-one
correspondence by physical separation, and I pay attention. Do I learn from
this something numerical, or just see As running out before Bs or vica
versa or a running out simultaneously? And if I did have an inkling of what
was going on, would it be a purely cardinal intuition, or rather an
understanding that you are 'counting-together', like one-ein, two-zwei,
three-drei, etc., even though I do not have any words
for this process? The whole business of putting two sets in one-to-one
correspondence, even though it does not employ a count
explicitly, is 9-months pregnant with the fully fledged notion of counting.
The pure cardinal is plausible, insidiously, precisely to the degree that
its one-to-one correspondence suggests and strains toward counting.
We underestimate how extraordinarily primitive counting is,
anthropologically, and by association conceptually. As if we could just
extract the concept of number and every other concept would stay the same.
As if was number was not deeply embedded in a broad linguistic and
conceptual web, and associated with it the layers of cultural acquisition
upon which apparently ordinary practical human activities depend. It is not
the one-to-one correspondence itself, as it were the focus
to this test for equinumerosity, which suggests number, but the whole
situation, the treating of something as a collection of
objects, the treating of those objects as aggregatable units, the stepwise
transfer of objects from one pile to another. There is a submerged
sophistication in this act of placing things in one-to-one correspondence,
from which it is virtually impossible to imagine that number be excluded.
(The same submerged sophistication, of course, in the notion of a counting
process. Does this make it useless for our purposes? But we do not present
it as if it were some cornerstone of arithmetic. Rather we say something
like: See how closely related are counting-with-number and the counting
process.)
How could we have been fooled, into this notion of the purely
cardinal number, for so long? After all this time, it seems to me, the dead
hand of logicism still holds sway.That Jupiter has 4 moons is held to be
plain matter of fact. It would have had 4 moons whether life had sparked
off on the Earth or not. Anything not analytically true is purely
contingent. The separation of logic and psychology is as sure as that
between heaven and hell. And without human beings, would there have
been noise and heat, red things and speeding objects, rocks bigger than a
bus, and vast desolate stretches of empty space? Everything just as we
humanly understand it, but no humans, and four virtual fingers raised,
corresponding to the four moons of Jupiter.
Wise writers on number (like Dantzig) acknowledge the role of
analogy, the analogy (that we now understand as number) that exists
between, say, two pheasants and two pebbles. This is not simply a matter of
the right mode of pattern recognition clicking in. This kind of analogy is
quite general in language: it is the very power of language to rule over
and subdue the world. In any case, unless this analogy has been made, so
that number is essentially given, there simply is no relevant one-to-one
correspondence between pheasant and pebble for someone to talk about in
their Introduction to Set Theory.
The major immediate result of these reflections is that there is no
infinite set of finite numbers. Ordinality and cardinality go hand in hand.
Numbers are best thought of as a rake or net. Somehow the net analogy seems
more comfortable, though we only need one dimension (complex numbers
apart). Indefinitely extendable in reach but also indefinitely flexible to
a finer and finer mesh. There is no ultimate mesh, no infinite set of
rational numbers, from which only the irrational numbers are excluded.
However fine the mesh, there is a still finer mesh, and the capture of all
the rationals can no more be achieved than the netting of the irrationals.
The notion of an infinite decimal expansion is an attempt to close the
mesh, to complete the catch, but we can no longer take such a notion for
granted.
How then are we to regard the irrationals, numerically? Are they
numbers at all? How do they fit in with the ordinary, well-behaved rational
numbers? Since we have rejected an infinite set of rational numbers, the
issue of completeness (the creation of a happy family of all possible
numbers, all sitting at the same table regardless of how recalcitrant some
of them are) is no big deal. But there is something putative about the
numerical status of an irrational. It is not just that if sq.rt.2 (say)
were a number, then it would be between 1.41 and 1.42. We have the
understanding that, whatever the sq.rt.2 is, we are getting closer and
closer to it in calculating the successive decimal places. Is sq.rt.2 some
sort of limit to which the calculations approach, like 2 is the limit of
the sequence of the initial sums of 1 + 1/2 + 1/4 +.............? The
trouble is that we have no other way of getting at what sq.rt.2 is,
numerically, other than these finer and finer calculations. Here, the
endlessness of the calculation is the absence of a limit. The conventional
notion of a limit, as it might be used here, is predicated on the existence
of infinite decimal expansions, and these we reject.
We must remember that entities like the sq.rt.2 were originally
thought or assumed to be rational, by the Greeks to whom we owe the origins
of our mathematics (and much else). It as though we had a box full of
diamonds, but it turned out that some of them were fake. The fakes look
like diamonds, feel like diamonds, and perhaps function like diamonds in
various other ways, but they are not the genuine article. Yet though they
are fake, they are after all fake diamonds, not fake emeralds or fake
passports. We will be more precise about irrationals later, bui in pretty
much the same way, I believe, we have all the numbers, indiscriminately
considered, but these must be understood to contain the fake alongside the
genuine.
The big issue to be addressed is the relationship between
arithmetic and geometry, between our understand of number and our
understanding of space. There is a close correspondence between the left
and right (or up and down) along the spatial line, and the greater-than or
lesser-than relation of the number line. Both may be thought of as dense,
in so far as between any two points or numbers, another point or number can
be found. The difference between them, and what alone makes the spatial
line continuous, is that in the spatial case the points are thought of as
being already there. In the spatial case, it is as if an infinite
subdivision has already been made. Given a unit length, sq.rt.2 or pi has a
definite position on the line (in the latter case, the length rolled out by
circle of unit diameter after one revolution). There is no compulsion to
think of number in that way, and it obscures the difference between number
and space to do so. The existence of a number between any two given numbers
is the existence of a method for determining such a number, just as the
existence of larger and larger numbers without limit is the existence of a
rule in (typically) a positional number system for producing a successor.
We have not escaped the net.
We should be aware that the notions of continuity provided by
analysis or analytic geometry are useless in so far as they rely upon the
conventional notion of the real number. For example, it is useless to talk
about how a function may take on all value between f(a) and f(b) if we do
not have the right understanding of the continuity of the domain a to b.
The whole issue of continuity needs to be re-thought. If Euclid did skip an
axiom of continuity in Proposition 1, Bk 1 of the Elements, it would not be
the axioms of Archimedes and Completeness as proposed by Hilbert. It seems
to me part of our understanding of a spatial line, in contradistinction to
the number line, is that it is filled with points. It has infinity built
in. Whatever bizarre functions one may come up with which mimic the line
while containing discontinuities, one must first have the notion of a
simple line as a plenum of points, so that two lines which cross (like
Euclid's circles) necessarily have a point in common.
The spatial and number lines can be understood as exact opposites,
in respect of continuity. A spatial line begins as a plenum and successive
subdivision aims at discreteness in a fundamental, exhaustive numbering --
hopelessly, since only smaller and smaller spatial intervals are produced
which remain plenary in character for ever. Numbers are inherently a finite
and discrete structure, and successive subdivision of a number interval
aims at plenary completion -- hopelessly because only finer and finer
discontinuities are produced forever, and the numbering remains finite and
discrete. Spatializing the number line and discretizing the spatial line
are opposite sides of the same dud coin. The irrationals stand like
reminders of translational indeterminacy between the two great languages of
Arithmetic and Geometry.
Let us suppose some purpose is served by marrying the two,
Arithmetic and Geometry. We will need to get the shotgun out, and I can
already hear the voice of Zeno responding to the vicar's challenge to the
congregation. We want a ruler equal to the task of measuring any length
whatsoever, however large or small, and however it arises, for example as
the length of the diagonal of a unit square, or the length rolled out after
one revolution of a circle with unit diameter. There are, on the face of
it, two ways in which one could introduce the infinite into arithmetic.
Firstly one could have (with & as the infinite term):
1, 2, 3, ....&, & + 1, & +2, ...
Or (with I as the infinite term):
1, 2, 3, .......I,
where I is understood to be unexceedably infinite.
We should stress that both of these are fictions. There can be no
place in either sequence where the numbers first become infinite, since
&-1, &-2, etc. (or I-1, I-2, etc.) are infinite too, so there can be no
infinite numbers in the sequence of numbers. There is only an endless
succession of finite numbers. But we want to treat & and I as somehow fixed
in size, though infinite; we want to treat them as number-like.
The first option seems to me to be self-defeating, as if to say the
infinite is after all unattainable, and that at the infinitesmal level
infinite precision is impossible. We can form the series:
1&, 2&, 3&, ....
or: &(1, 2, 3, ...).
So it seems that we are back to square one as far as denumerating
the infinite is concerned. It is the second option that I want to explore.
Accordingly I present what I will call the Unreal numbers. (The
conventional real numbers are already pretty unreal, but not perhaps unreal
enough.)
We express the positive whole numbers (assuming the denary system)
as follows, starting with I:
999....999
999....998
:
:
998 999
:
:
000....002
000....001
We understand there to be a maximum string length L, less than I
but still infinite. We can relate L and I (so far, at least) as: I = 10^L -
1.
We want to have a decimal point ranging anywhere over the L + 1
places in the number string. We need a minus sign, and a zero. There will
be a smallest positive number, which I shall call i.
i = .000....001 (string length L)
Some examples:
637.246........388
637.246000...000
where in each case the string length of the decimal part is L - 3.
930041.....242.39,
where the string length of the whole number part is L - 2.
Obviously a whole number with string length L can have no decimal part.
With the additions of the decimal point, zero, and the minus sign,
inevitably the number of numbers exceeds I, were that possible. For any
given string length I can only be the count of the positive whole numbers.
We have to accept this as part of the conceit of the whole system., and we
deal with numbers greater than I by equating them with I i.e. I + n = I
What is important that we have an infinite number I, a smallest number i,
and a fixed string length which is part and parcel of thinking of I as
fixed and which acts like a limit on the number of significant figures.
Moreover, it is important to point out that the unit interval has exactly I
intervals (without invoking I + n = I) of 'width' i.
One of the attractions of this system is that it restores the
infinite and the infinitesmal to a proper relationship, as opposite ends of
the same teleoscope. We may observe the specific relationships:
I . i = 1 - i (.999....999)
1/I = 0
1/i = I (actually I + i, but this = I)
The irrational numbers may be supposed to have a representation on
the number line by virtue of having some definite terminal expansion. Pi,
for example, has L - 1 decimal places, and the ultimate decimal place will
be one of the numbers 0-9. Of course no sense can be attached to
discovering the terminating decimal expansion of pi, but neither can it be
attached to knowing a middle portion of the decimal expansion of pi. We
will only ever know the first few digits. There being some definite
terminal part of the expansion of pi should be no more shocking than the
idea of there being some definite sequence of digits in the (infinite)
middle of its expansion. If we want pi to be a number, we have to pay the
price. All we have done is remove the elliptical infinity dots (....) from
the end of a number to its middle, where it does less harm.
You will notice that powers of i, that is, i^n where n > 1, are
automatically zero. It is also true that the addition of i (or multiples of
i) can have zero effect. For example, 3.6 + i = 3.6, since there are only L
- 2 decimal places available to the right of the 3.6. This is reminiscent
of some of the old approaches to the calculus -- more Leibnitz or some
of his followers, than Newton, I think. Why might that be interesting? As
already suggested, without an established alternative to the defective real
number system it is not just the Calculus that has its trousers round its
ankles. I do not say that the notion of a limit is necessarily wrong in the
calculus. But let me quickly look at a subject where the notion of a limit
requires refinement, and where the infinitesmals might be relevant, namely,
the subject of convergent infinite sums, such as 1 + 1/2 + 1/4 + ..... The
conventional understanding, I take it, is that this is an actually infinite
series whose limit is 2. Is this tired, old distinction between potential
and actual infinity relevant? Very much so. We now know it is a fallacy
that there are an infinite number of finite rationals. Moreover, it is an
often acknowledged intuition (and just as often dropped, since the
conventional view of the reals forces one to reject it) that an infinite
number of finite numbers ought to sum to infinity. This intuition is
perfectly sound, I believe. The sum 1 + 1/2 + 1/4 + ... should sum to
infinity if it was really an infinite set of finite numbers. Now I do not
doubt that the sum in fact does not exceed 2, but there are two ways to
look at this. Either we interpret it in strictly finite terms as the limit
of the succession of initial sums, in which case the impossibility of
exceeding 2 (or indeed reaching 2) is identifiable with the impossibility
of summing an actually infinite number of terms, or, we understand that as
the number of terms becomes infinite, the terms themselves become
infinitesmal. You cannot have it both ways. For example, there is no real
number with an infinitely long decimal expansion which is not in the
remotest parts infinitesmal, as opposed to finitely small.
Do the ordinary operations of arithmetic go through OK with the
Unreal numbers? In fact the zero effect ( a + b = a; a, b > 0) does cause
problems with associativity and distributivity. Thus from the fact that a +
b = a it does not follow that na + nb = na. For example, let a be
111.111....111 and b be 333i. Then a + b = a. But 5a + 5b is not= to 5a.
One can see this as a problem with the zero effect equality itself, rather
than a failure of associativity and distributivity as such. It can be
avoided by treating the zero effect equality as a rounding off operation,
only to be permitted at the end of a calculation, not in the middle.
Let us turn our attention to the spatial line. Before we apply our
Unreal number line ruler to it, we should ask ourselves what is known, if
anything, about the relative number of points in one line as compared to
another. For example, it has been known for ages that the points of two
concentric circles can be put into one-to-one correspondence by associating
points on a common radius.
Also well known is that if lines are drawn joining the
corresponding end points of two straight lines and extended to a vertex
(unless the lines are parallel), forming a triangle, lines drawn from the
vertex through the one line and to base of the triangle (the other line)
effect a one-to-one corresondence between the points of the line. (It is
not strictly necessary for the two lines to be straight provided each line
is cut only once per line.)
We can go further than this. We should now accept that the infinite
subdivision of any line fills the (1 - dimensional) space of the line,
leaving no gaps. For we have rejected the fallacy of infinite numbers of
finite rationals, and an infinite division of a line is a division into
infinitesmals, so there is no longer a reason to believe any points are
left out. If we consider a line as infinitely subdivided into (say) its
half, quarter, eighth, sixteenth.... parts, then by joining the
corresponding points (assuming we know how to find a midpoint) of any two
lines whatsoever we effect a one-to-one correspondence between them.
So the one-to-one correspondence does not appear to tell us
anything beyond that all lines consist of an infinite number of points. Of
course we already know from the first part of this work that the one-to-one
correspondence does not tell us anything about number independently of the
possibility of a count, so this is hardly surprising. However, a count, a
numbering of the infinite points in a spatial interval is precisely what we
are proposing. The situation here is different from, for example, Galileo's
puzzle over the natural numbers and their squares. There, the short answer
is that there is no 'all the naturals', nor 'all the squares', and the
deeper paradox, or rather irony, is that we are trying to count the very
thing we use to count with. Here we acknowledge an infinite number of
points, and are proposing an (admittedly artificial) means of numbering
them. So there is something to be reconciled.
What is true is that any line regardless of relative length can be
split into an infinite number, the same infinite number, I, of intervals of
infinitesmal length (unless the line is infinitely long, when it will have
unit parts). In any geometric diagram, with perhaps a circle, various
straight lines, perhaps a triangle and etc., we have the freedom to specify
anything as the unit length; it could be the base of the triangle, the
diameter or radius or circumference of the circle, or anything at all. We
take our unit arithmetic interval, from 0, i, 2i,.....to .999....999, then
1, which has I intervals of difference i, to correspond to the (arbitrarily
or freely) chosen unit length of the space line. Once we have settled on a
unit length, say the radius of the circle, then there are more points, for
example another radius' worth to complete the diameter, but we invoke I +
n = I to disallow the total number of points in the diameter exceeding I.
That the number of points in radius and diameter is the same is forced upon
us, not because of some nonsense about one-to-one correspondence, but by
the conceit of setting an infinite maximum, by the conceit of the whole
project of numbering the infinite. But the system is scaleable,
nevertheless. The first radius has its infinite number of points 0 to 1,
the second its infinite number of point 1 to 2, and so on. And the whole 1
- dimensional space is extendable out to I, where decimal places have
disappeared, and divisible down to i, where decimal places are all there
is.
Let us leave aside now the Unreal numbers, and pursue the notion of
a finite number with a view to understanding the irrationals, and better
understanding the difference between number and space. No infinite number,
whether I or &, can actually be in the number line. Numbers are finite,
ordinally and cardinally. In fact the distinction between ordinal and
cardinal is unnecessary, or at least dubious, for there is no such thing as
a purely ordinal number, for this would be merely a sequence without
connotation of 'magnitude', and as we have seen, the purely cardinal number
is a myth. Of course there is an ordinal or sequential aspect to number,
but its cardinal aspect exists only in the context of that ordinality. So
that whereas the number sequence is an example of the more general notion
of a sequence, cardinality is nothing different from number itself,
including its ordinal nature. Numbers are finite, and ipso facto rational.
They are discrete, like beads on a string. We can put beads between beads
indefinitely, and beads at the end of the string indefinitely, but they are
always finite strings of beads. Necessarily there are gaps between the
beads. Even so the distinction between the number line and space line might
be unimportant if we were confined to a single dimension. But crucially
space comes in two, three and even more dimensions. In two dimensions there
are angles and arcs, and there are lengths defined in terms of those two
dimensions, such as the hypotenuse of a right-angled triangle. We can put
up an array of beads (ordered pairs of beads), but inevitably there will be
places between the mesh, places corresponding to no intersection of number,
ratios of relative lengths which cannot be aligned with any subdivision of
a single axis.
The usual proof of the irrationality of sq.rt 2 is, like many
reductio proofs, both neat and thoroughly unperspicuous. It is not as if
the irrationality of sq.rt 2 has anything to do with the properties of odd
and even numbers. It's a curiousity, a showpiece, as if to say: See what we
can do by purely algebraic means. From a geometric point of view it's quite
plain that no square root can be a rational number unless the number whose
root is to be taken is a perfect square, or ratio of perfect squares. There
are, after all, only so many perfect sqares, one for each of the natural
numbers.
You cannot make a (larger) square out of 3 squares, or out of 7
squares.
Of course you can make a square whose side is 1 and a half, but if
we divide our grid into halves, such a square is identical to the 3 X 3
square, and a square whose side is 1 and a sixth is equivalent to a 7 X 7
square in a grid split into sixths.
You cannot get more out of the notion of a square root than you put
into the notion of squaring. Any rational number has a square, and given
that function, the rational square roots are, inversely, the rational
numbers we started with, each associated with a unique square. Inevitably
if you want the square root of a non-square number to be a number, it is
going to be an irrational number. Similar considerations apply to cubic
roots in three dimensions, quartic roots in four dimensions, and so on.
Even if you take the existence of irrational numbers, of whatever
kind, as given (which obviously I don't, and indeed I am in the process of
trying to give an account of irrational number), so that we can talk about
their squares and square roots, it is clear that square roots of non-square
rationals are going to be a source of irrationals.
In any polynomial:
Cnx^n + Cn-1x^n-1 + .....+C1x + C0 (Ci integers),
where exactly the zeros lie will in general depend on all the coeffecients,
but it is known that the denominator of any rational zero must be a factor
of Cn. For any given polynomial, we only have to divide the unit grid into
Cnths, and zeros which miss those grid marks miss irrevocably all grid
points. For example, if Cn = 1, the zeros have to be integer values or they
are not rational at all.
So what is an irrational number? It's a point on a spatial line to
which there corresponds no actual number, but which can be ever more
closely approached by a number. It's a mismatch between number and space,
but such is the flexibility of our number system, we can find numbers for
positions arbitrarily close to it. For example pi is the length rolled out
on an axis by a circle of unit diameter after one revolution. (It is not
important how pi is 'defined', what is its canonical description, as ratio
of circle circumference to diameter or otherwise, only that it has this
propery.) The end point is a definite position on the (spatial) line,
marking off a precise magnitude. (The usual treatment of book V of the
Elements is to interpret the propositions algebraicly, and decry Euclid's
avoidance of the concept of the irrational number. But on the contrary,
unencumbered by the modern myth of the irrational number, he is wiser than
his critics, and well might he ask what is to be the base currency of this
algebra, since he knows that number and irrational quantities can only
congregate on the spatial line.) There is no number for this position, this
magnitude. It is the very finitude of the number system, in contrast with
the accepted infinities of space dimensions, which gives rise to irrational
numbers. By all means put a bead on the string there, and call it pi. But
it can correspond to no systematic numerical division of the spatial line.
By all means let's not bother, for practical mathematical purposes, to
distinguish rational from irrational. But let's not pretend there is some
uniform definition of number catering for both rational and irrational (in
which, for example, the difference depends merely on the existence or
otherwise of a certan upper bound), which destroys the very basis upon
which irrational numbers arise. The notion of a limit is quite useless.
There is no infinite decimal expansion of pi. Unless we allow that it is
infinitesmal in its infinitely remote parts. For otherwise all this can
mean is that the expansion goes on without limit, without end. There is not
something numerical the expansion is approaching. It is something
positional that is being approached, and the infinite expansion of pi
expresses the hopelessness of finished numerical representation.
It can seem as if the decimal expansion of pi must somehow be
already there ahead of any calculation. Like the future in a deterministic
universe, which exists somehow alongside the past and the present, for time
is an illusion of perspective. Isn't mathematics timeless in just this way?
Don't all the digits of pi have to be already there for us to discover
them?
All of them? So at what point, after the tenths in the first
column, the hundredths in the second column, the thousandths in the third
column, and so on, do we reach the infiniteth column? There is no infinite
set of natural numbers, so no infinite set of the digits of pi. Our
capacity to subdivide the number line is endless. That capacity exists.
There is a method for producing the successor of any natural number. That
method exists. The numbers produced form a great chain of
..next, next, next,..... That chain exists. But it's a chain tied at only
one end.
Number has a relatively simple structure, which obviously derives
from counting. Or it would be obvious, had not people like Frege and
Russell wiped away the fingerprint evidence of the human ownership of the
concept of number. Though it is relatively simple, that it is not to say
it doesn't have a deep semantics. As we have already hinted, every concept
is part of a tangled web of other concepts. For example, counting, or at
least the possibility of counting, seems intimately bound up with the
understanding of a thing as a certain kind of thing (and so the
understanding of something as an another example of that kind of thing, and
so the understanding of multiple examples of that kind of thing, and so
on), but this has to do with the very possibility of, not merely language,
but perception. For unless, in the forest (millenia before the herding of
sheep, we may suppose), I can see a tree as another tree (even if I don't
have a word for 'tree', or any words at all), can see a wild boar as
another wild boar, in the nexus of activities and practical possibilities
which underpin the meaning of tree for me (tree for climbing, for shelter,
for hiding behind), and of boar (good to eat, hard to catch, tusks for
trophy, for body ornament), then I cannot properly be said to see at all.
So some sort of distinction between single and multiple seems to be
implicit in human (or indeed animal) experience. What then, shall we reason
about the One and the Many, like Parmenides? The equivalence class of
equinumerous sets, the equivalence class of infinite sequences of
rationals, the transfinite ordinals and cardinals, these are the Ones and
Manys of our own time, the products of reasoning in free-fall, unrestrained
by any principle of reflexion whereby the critical scrutiny to which the
world is subjected would be applied to the reasoning itself, and which
would necessarily locate its terms of discourse in relation to the ordinary
human activity of counting.
There is not something peculiarly mysterious about the ontological
status of numbers. One might equally fret about the ontological status of a
bus-ticket. (No mere piece of paper.) It's all part of the human traffic.
As bus-tickets go, numbers happen to be rather ancient.
Leon Street
april 2008
Dave L. Renfro
> We must remember that entities like the sq.rt.2 were
> originally thought or assumed to be rational, by the
> Greeks to whom we owe the origins of our mathematics
> (and much else).
You need to do some more background reading on the topic
you're writing about, as this is wrong in several ways.
For one thing, the Greeks didn't really think of sqrt(2)
as a "number". Instead, they dealt with the idea of
line segment lengths being commensurable or not (and
the early Greeks almost certainly never considered the
possibility of non-commensurable line segment lengths).
Dave L. Renfro
Michael Press
<29551283.1208524802...@nitrogen.mathforum.org>,
Where is the division between early and late?
Hippasus of Metapont circa fifth century BC?
He showed that in a regular pentagon
a side and a diagonal are incommensurate.
--
Michael Press
leonstreet
wrote:
>leon street wrote (in part):
Which is why I deliberately referred to sqrt(2) as an 'entity' in
the passage you quoted. I am aware of the limitations in the early Greek
concept of number. (It's the fallacies in the modern concept of number
which concern me.) To talk of the irrationality of sqrt(2) is a reasonable
paraphrase of the incommensurability of the diagonal and side of a unit
square. In this sense, I take it, the Pythagoreans did expect sqrt(2) to be
rational. I do not myself believe that sqrt(2) is a number (because of the
arguments in the whole post), but it is convenient to characterize it as
irrational.
Thanks,
leon
Gerry Myerson
wrote:
Long ago, mathematics discarded what things are in favor of
what things do. sqrt(2) does things a number ought to do, so
whether it is a number or not is semantics, or philosophy, or
religion, but not mathematics.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
leonstreet
<ge...@maths.mq.edi.ai.i2u4email> wrote:
>In article <qbfo049ba9i2ibnr2...@4ax.com>, leon street
>wrote:
>
>> On Fri, 18 Apr 2008 09:19:32 EDT, "Dave L. Renfro" <renf...@cmich.edu>
>> wrote:
>>
>> >leon street wrote (in part):
>> >
>Long ago, mathematics discarded what things are in favor of
>what things do. sqrt(2) does things a number ought to do, so
>whether it is a number or not is semantics, or philosophy, or
>religion, but not mathematics.
Hmm. That's an interesting piece of ...............Semantics?
Philosphy? Religion? Mathematics?
I now have a new picture of the sqrt(2), as the John Wayne of the
number world, getting off his horse and doing things a square root has
gotta do.
If you would like to follow the preceding gentleman to the First
Aid Tent, where gnat removal and broken toes can be attended to.
leon
Angus Rodgers
> I now have a new picture of the sqrt(2), as the John Wayne of the
>number world, getting off his horse and doing things a square root has
>gotta do.
Yep, I gotta corral me them rationals. <spit>
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
Gerry Myerson
wrote:
> On Mon, 21 Apr 2008 23:51:19 GMT, Gerry Myerson
> <ge...@maths.mq.edi.ai.i2u4email> wrote:
>
> >In article <qbfo049ba9i2ibnr2...@4ax.com>, leon street
> >wrote:
> >
> >> On Fri, 18 Apr 2008 09:19:32 EDT, "Dave L. Renfro" <renf...@cmich.edu>
> >> wrote:
> >>
> >> >leon street wrote (in part):
> >> >
>
> >Long ago, mathematics discarded what things are in favor of
> >what things do. sqrt(2) does things a number ought to do, so
> >whether it is a number or not is semantics, or philosophy, or
> >religion, but not mathematics.
>
> Hmm. That's an interesting piece of ...............Semantics?
> Philosphy? Religion? Mathematics?
>
> I now have a new picture of the sqrt(2), as the John Wayne of the
> number world, getting off his horse and doing things a square root has
> gotta do.
> If you would like to follow the preceding gentleman to the First
> Aid Tent, where gnat removal and broken toes can be attended to.
OK, so you're interested in making jokes, not mathematics. Bye.
leonstreet
wrote:
>On Tue, 22 Apr 2008 12:43:34 +0100, leon street wrote:
>
>> I now have a new picture of the sqrt(2), as the John Wayne of the
>>number world, getting off his horse and doing things a square root has
>>gotta do.
>
>Yep, I gotta corral me them rationals. <spit>
A potted version of the key initial argument.
1) The counting process is as I described it: the progressive,
stepwise transfer from the to-be-counted (or to-be-transferred) to the
has-been-counted (has-been-transferred).
2) The number (how many) of a collection is the stage of the counting
process reached when the collection is exhausted by the counting process.
3) The names for the stages of the counting process form a sequence,
but also and at the same time constitute the numerical magnitudes, the
(cardinal) numbers.
4) The process of pairing off the elements from two finite (finite to
begin with) collections in a one-to-one correspondence is supposed to
provide a separate avenue, independent of counting, to the relative
numerical magnitude of two collections, in a kind of direct-access,
information-effecient way.
5) But in fact the establishing of a one-to-one correspondence is the
performance of two counting processes in step with each other. Then of
course we know that if the elements from each of the collections are
exhausted at the same step then we have reached the same stage of the
counting process, whatever that stage is. Of course this does not estalish
the purely cardinal number independently of counting.
6) Perhaps part of the plausibility of the purely cardinal number is
not merely that we can say two collections are equinumerous without
counting them, which is merely to say that in the double counting process
we do not need to register at any stage which stage we are at, but in the
fact that in principle we could say that two collections are equinumerous,
the same stage of the counting process is reached, before we have even
invented the names for the stages of the counting process.
Even so, we clearly have not reached a notion of the cardinal
number independent of the ordinal number. The sequential and cumulative
character of the counting process guarantees that the names for its stages
simultaneously and indissolubly combine the familiar cardinal and ordinal
aspects. Nothing more is needed.
Is this completely crazy, or am I just in the wrong market-place?
BUY STREET'S UNREAL NUMBER ELIXIR
Cures divers Aritmetikal ills, also baldness, ear-ache, leprosy,
constipation, prairie foot and piles.
Street
Tim Little
> 5) But in fact the establishing of a one-to-one correspondence is
> the performance of two counting processes in step with each other.
How can I count without having a one-to-one correspondence from each
number in my count to the next?
If I haven't counted up to a billion yet, is it then possible that one
of those numbers I haven't yet counted may have two different next
numbers, or none? According to what you say, I should have such
uncertainty, since "the establishing of a one-to-one correspondence is
the performance of two counting processes in step with each other". I
haven't done the counting yet, so presumably I can't conclude the
correspondence until I have.
Is this a reasonable conclusion from your position?
- Tim
Angus Rodgers
For what it's worth, I don't think your ideas are crazy at all, and
I even have rough notes of similar ideas* about "counting processes"
from a couple of decades ago (although perhaps this doesn't prove
anything about them not being crazy!), but: (a) for reasons of my own,
I'm reluctant to think too much about the foundations of mathematics
until I've learned quite a lot more real mathematics; and (b) although
I am inclined to agree with the positive part of what you say (about
the process of counting finite sets), I don't agree with the negative
part (about irrationals not being numbers, and so on). For both these
reasons, I didn't bother to reply, although I thought seriously about
doing so. You may wonder why (b) would constitute a reason for not
replying! It is that for me, the existence of "mathematical reality"
is somewhat "ontologically insecure" (as R. D. Laing might have put
it), and I greatly prefer to cultivate my insecure relationship with
mathematics and try to make it more secure, rather than arguing about
whether what I need so desperately to exist actually exists or not -
however valid such arguments are. In other words, for me, it is a
quasi-religious issue. I can't speak for others, of course (and it
is obvious to me as well as others that my feelings and intuitions
about mathematics are distinctly odd, eccentric ones - although my
understanding of the content of mathematics seems 'normal' enough),
but for what it's worth (he said again, defensively), I suspect that
"quasi-religious" feelings about the foundations of mathematics are
not all /that/ unusual, and you may be treading on people's dreams.
I think it's fair to say that mathematicians tend not to be terribly
enthusiastic about debating the foundations of the field, especially
against someone taking a dismissively sceptical position towards most
of what makes mathematics so interesting. But what do I know ...
*Anyway, here is a snippet of a transcription of some of my ancient
handwritten notes, written during a painfully long period of exile
from Cantor's paradise. I haven't the heart to trawl through any
more of them at the moment, as I not only forget, but actively prefer
to forget, what it was like to view mathematical reality from the
perspective of a complete outsider. But I offer this as reassurance
that, even if you are talking total bollocks, you aren't completely
alone in thinking that way! :-)
I later, in 1988, went on to relate these sort of ideas about counting
finite sets to what could be described as a finite version of Peano's
axioms (second-order), in which the successor relation has a "last" as
well as a "first" element. There's nothing complicated about it, and
it can be related to quite standard ideas about finiteness (such as
the criterion that a set is finite iff it can be doubly well-ordered).
I think it is probably worth writing up in some way, but I've never
got around to doing all of it systematically - mainly for the reason
I've already indicated, which is that I prefer to have a more secure
and extensive understanding of the field that I'm trying to provide a
"foundation" for before building a supposed "foundation" which doesn't
actually support anything that needs supporting.
(Also, as I was vaguely reflecting only today, I have less interest
than I used to do in finding wholly convincing rigorous foundational
arguments for areas of mathematics that already feel fairly secure
to me - see e.g. <http://www.apronus.com/math/math.htm>, which not
coincidentally is written by two Roman Catholic graduate students of
mathematics! - than in consolidating areas of my knowledge which feel
intuitively weak to me, and, thus fortified, moving on to learn the
stuff which feels most /interesting/. I do mean to come back to all
my foundational worries eventually, however.)
I'm very worried about posting this, because: (i) I (deliberately)
haven't thought about any of it afresh; (ii) there's so much more
stuff connected with this which I could ramble on endlessly about,
thus never getting any mathematics done, and going quietly nuts;
(iii) ... I'm sure there was a (iii)! Perhaps (iii) was that, even
while trying my utmost /not/ to actually think about any of this, I
can't help realising that it connects with areas of set theory that
I haven't thought about yet, and don't feel ready to start thinking
about yet. And other areas of mathematics as well, e.g. structures
of lattices of subobjects of objects other than finite sets, e.g.
finite-dimensional vector spaces, as well as objects with infinite
lattices of subobjects. And ordinal numbers and the well-ordering
theorem, and other stuff I haven't come to yet, in the textbook on
set theory that I am (or was!) reading. (Counting infinite sets
looks like fun, too.) Oh, well.
===
Wed 5 Nov 1986
546.5
If S is a set you are interested in counting, let P be the set of
all subsets of S. Then P is a "space", between whose "points"
(the subsets of S) [...] there is the relation of differing by a
singleton [...] Define an "enumeration" of S as any minimal
subset of P with the following properties: (i) {} is an element;
(ii) if A is an element which is != S, then there is just one
element in the set which consists of A plus a singleton. I think
this formal definition captures the idea of an attempt to count
the elements of S.
Wed 17 Dec 1986
576.19
[...] The part played by a number system in counting is essentially
as a /record/ or /trace/ of a process being effected on the set
being counted. You could carry out the process without involving
a number system; only at the end you would have no record of its
history.
Here is how to count a set:
Set "elements counted" to the empty set.
Set "elements not yet counted" to the set to be counted.
Set "number" to zero.
*WHILE* there are still elements to be counted:
Remove one element from the set of those still to be counted.
Add it to the set of those counted.
Replace number by its successor.
"Number" is the number of elements in the set.
576.23
The process of counting requires a *counter* [...] - a device
which can be in (countably many!) distinct states. [...]
Imagine trying to design a robot which could count collections
of objects.
Anyway, thinking about the process of counting forces you to
think about connecting "mathematics" to "the real world".
There should be a theory, applicable to processes such as counting,
and containing propositions linking concepts such as the termination
of a process with concepts relating to mathematical induction, such
as Dedekind's "chains".
576.24
There are a lot of things you can do while counting through a set.
I have already mentioned two: (i) incrementing a counter, starting
from zero; (ii) doing nothing. You can do other things, such as
counting another set repeatedly at the same time. [...]
===
So you see, I do (even if no-one else does!) think you have a valid
point to make, about a one-to-one correspondence between two finite
sets being factorisable into two counting processes, being run in
parallel with each other. I'd vaguely like to encourage you to keep
thinking about such things - but not to be so quick to dismiss the
reality of parts of mathematics which don't fit some philosophical
framework (which may itself be a lot less "certain" than what it
ventures to dismiss). That way lies fanaticism. (And we see rather
a lot of anti-Cantorian fanaticism in sci.math - which may lead to
some rather jaundiced reactions to your article, reasonable though
much of it is.)
leonstreet
I'm not sure I understand this, Tim. Your reference to a one-to-one
correspondence from one number to the next confused me.
The counting process is set up so that a single unit is transferred
at a time, until all units are exhausted. There is no limit on the size of
the collection.
The positional number system generates an inexhaustible supply of
successive numerals to mark the stages of the counting process.
These are mechanical procedures. It is presupposed that we know
what a collection is, and presupposed that we can distinguish between a
single unit of a collection and multiple units of a collection. These might
be rather large presuppositions, but we are in respect of them in the same
boat when it comes to describing the counting process as we are in
describing the one-to-one correspondence.
I take it your objection is something along the lines of the idea
that the natural numbers must pre-exist any actual counting process. But
this only makes sense, I think, if the numbers are thought of platonically,
as independent of human device. I would claim that our concept of number is
tied to the counting process. The natural numbers are a string tied at one
end, adequate for the counting of any size collection. But this does not
mean there is an infinite set of natural numbers. No more does a sausage
machine have to be able to produce an infinite number of sausages in order
to be able to produce as many sausages as you want.
Forgive me if I've misconstrued your meaning,
leon
leonstreet
wrote:
>On Thu, 24 Apr 2008 11:46:05 +0100, leon street wrote:
>
>>On Tue, 22 Apr 2008 17:05:04 +0100, Angus Rodgers <twi...@bigfoot.com>
>>wrote:
This was a truly helpful post, and helpful in various different
ways, and more sympathetic than I expected or deserved. You express
yourself eloquently and honestly, and bravely (I have in mind the
presentation of your un-reexamined notes). I understand your sentiments,
though the driving force behind my thinking is different. Though there are
problems in mathematics which, from time to time, interest me for their own
sake (I once spent about 3 yrs in my spare time trying to enumerate the
n-step self-avoiding random walks, only to suddenly, almost overnight, lose
interest, though I probably learnt a fair amount of mathematics in the
process), these days it tends to be the philosophical issues which act as
the spur, and I tend to lazily just learn enough mathematics to be able (I
hope) to think about them clearly. It's a bad approach, I know, and all I
can say in mitigation is that I make some effort to ensure my woeful
ignorance of key areas of mathematics is slightly less woeful, month upon
month. It's as if I'd gone back to school, only this time I demand an
answer to every puzzle and question which even half-forms in my mind before
I will proceed. Again, I know it's a hopeless way of learning, even for
someone of mature years. (For a start, who will be my teacher?) There is an
in-built scepticism to this approach, which yet doesn't have to be, I
trust, dismissive and cranky, but can still be humble and honest.
It occurred to me since I last posted (this is not in response to
your post, which I've only caught up with today, and I'm still thinking
about your notes) that the natural way to 'reinvent' the real numbers, as
opposed to the probably ludicrous unreal numbers of the original post, is
to frankly admit the (as I see it) 'spatial borrowing' and define them as
ratios of lengths, or lengths relative to a unit length. Thus sq.rt(2) is
the length of the diagonal of a unit square; 5, for example, is of course
just the ration of 5 unit lengths to a unit length; and so on. In this way
the usual real numbers are 'saved', and the Eudoxus/Euclid stuff on
magnitudes can be seen as the beginning of a theory of the real numbers.
The number line thus becomes, not a disposable geometrical representation
of the real numbers, but their actual geometrical foundation. I know this
goes against the grain. And if (infinitary) arithmetic is based upon
geometry, the question (since say the 1800s) could be: Which geometry?
Anyway, this was just in the way of updating my thinking, if you'll
forgive the presumption, and I appreciate the better since your post that I
have some way to go in proving the modern concept of real number
deffecient.
Much obliged, Leon.
Tim Little
> The positional number system generates an inexhaustible supply of
> successive numerals to mark the stages of the counting process.
What is a succession of numerals, if not a correspondence from one
numeral to the next? You don't have to count up to 180476315 to know
that there is one and only one next numeral, 180476316. You know this
because you have a 1:1 correspondence that determines the next
numeral.
Your claim was that determining a 1:1 correspondence requires
counting. I deny that claim, by virtue of the fact that the means you
use to carry out that count *is* a 1:1 correspondence.
> I take it your objection is something along the lines of the idea
> that the natural numbers must pre-exist any actual counting process.
Not at all. It's that the process of counting implicitly requires a
1:1 correspondence, not the other way around.
We could consider very primitive pre-counting methods. For example:
1, 2, many, many, etc. This is not a 1:1 correspondence, since both
"2" and "many" lead to the same next marker "many", and so becomes
useless for counting. Or a very young child's "1, 2, 3, Um I don't
know what's after 3". Once again, no 1:1 correspondence between each
marker and the next, and so useless for counting.
- Tim
Angus Rodgers
> It occurred to me since I last posted (this is not in response to
>your post, which I've only caught up with today, and I'm still thinking
>about your notes) that the natural way to 'reinvent' the real numbers, as
>opposed to the probably ludicrous unreal numbers of the original post, is
>to frankly admit the (as I see it) 'spatial borrowing' and define them as
>ratios of lengths, or lengths relative to a unit length. Thus sq.rt(2) is
>the length of the diagonal of a unit square; 5, for example, is of course
>just the ration of 5 unit lengths to a unit length; and so on. In this way
>the usual real numbers are 'saved', and the Eudoxus/Euclid stuff on
>magnitudes can be seen as the beginning of a theory of the real numbers.
>The number line thus becomes, not a disposable geometrical representation
>of the real numbers, but their actual geometrical foundation. I know this
>goes against the grain. And if (infinitary) arithmetic is based upon
>geometry, the question (since say the 1800s) could be: Which geometry?
(I really must stop reading sci.math at bedtime!) :-)
My initial long reply to this article was far too long, so for the
moment I'll just say: (1) I could reply to this in exactly the same
way as to your remarks on counting, i.e. by saying that either you
are not crazy, or else I have been crazy in exactly the same way (as
a transcription of more of my notes would confirm!); (2) I strongly
recommend reading Dedekind's "Essays on the Theory of Numbers" (in
English translation, in a cheap Dover paperback edition - presumably
it's also available online at Project Gutenberg), for inspiration. (I
do tend to go on about that book!) I agree: Eudoxus is also inspiring.
If I say any more, I'll just end up writing my first reply all over
again. These are really deep waters, with strong currents.
I just checked, and my first reply was only a little over 130 lines
long (including quotes). I think the problem was not really so much
with its length as with the sheer number of half-baked ideas I threw
into it. I don't often (ever?) bump into anyone who is (or seems to
be) bothered by the same things that have bothered me for so long
(but which I'm trying not to bother too much about at the moment!).
It's probably better if I chuck in only one or two half-baked ideas
at a time (with some checking to see if they're really relevant). I
recognised a lot more in your article than I've actually replied to
here. I hope that will do for the moment!
leonstreet
Ok, but this isn't the usual way that a one-to-one correspondence
would be appealed to in explaining counting, which would be in terms of a
correlating of numeral and 'object'. You seem to be talking about numeral
production as such.
Your
1, 2, many (more than 2)
is not really an abstract sequence, a sequence of abstract
numerals, but the description of a (degenerate case of a) number system in
which there are no unique symbols for numbers greater than two. But this is
exactly what we do have.
Rule for producing a successor (roughly):
0,1,2,3,4,5,6,7,8,9
is a chain of successors, originating, left to right, from 0.
abc......ijk is some numeral (base 10)
EITHER k < 9 i.e. k = one of 0-8
Then replace k by its successor.
OR k=9
Then replace k by 0
AND
EITHER j < 9
Then replace j by its successor
OR j=9
Then replace j by 0
AND
etc
etc
OR
................
replace a by 0 and put 1 in front.
I think an element of a sequence has a unique successor (if it has
a successor at all, i.e. it is not the last element) by definition. After
Matthew, Mark, Luke , John, Act, Romans, there are two books of the
Corinthians, distinguished as first and second, but also of course as
distinct books distinguishable by objective means (for example the first
one is the one with the oft-quoted passage extolling the gift of Love). If
we did not distinguish an order between the Corinthian epistles, we would
have a different sequence, namely: the Matthew books, the Mark books, the
Luke books, ..... the Corinthian books, etc, where the individual elements
were no longer single books but groups of books, some of which may happen
to have a single member. In any case, there are clearly no such
complications in the case of the succession of numerals.
Numerals are abstract symbols. My claim is that in the context of
the counting process, understood as this operation upon a collection of
'objects', the step-wise transfer from the (abstract) pile to-be-counted to
the (abstract) pile has-been-counted, the symbols become representative of
numerical quantity, but they remain (their meaning as cardinal numbers
depends upon their remaining) symbols in a sequence.
Leon
leonstreet
wrote:
>On Sun, 27 Apr 2008 01:12:39 +0100, leon street wrote:
That's absolutely fine. Many thanks for all your comments.
Leon
Tim Little
> Ok, but this isn't the usual way that a one-to-one correspondence
> would be appealed to in explaining counting, which would be in terms
> of a correlating of numeral and 'object'.
True, it isn't. I was merely arguing that since you need a 1:1
relation to even generate successive numerals, it is pointless to try
to paint counting by numbers as being logically prior.
Given the newsgroup, it is also worth pointing out that there are many
mathematical entities for which counting is impossible, but 1:1
correspondence can still be established.
- Tim
Brian Chandler
<snip>
> You cannot make a (larger) square out of 3 squares, or out of 7
> squares.
Curious claim... I wonder what it means?
> Of course you can make a square whose side is 1 and a half, but if
> we divide our grid into halves, such a square is identical to the 3 X 3
> square, and a square whose side is 1 and a sixth is equivalent to a 7 X 7
> square in a grid split into sixths.
Two questions.
(1) Can you make a square out of 4 squares, and if so, how?
(2) Can you make a square out of 5 squares?
Brian Chandler
http://imaginatorium.org
Gerry Myerson
<af61384e-ae70-4ab5...@p25g2000pri.googlegroups.com>,
Brian Chandler <imagin...@despammed.com> wrote:
> leonstreet wrote:
>
> <snip>
>
> > You cannot make a (larger) square out of 3 squares, or out of 7
> > squares.
>
> Curious claim... I wonder what it means?
It means what it says. If you have three (geometric) squares, or 7,
you can't put them together to form a square. But the part about 7
is wrong.
> > Of course you can make a square whose side is 1 and a half, but if
> > we divide our grid into halves, such a square is identical to the 3 X 3
> > square, and a square whose side is 1 and a sixth is equivalent to a 7 X 7
> > square in a grid split into sixths.
>
> Two questions.
>
> (1) Can you make a square out of 4 squares, and if so, how?
If the 4 squares are the same size, I'm sure you can figure out
how to put them together to form a square.
(Then take one of the 4 squares, and decompose it into 4 squares,
and, voila! 7.)
> (2) Can you make a square out of 5 squares?
The way I've seen the problem stated, it says you can decompose
a square into n squares for n = 4 and for all n at least 6. I don't
recall seeing anyone write out a proof that you can't do it for
n = 2, 3, and 5.
Bill Taylor
> The way I've seen the problem stated, it says you can decompose
> a square into n squares for n = 4 and for all n at least 6. I don't
> recall seeing anyone write out a proof that you can't do it for
> n = 2, 3, and 5.
Well 3 (similarly 2) is very easy, surely.
You need at least 4 squares to fill up the spaces near
the corners, as it's clear that you can't cover two corners
with one square (else it's the only one).
I expect there's something vaguely similar for 5, but it's
going to be at least a bit more complicated!
--------------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
--------------------------------------------------------------
Yesterday, I thought I had a new deja vu experience.
But on second thoughts I realized I'd had it before.
--------------------------------------------------------------
Brian Chandler
> In article
> <af61384e-ae70-4ab5...@p25g2000pri.googlegroups.com>,
> Brian Chandler <imagin...@despammed.com> wrote:
>
> > leonstreet wrote:
> >
> > <snip>
> >
> > > You cannot make a (larger) square out of 3 squares, or out of 7
> > > squares.
> >
> > Curious claim... I wonder what it means?
>
> It means what it says.
Hmm, I think this is actually wrong. It says what it says (trivially),
and it means what it means (trivially), but while I can see what it
says, I'm not sure what it means, hence my question.
> If you have three (geometric) squares, or 7,
> you can't put them together to form a square. But the part about 7
> is wrong.
Ah, you think he's referring to an interesting problem about tiling a
number of squares of possibly different sizes to make a larger square.
I thought - given that the OP is "worrying" about whether irrationals
"exist" - that he meant one could not cut up three squares of the same
size and put the bits together to make a single square. I might be
wrong, but it seems unlikely he would suddenly refer to an unrelated
and interesting problem.
Brian Chandler
http://imaginatorium.org
Gerry Myerson
<88ad6b59-4198-48de...@i36g2000prf.googlegroups.com>,
Brian Chandler <imagin...@despammed.com> wrote:
All I have to go by is what you included in your post,
and there was nothing there about irrationals. I'm sure
that you can cut up three squares, of the same area or not,
and put the bits together to make a single square. That is,
I'm sure it can be done, because there's a theorem that says so;
I'm not sure I could do it personally, but I'm sure it can be done.
Gerry Myerson
> <88ad6b59-4198-48de...@i36g2000prf.goog
Now that I think about it, it's actually very easily done.
Given two squares, of sides a and b, a >= b, align both squares with
sides vertical and horizontal, the left side of the smaller abutting the
right side of the larger, the bottoms forming a line segment. Like this:
Oo
only with squares instead of circles. Locate the point P on the bottom
of the larger square, at distance b from the lower left corner of that
square. Cut along a straight line from P to the upper left corner of the
bigger square, and from P to the upper right corner of the smaller
square. You now have 5 pieces (4, if a = b) which are easily re-arranged
to form a single square.
I first saw this construction in Dudeney's Amusement in Mathematics.
By induction, you can cut up any finite number of squares, of any sizes,
into a finite number of pieces which can be re-assembled to form a
single square.