A Most Basic Question
tiglath
I have had this surreal conversation with this guy I met first in a
religious newsgroup, I don't have his permission to print what he said to
me, in private email, so I will paraphrase what he said by changing a couple
of inessential adverbs and adjectives with synonyms. The meaning remains
unchanged.
The author of the paragraph below, The Professor, teaches physics in the
Wake Forest University in North Carolina. I pity his students.
The Professor claims that:
No, mathematics is not science. It's used a lot in science, but the
processes are different. Basically, mathematics is applied logic. It lacks
anything comparable to experiment. It also lacks the step of abduction,
which is part of theory-formation, an essential part of science.
He defines abduction as "borrowing ideas from a known situation that is
similar or analogous to one the you're wondering about, and use them as a
source of hypotheses for the present situation."
Whereas, I thought that abduction is a process that proceeds by inferences
where the major premise is certain and the minor one only probable.
I
How can a Physics teacher write that Mathematics is not science? Is this
another Professor On Acid in the Silly Season at American Universities?
Comments welcome.
Nathan Urban
> How can a Physics teacher write that Mathematics is not science?
I don't think mathematics is science. Science _uses_ mathematics to
formulate its theories, but ultimately science is all about producing
models that describe the outcomes of experiments, while mathematics has
no such concern.
james d. hunter
>
> tiglath (tig...@usa.net) wrote:
> :
> : How can a Physics teacher write that Mathematics is not science?
>
> science.
>
> You might start off by identifying for yourself the criteria that
> characterize a science. Most of all, what does it strive to
accomplish?
>
> Contrast this with the conventions, definitions and relationships
that
> define mathematics. What does it strive to accomplish?
>
> Finally, think about the conventions, definitions, and relationships
> that characterize a language.
>
> Now, ask yourself: Is mathematics a science, or a close sister to a
> language?
Math is a science, maths is not. First order logics of numbers
are sciences. If they weren't, all the "scientists" would be
working the night shifts at Burger King.
D.E.J.A.
Re; Maths (science or not?)
Not this topic again...
:o)
Is this in the FAQ?
If so, I direct you to that.
If not, expect like 50 postings about this topic...all the same stuff. ;)
DEJA
Uncle Al
There was a time when mathematics was not experimental - the purest
self-consistent thoughts uncontaminated with real world testing, then
only applied in retrospect. Computers have made it possible to test
hypotheses by direct experiment, as in the Four-Color Theorem. If you
compare theory with empirical data, that is science.
It may not be a gusher, but the wall has been breached.
--
Uncle Al Schwartz
http://www.mazepath.com/uncleal/
http://www.ultra.net.au/~wisby/uncleal/
http://www.guyy.demon.co.uk/uncleal/
http://uncleal.within.net/
(Toxic URLs! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?" The Net!
Don Redmond
> I have had this surreal conversation with this guy I met first in a
> religious newsgroup, I don't have his permission to print what he said to
> me, in private email, so I will paraphrase what he said by changing a couple
> of inessential adverbs and adjectives with synonyms. The meaning remains
> unchanged.
>
> The author of the paragraph below, The Professor, teaches physics in the
> Wake Forest University in North Carolina. I pity his students.
>
>
> The Professor claims that:
>
> No, mathematics is not science. It's used a lot in science, but the
> processes are different. Basically, mathematics is applied logic. It lacks
> anything comparable to experiment. It also lacks the step of abduction,
> which is part of theory-formation, an essential part of science.
>
>
> He defines abduction as "borrowing ideas from a known situation that is
> similar or analogous to one the you're wondering about, and use them as a
> source of hypotheses for the present situation."
>
> Whereas, I thought that abduction is a process that proceeds by inferences
> where the major premise is certain and the minor one only probable.
>
> I
> How can a Physics teacher write that Mathematics is not science? Is this
> another Professor On Acid in the Silly Season at American Universities?
> Comments welcome.
I'll say a few words.
First, I always think of mathematics as being an art and not a science, and
so I guess I agree.
On the other hand, I, and I'm sure other mathematicians, do experimental
mathematics in the sense that we try out theorems to see if we can prove
them. If one is in a particularly numerical field (like number theory or
numerical analysis or (fill in your own here)), then one can do science
like experiments by producing a bunch of special cases. I don't know if
the physicist would consider this close enough to the scientific method
for his tastes or not.
Finally, I don't know how many mathematicians just conjure up theorems out
of the rules of logic. We use logic as a tool not as the main road (unless
you're a logician, but they may not be mathematicians either.)
Those are my thoughts. But as remember YMMV.
Don
Virgil
no more science than grammar is literature.
--
Virgil
vm...@frii.com
Nicolas Bray
This thread suffers from a problem which mathematics abhors: not defining
terms. Define science and math. Then we can discuss the question
rationally.
(of course, agreeing upon those definitions will probably require a thread
10 times longer)
Harry H Conover
:
He can do so easily and with confidence, because mathematics is not a
science.
You might start off by identifying for yourself the criteria that
characterize a science. Most of all, what does it strive to accomplish?
Contrast this with the conventions, definitions and relationships that
define mathematics. What does it strive to accomplish?
Finally, think about the conventions, definitions, and relationships
that characterize a language.
Now, ask yourself: Is mathematics a science, or a close sister to a
language?
Harry C.
Harry H Conover
: There was a time when mathematics was not experimental - the purest
: self-consistent thoughts uncontaminated with real world testing, then
: only applied in retrospect. Computers have made it possible to test
: hypotheses by direct experiment, as in the Four-Color Theorem. If you
: compare theory with empirical data, that is science.
Indeed, but what empirical data exists in mathematics to test theory
against? (Sadly, I am not acquinted with the Four-Color empirical
obseration. I would remind you that paradoxes and conundrums within
a language do not a science make.)
Mathematics has faces that are both trivial and subtle, but neither
make it a science because the object of all science is to understand
and explain the mechanisms of nature. By contrast, mathematics
reveals nothing beyond it's own rules and behavior. By any standard,
that isn't science.
Harry C.
:
: It may not be a gusher, but the wall has been breached.
me...@cars3.uchicago.edu
>
>
>I have had this surreal conversation with this guy I met first in a
>religious newsgroup, I don't have his permission to print what he said to
>me, in private email, so I will paraphrase what he said by changing a couple
>of inessential adverbs and adjectives with synonyms. The meaning remains
>unchanged.
>
>The author of the paragraph below, The Professor, teaches physics in the
>Wake Forest University in North Carolina. I pity his students.
>
>
>The Professor claims that:
>
>No, mathematics is not science. It's used a lot in science, but the
>processes are different. Basically, mathematics is applied logic. It lacks
>anything comparable to experiment. It also lacks the step of abduction,
>which is part of theory-formation, an essential part of science.
>
>
>He defines abduction as "borrowing ideas from a known situation that is
>similar or analogous to one the you're wondering about, and use them as a
>source of hypotheses for the present situation."
>
>Whereas, I thought that abduction is a process that proceeds by inferences
>where the major premise is certain and the minor one only probable.
>
>I
>How can a Physics teacher write that Mathematics is not science? Is this
>another Professor On Acid in the Silly Season at American Universities?
>Comments welcome.
>
not.
Mati Meron | "When you argue with a fool,
me...@cars.uchicago.edu | chances are he is doing just the same"
tiglath
Virgil <vm...@frii.com> wrote in message
news:vmhjr-30089...@ftc-0409.dialup.frii.com...
> Mathematics is a large part of the language of science, but mathematics is
> no more science than grammar is literature.
How about people who broke the Enigma. Didn't they do thousands of
experiments till the found the right set of rotors to break the cipher? Is
this not science?
And what about the people who are doing experimental research to factor
bigger and bigger numbers into large prime factors, are this peoples
"grammarians."
What about the cryptographic products produced by the NSA and commercial
firms. Have this purely mathematical products have been through a process
any less scientifc than producing a drug, or a new plastic?
Kyle R. Hofmann
> Virgil <vm...@frii.com> wrote in message
> news:vmhjr-30089...@ftc-0409.dialup.frii.com...
> > Mathematics is a large part of the language of science, but mathematics is
> > no more science than grammar is literature.
>
> How about people who broke the Enigma. Didn't they do thousands of
> experiments till the found the right set of rotors to break the cipher? Is
> this not science?
Yes, it's applied cryptography, not math.
> And what about the people who are doing experimental research to factor
> bigger and bigger numbers into large prime factors, are this peoples
> "grammarians."
If they're working on new algorithms (as opposed to just throwing lots of
hardware at a big number), then under the above categorization, they would
be a subset of all "grammarians".
> What about the cryptographic products produced by the NSA and commercial
> firms.
Applied cryptography again. It's an application of mathematics, not pure
mathematics.
> Have this purely mathematical products have been through a process
> any less scientifc than producing a drug, or a new plastic?
Yes; they've been subject to a logical process rather than a scientific
one.
--
Kyle R. Hofmann <rhof...@crl.com> | "...during the years between 960 and
1000 there was great activity in the production of homilies ... [ The
Blickling Homilies ] voice the almost universal belief that the world would
end in the year 1000." -- The Concise Cambridge History of English Literature
jsa...@ecn.ab.ca
: He defines abduction as "borrowing ideas from a known situation that is
: similar or analogous to one the you're wondering about, and use them as a
: source of hypotheses for the present situation."
: Whereas, I thought that abduction is a process that proceeds by inferences
: where the major premise is certain and the minor one only probable.
Abduction is what happened to the Lindbergh baby.
His definition corresponds to _induction_, and yours corresponds to
_deduction_.
He is quite correct - as far as it goes. Mathematics, when pure, is spun
off from the human mind, whereas physics and chemistry are in unremitting
contact with Nature.
Yet the fact that logic "works", or that counting is a relevant and
meaningful activity, _are_ inferences made from the observation of the
natural world around us. Mathematics is not in the position of the
powerless Antaeus (a legendary Greek hero who had great strength, but only
when in contact with the Earth) - it does rest upon a point of contact
with observable reality, however slender (and, in a sense, _inessential_
to the activity of mathematics itself, however essential it was to our
motivation to engage in it) it may be.
John Savard
tiglath
Kyle R. Hofmann <rhof...@crl.com> wrote in message
news:slrn7smr28....@crl4.crl.com...
> Yes; they've been subject to a logical process rather than a scientific
> one.
If mathematics is the language of science. How can the language of science
not be part of science.
It like saying that human communication involves evoking ideas in each
others brains; the best one to do that is by means of language, but language
is not part of human communication.
Until we can put ideas directly into each other's heads by telepathy,
language will be part of communication.
Remove mathematics from all science, can science exists, prosper, prevail?
If yes, then mathematics is not science, if not, we can say mathematics is
tiglath
<jsa...@ecn.ab.ca> wrote in message news:37cb6...@ecn.ab.ca...
> tiglath (tig...@usa.net) wrote:
> : He defines abduction as "borrowing ideas from a known situation that is
> : similar or analogous to one the you're wondering about, and use them as
a
> : source of hypotheses for the present situation."
>
> : Whereas, I thought that abduction is a process that proceeds by
inferences
> : where the major premise is certain and the minor one only probable.
>
> Abduction is what happened to the Lindbergh baby.
>
> His definition corresponds to _induction_, and yours corresponds to
> _deduction_.
Abduction, not induction. Look it up.
>
> He is quite correct - as far as it goes. Mathematics, when pure, is spun
> off from the human mind, whereas physics and chemistry are in unremitting
> contact with Nature.
If that it is all it takes to qualify, then the Human Mind is part of
Nature. Thoughts are natural processes.
Torkel Franzen
> Abduction is what happened to the Lindbergh baby.
In the present case, what is at issue is logical abduction in the
sense of the word introduced by Peirce.
Robert O'Dowd
Doesn't that depend on what you call "science"?
The scientific method, for example, makes no specific mention of
mathematical
concepts.
Applying the scientific method, I can hypothesise that your dog will not
yelp when I kick it (I won't kick my own dog). I can conduct an
experiment.
When I kick your dog, he yelps. My conclusion is that my hypothesis is
disproven. That is the scientific method. I have formed a hypothesis,
tested it in an experiment, and found a counter example that disproves
my hypothesis. Where is the mathematics in that?
Mathematics, as a language, is an important tool for scientists. It is
not
the only one. There is fundamentally nothing I can describe
mathematically
that I can not describe in English (although the description may not be
as
concise as the mathematical description). So, we could use the English
language as a substitute for mathematics in a scientific discussion.
That
suggests the language of mathematics is not essential to science.
-<Automagically included trailer>
Robert O'Dowd Ph +61 (8) 8259 6546
MOD/DSTO Fax +61 (8) 8259 5139
P.O. Box 1500 Email:
robert...@dsto.defence.gov.au
Salisbury, South Australia, 5108
Disclaimer: Opinions above are mine and may be worth what you paid for
them
tiglath
Robert O'Dowd <nos...@nonexistant.com> wrote in message
news:37CB7917...@nonexistant.com...
>
> Applying the scientific method, I can hypothesise that your dog will not
> yelp when I kick it (I won't kick my own dog). I can conduct an
> experiment.
> When I kick your dog, he yelps. My conclusion is that my hypothesis is
> disproven. That is the scientific method. I have formed a hypothesis,
> tested it in an experiment, and found a counter example that disproves
> my hypothesis. Where is the mathematics in that?
Where is the Medicine in that, where is the astronomy in that? Just because
they do not apply to your simple scientific experiement does it mean that
they are not science?
If one tries to come up with a better method to factor large numbers and try
each new approach in a computer and through this experimental research he or
she finally adds a new such method to Number Theory, as it has happened .
Is that not science? Isn't it akin to developing a new chemical process,
or develop a Physics theory? Hypothesis, test, refine, test, refine,
test... Bingo!
>
> Mathematics, as a language, is an important tool for scientists. It is
> not
> the only one. There is fundamentally nothing I can describe
> mathematically
> that I can not describe in English (although the description may not be
> as
> concise as the mathematical description). So, we could use the English
> language as a substitute for mathematics in a scientific discussion.
> That
> suggests the language of mathematics is not essential to science.
If you did such thing you would still be handling mathematical concepts. It
makes no difference you write them in Mathematical notation or in full
English prose. Differentiation is still differentiation, and geometry is
still geometry.
Steven B. Harris
In <7qfb0s$9ne$1...@autumn.news.rcn.net> "tiglath" <tig...@usa.net>
writes:
>
>
>
>I have had this surreal conversation with this guy I met first in a
>religious newsgroup, I don't have his permission to print what he
said to
>me, in private email, so I will paraphrase what he said by changing a
couple
>of inessential adverbs and adjectives with synonyms. The meaning
remains
>unchanged.
>
>The author of the paragraph below, The Professor, teaches physics in
the
>Wake Forest University in North Carolina. I pity his students.
>
>
>The Professor claims that:
>
>No, mathematics is not science. It's used a lot in science, but the
>processes are different. Basically, mathematics is applied logic. It
lacks
>anything comparable to experiment. It also lacks the step of
abduction,
>
>
>He defines abduction as "borrowing ideas from a known situation that
is
>similar or analogous to one the you're wondering about, and use them
as a
>source of hypotheses for the present situation."
>
>Whereas, I thought that abduction is a process that proceeds by
inferences
>where the major premise is certain and the minor one only probable.
If you'll look closely, you'll see that it's the same thing. Except
that he's also telling your where the minor premise ideas come from.
>
>I
>How can a Physics teacher write that Mathematics is not science?
"Science" is short for "natural science". The word "science" itself
is derived from "knowledge" and historically means a field with a good
defined knowledge content and algorithmic approach to getting work
done-- a craft as opposed to an art. As in political science and
computer science. What we now called natural science used to be called
natural philosophy until they finally decided they knew what they were
doing enough to teach it as a craft, and not a philosophy. The
shortened term came later, and now some people seem to think it coopts
all the other historical meanings.
> Is this
>another Professor On Acid in the Silly Season at American
Universities?
>Comments welcome.
Try being less provincial. That's what a university education is
supposed to do for you, you know.
Steven B. Harris
writes:
In <7qg1tf$e5a$1...@autumn.news.rcn.net> "tiglath" <tig...@usa.net>
writes:
It's the scientific method, but it's not natural science, due to
the fact that nature is not being studied, but rather a particular
logical system or language, and its implications. One just as well be
studying painting, ballet, or German in this hypothetico-deductive
fashion (all quite possible to do, I assure you). This does not make
any of them natural sciences.
When we say "science" in standard American English we *mean* it as
a shorthand for natural science. One can see the "scientific method"
used in, say, political science-- but "science", as we use the word as
shorthand for waht was historically "natural philosophy," is more than
the method. It's also the target of study, and the body of results.
This is a good example of why.
Steven B. Harris
writes:
>> : He defines abduction as "borrowing ideas from a known situation
that is
>> : similar or analogous to one the you're wondering about, and use
them as
>a
>> : source of hypotheses for the present situation."
>>
>> : Whereas, I thought that abduction is a process that proceeds by
>inferences
>> : where the major premise is certain and the minor one only
probable.
>>
>>
>> _deduction_.
>
>Abduction, not induction. Look it up.
Abduction is not a very useful word, though it was used by the
Greeks for certain kinds of deductive arguments with a "minor" (hah)
premise that was missing (enthymeme) or uncertain. Alas, the question
of whether or not the premise is correct makes all the difference,
because you have to face the logical problem before you can do anything
else. So the premise that seems minor is not so minor after all.
Deciding on the truth of a premise about nature (or indeed the truth
about any synthetic premise) puts you squarely up against the problem
of INDUCTION. You can hide the problem of induction, but you can't
make it go away. And since it will bite you in the rear one way or the
other, you might as well get it out in the open where the danger(s) to
your conclusions are open for all to see.
So yes, let us keep "abduction" as a term for what happened to the
Lindbergh baby, and let the other definition fade into a part of
ancient thinking which is best forgotten.
Steven B. Harris
writes:
>
>Kyle R. Hofmann <rhof...@crl.com> wrote in message
>news:slrn7smr28....@crl4.crl.com...
>> Yes; they've been subject to a logical process rather than a
scientific
>> one.
>
>If mathematics is the language of science. How can the language of
>science not be part of science.
Any language which encompases logic and some ideas about sets which
can correspond with physical measurement can be used as a a language to
describe natural science. Mathematics (here I include also
mathematical notation, but this is not strictly necessary either) is a
good shorthand for of one of these descriptive languages. But you can
say anything in English words that you can say in an equation, if you
want to take long enough. The only question is whether or not you want
to defnine that sort of thing as "mathematics" also, and make the thing
a tautology.
Do not confuse compactness with uniqeness. It's much the same with
musical notation. There have been many great musicians who could not
read music at all. It dosen't follow that they didn't know music, or
hadn't had music communicated to them, or know how to communicate it to
others. Well, how did they? Obviously by means of some language.
What language? We don't have a name for it. The language of music,
which isn't necessarily notes on a page, don't you know.
It's the same with mathematical notation. It's sympbols standing in
for ideas of logic and (when it comes to natural sciences) also nested
set and category sequences which stand in for physical measurement.
That's all. When you talk of pure mathematics, you leave out the
"stand in for physical measurement" part, but keep the other ideas.
That's all.
Steven B. Harris
>
>Yet the fact that logic "works", or that counting is a relevant and
>meaningful activity, _are_ inferences made from the observation of the
>natural world around us.
That a particular form of logical rules "work," and that counting as
we choose to do it, "works" may have been inferances from the natural
world, but they also reflect our bias in choosing systems in math which
are useful in describing the natural world. You don't HAVE to. We
certainly can construct math systems which are self-consistant but not
useful in describing the world. Some geometries, for instance (say
what you will, one of those possible non-Euclidean geometries is not
descriptive of nature-- we just don't know which, quite yet).
> Mathematics is not in the position of the
>powerless Antaeus (a legendary Greek hero who had great strength, but
>only when in contact with the Earth) - it does rest upon a point of
>contact with observable reality, however slender
Yes, but...
> (and, in a sense, _inessential_ to the activity of mathematics
>itself, however essential it was to our motivation to engage in it) it
>may be.
RIGHT. Good post.
Steven B. Harris
scientific
>one.
They may have been through a "scientific" process, if by science you
mean only the scientific method used by *scientists*, while doing their
work to discover how nature works. But "science" (natural science) is
NOT just the method, but also the body of knowledge and the target of
it (nature).
Sorry to repeat myself ad nauseam here, but this ought to be part of
the FAQ. Some people actually don't believe the last sentence of the
paragraph above, and will tell you that "science" is *only* the method.
But those people are forced to then consider anything which can be
studied by the methods of science, from math to how to congugate French
verbs to why the first world war started, as "science." Which does
great damage to the purpose for which we use this word "science" in
American English, which is as a shorthand for "natural science." So
let's not polute the language, okay? We need a word for natural
science, and "science" is a very convenient and colloquial one which is
understood by most people. Let's keep it that way. If people can't
seem to get the idea, I'm going to aggitate to go back to saying
"natural philosophy" so we don't all talk at cross purposes.
bill b
>
> .... The Professor claims that :
>
> No, mathematics is not science. It's used a lot in science, but the
> processes are different. Basically, mathematics is applied logic. It lacks
> anything comparable to experiment. It also lacks the step of abduction,
> which is part of theory-formation, an essential part of science.
>
> How can a Physics teacher write that Mathematics is not science? Is this
> another Professor On Acid in the Silly Season at American Universities?
> Comments welcome.
>
IMHO Mathematics is a science - a subclass of Logic - but not a Physical
science, such as Physics.
The Physical Sciences are driven by the observation of physical objects and
phenomena (so-called "physical realities").
Logic is driven by the natural relationships which occur when there is existence
of any kind - whether "real" (in the "physical" sense) or hypothetical.
Mathematics is concerned with the quantitative aspects of these relationships.
Bill B
tiglath
Steven B. Harris <sbha...@ix.netcom.com> wrote in message
news:7qg8in$s...@dfw-ixnews6.ix.netcom.com...
I must be making headway, because now you need to add the word "natural" to
keep mathematics out of "science." Ever heard of the natural numbers? Or
the
natural logarithms?
Of the Fibonacci numbers or the Golden Ratio which represent numerical
relationships inherent in Nature? The Golden Ration is to the shell of the
nautilus what the model of the atom is to whatever real atoms are like.
>
> When we say "science" in standard American English we *mean* it as
> a shorthand for natural science.
Science has a broader meaning that that, see a dictionary.
One can see the "scientific method"
> used in, say, political science-- but "science", as we use the word as
> shorthand for waht was historically "natural philosophy," is more than
> the method.
I agree it is more than a method. Is a method, lots of theoretical models,
axioms, and a language called mathematics, which in itelf is an abstract
science, in which there is research, study, methodologies and applications.
tiglath
> In <7qfr73$j2k$1...@autumn.news.rcn.net> "tiglath" <tig...@usa.net>
> writes:
> >
> >
> >Kyle R. Hofmann <rhof...@crl.com> wrote in message
> >news:slrn7smr28....@crl4.crl.com...
> scientific
> >> one.
> >
> >science not be part of science.
>
>
> Any language which encompases logic and some ideas about sets which
> can correspond with physical measurement can be used as a a language to
> describe natural science. Mathematics (here I include also
> mathematical notation, but this is not strictly necessary either) is a
> good shorthand for of one of these descriptive languages. But you can
> say anything in English words that you can say in an equation, if you
> want to take long enough. The only question is whether or not you want
> to defnine that sort of thing as "mathematics" also, and make the thing
> a tautology.
>
> for ideas of logic and (when it comes to natural sciences) also nested
> set and category sequences which stand in for physical measurement.
> That's all. When you talk of pure mathematics, you leave out the
> "stand in for physical measurement" part, but keep the other ideas.
> That's all.
Mathematics is more than a notation. The notation is only a convenient way
to express mathematical concepts. These could be expressed, and are
expressed, in any human language. You cannot replace the concept of
division with anything equivalent that is not mathematical.
>
Larry Mead
: I have had this surreal conversation with this guy I met first in a
: religious newsgroup, I don't have his permission to print what he said to
: me, in private email, so I will paraphrase what he said by changing a couple
: of inessential adverbs and adjectives with synonyms. The meaning remains
: unchanged.
: The author of the paragraph below, The Professor, teaches physics in the
: Wake Forest University in North Carolina. I pity his students.
: The Professor claims that:
: No, mathematics is not science. It's used a lot in science, but the
: processes are different. Basically, mathematics is applied logic. It lacks
: anything comparable to experiment. It also lacks the step of abduction,
: which is part of theory-formation, an essential part of science.
So far nothing controversial.
: How can a Physics teacher write that Mathematics is not science? Is this
: another Professor On Acid in the Silly Season at American Universities?
: Comments welcome.
Because it is not. Science depends on the process of "induction":
generalization from specific to the general case. But most importantly,
science depends on *experimentation* . Clearly mathematics lacks that.
The professor is quite right.
--
Lawrence R. Mead Ph.D. (Lawren...@usm.edu)
Eschew Obfuscation! Espouse Elucidation!
www-dept.usm.edu/~physics/mead.html
Larry Mead
: Virgil <vm...@frii.com> wrote in message
: news:vmhjr-30089...@ftc-0409.dialup.frii.com...
:> Mathematics is a large part of the language of science, but mathematics is
:> no more science than grammar is literature.
: How about people who broke the Enigma. Didn't they do thousands of
: experiments till the found the right set of rotors to break the cipher? Is
: this not science?
: And what about the people who are doing experimental research to factor
: bigger and bigger numbers into large prime factors, are this peoples
: "grammarians."
: What about the cryptographic products produced by the NSA and commercial
: firms. Have this purely mathematical products have been through a process
: any less scientifc than producing a drug, or a new plastic?
Good mathematics, yes. Science, no. The experiments must be with
real-world objects, not abstract numbers.
james d. hunter
>
> tiglath wrote:
> >
> > I have had this surreal conversation with this guy I met first in a
> > religious newsgroup, I don't have his permission to print what he said to
> > me, in private email, so I will paraphrase what he said by changing a couple
> > of inessential adverbs and adjectives with synonyms. The meaning remains
> > unchanged.
> >
> > The author of the paragraph below, The Professor, teaches physics in the
> > Wake Forest University in North Carolina. I pity his students.
> >
> > The Professor claims that:
> >
> > No, mathematics is not science. It's used a lot in science, but the
> > processes are different. Basically, mathematics is applied logic. It lacks
> > anything comparable to experiment. It also lacks the step of abduction,
> > which is part of theory-formation, an essential part of science.
> >
> > similar or analogous to one the you're wondering about, and use them as a
> > source of hypotheses for the present situation."
> >
> > Whereas, I thought that abduction is a process that proceeds by inferences
> > where the major premise is certain and the minor one only probable.
> >
> > How can a Physics teacher write that Mathematics is not science? Is this
> > another Professor On Acid in the Silly Season at American Universities?
> > Comments welcome.
>
> self-consistent thoughts uncontaminated with real world testing, then
> only applied in retrospect. Computers have made it possible to test
> hypotheses by direct experiment, as in the Four-Color Theorem. If you
> compare theory with empirical data, that is science.
>
That's a general misconception about computers, though.
It's only mathematicians who live under the delusion
that they invented computers.
Jim Carr
"D.E.J.A." <em...@address.com> writes:
>
>Is this in the FAQ?
Is there a sci.logic FAQ on the web?
Neither it nor sci.math has a web version linked from the sci.physics FAQ.
--
James A. Carr <j...@scri.fsu.edu> | Commercial e-mail is _NOT_
http://www.scri.fsu.edu/~jac/ | desired to this or any address
Supercomputer Computations Res. Inst. | that resolves to my account
Florida State, Tallahassee FL 32306 | for any reason at any time.
Jim Carr
} : inferences
} : where the major premise is certain and the minor one only probable.
}
}
} His definition corresponds to _induction_, and yours corresponds to
} _deduction_.
In article <7qfs31$mm1$2...@autumn.news.rcn.net>
"tiglath" <tig...@usa.net> writes:
>
>Abduction, not induction. Look it up.
It is not in the biggest dictionary I have in the office, so
either I need the OED or a more recent one. The only role of
'abduction' in math that I know of is the one described by
Tom Lehrer: "I asked a friend in Minsk, who had a friend ...."
and "Let no one's work evade your eyes ..."
How would you distinquish abduction from induction? I would say
the latter would be an example where some set of cases is used
to guess at the major premise.
Jim Carr
news:vmhjr-30089...@ftc-0409.dialup.frii.com...
}
} Mathematics is a large part of the language of science, but mathematics is
} no more science than grammar is literature.
In article <7qfnbk$p$1...@autumn.news.rcn.net>
"tiglath" <tig...@usa.net> writes:
>
>How about people who broke the Enigma. Didn't they do thousands of
>experiments till the found the right set of rotors to break the cipher? Is
>this not science?
The Polish mathematicians who *broke* the Enigma did so based
on mathematical methods, then used the result to design a machine
that could be used to determine the daily settings. From that
point on it ceased to be mathematics.
tiglath
bill b <bill...@email.msn.com> wrote in message
news:#W70xH78#GA.241@cpmsnbbsa03...
> > tiglath
> >
> > .... The Professor claims that :
> >
> > No, mathematics is not science. It's used a lot in science, but the
> > processes are different. Basically, mathematics is applied logic. It
lacks
> > anything comparable to experiment. It also lacks the step of abduction,
> > which is part of theory-formation, an essential part of science.
> >
> > How can a Physics teacher write that Mathematics is not science? Is
this
> > another Professor On Acid in the Silly Season at American Universities?
> > Comments welcome.
> >
>
> science, such as Physics.
>
> The Physical Sciences are driven by the observation of physical objects
and
> phenomena (so-called "physical realities").
>
> Logic is driven by the natural relationships which occur when there is
existence
> of any kind - whether "real" (in the "physical" sense) or hypothetical.
> Mathematics is concerned with the quantitative aspects of these
relationships.
>
I think so too.
People are so caught in the artificial separation of scientific disciplines,
which aid analysis and systematic classification, but do not represent
reality. Physics is repleat with Mathematics. It could not exist without
it. Much of mathematics can be used to describe natural phenomena and
natural relationships in the physical world. Mathematics are so
intertwined with other scientific disciplines that I find it incredible that
people say that is not science. So far, all attempts to do so have
hinged on narrowing the meaning of science to natural science, or just a
method. But as someone has pointed out science is more than a method and
mathematics entities are inherent to the natural world, e.g., Fibonaccy
numbers, natural logarithms, the Golden Ratio, et cetera. Mathematical
research can follow the scientific method just as any other discipline.
Also useful but artificial is the separation of "pure" and "applied." Are
the results of medicine not medicine. I guess Anatomy and Physiology are
Pure Medicine, and Surgery and Therapy are applied medicine. Is surgery
not medicine? Likewise, although Pure Mathematics is the great example of
Deductive Logic, the moment you apply it is always combined with other
premises which have been obtained by induction. Why must it cease to be
mathematics at this point?
Other attempts are using the fallacious tecnique of "counting the misses and
not the hits," and contriving some example in which some mathematical
exercise is far removed from any application and is only a mental game.
This ignores that there is still a huge body of mathematical knowledge which
differs from such games by the fact that it has applications.
me...@cars3.uchicago.edu
>
>I must be making headway, because now you need to add the word "natural" to
>keep mathematics out of "science." Ever heard of the natural numbers? Or
>the natural logarithms?
"Natural numbers" are a name, which we made. Same about "natural
logarithms". We could have called them Gewurtzaminer logarithms
without making any difference. It is not the naming that matters.
Natural numbers are whatever we define them to be. We choose a systme
of axioms, then we study what follows from them. We can choose a
different system of axioms and we get entities with different
properties and relationships. And that's what mathematics is, the
study of properties and entities which we create, by the defining the
governing axioms.
Science is different. It deals with things which exist independent of
us. We're not free to define their relationships any way we wish. We
tend to believe that they too can be described using axiomatic systems
but first, it is only a belief (not provable) and second, even if
true, we do not know what the governing axioms are (and we'll never
know for sure). All we can do is to come with educated guesses and
check whether their consequences seem to much what we observe.
None of this, of course, negates the value of mathematics as tool for
science. But, don't confuse the tool with the activity. When you use
mathematics in science, you are not doing mathematics, just using it.
When you do mathematics, you're not doing science.
George Greene
: Because it is not. Science depends on the process of "induction":
: generalization from specific to the general case. But most importantly,
: science depends on *experimentation* . Clearly mathematics lacks that.
That is patently ridiculous.
Every time you make a conjecture that you can't YET
prove, you are beginning an experiment.
You can either add it as an axiom and see if any
proofs of contradictions result, or you can go
actively in pursuit of counterexamples. BOTH of these
searches are experiments because there are well-defined
rules for conducting them and measuring/recording the outcome.
And you DON'T usually know until AFTER you perform the calculation/
derivation/EXPERIMENT what the OUTCOME is going to be.
In physical science, an axiom of your proposed theory can
be contradicted by predicting an experimental outcome that
(to within error of execution) seems not to be occurring.
In math, an axiom of your proposed theory can be contradicted
by leading to a contradiction. But finishing the proof that it is
in fact contradictory and designing&conducting the experiment that
provokes reality to refute your theory are the SAME kind of experimentation
and research. In both cases you are trying to find some axioms that
correctly describe an intended model. In the math case, the model is
abstract; in the physics case, it is concrete. That is the ONLY difference.
It is NOT more important than ALL the similarities.
bo...@my-deja.com
me...@cars3.uchicago.edu wrote:
> Natural numbers are whatever we define them to be. We choose a systme
> of axioms, then we study what follows from them. We can choose a
> different system of axioms and we get entities with different
> properties and relationships. And that's what mathematics is, the
> study of properties and entities which we create, by the defining the
> governing axioms.
>
This is a myth.
Look at history. Calculus was being done for hundreds of years before
there was even an attempt at axiomization.
Some mathematics may be done by following axioms. But most (probably)
is trying to solve problems and the axioms come last to justify
the solutions.
The above is an admitted over-simplification.
Sent via Deja.com http://www.deja.com/
Share what you know. Learn what you don't.
me...@cars3.uchicago.edu
>In article <7qgija$vvu$9...@thorn.cc.usm.edu> Larry Mead <lrm...@orca.st.usm.edu> writes:
> : Because it is not. Science depends on the process of "induction":
> : generalization from specific to the general case. But most importantly,
> : science depends on *experimentation* . Clearly mathematics lacks that.
>
>That is patently ridiculous.
>
>
>Every time you make a conjecture that you can't YET
>prove, you are beginning an experiment.
No, because you're not dealing with anything external to you.
>In both cases you are trying to find some axioms that correctly describe
>an intended model.
Nope. In math you're free to decide on your axioms, as long as
they're not contradictory. In science you don't have this freedom.
tiglath
<me...@cars3.uchicago.edu> wrote in message
news:FHC4n...@midway.uchicago.edu...
> In article <7qgkhh$rqn$2...@autumn.news.rcn.net>, "tiglath" <tig...@usa.net>
writes:
> >
> >I must be making headway, because now you need to add the word "natural"
to
> >keep mathematics out of "science." Ever heard of the natural numbers?
Or
> >the natural logarithms?
>
> "Natural numbers" are a name, which we made. Same about "natural
> logarithms". We could have called them Gewurtzaminer logarithms
> without making any difference. It is not the naming that matters.
The naming is not arbitrary, it reflects that the e number is found in
natural relationships.
bo...@my-deja.com
me...@cars3.uchicago.edu wrote:
>
> Nope. In math you're free to decide on your axioms, as long as
> they're not contradictory.
>
That is not how math is done (very often).
Axioms are chosen to help explain or clarify some problem. They
come near the end of mathematical work, not at the beginning (except
in textbooks).
reference: history
me...@cars3.uchicago.edu
Everything is just as natural.
Base e is very convenient for calculus purposes as derivatives take a
simple and easy to remember form. But don't confuse convenience with
"naturalness".
tiglath
Huw Morgan <huw.m...@logical.com> wrote in message
news:936119294.8431.0...@news.demon.co.uk...
> A distinction between Mathematics and Physics (say) can be very easily
> identified - Mathematics requires a logically-deducted proof for any
theorem
> to be accepted. Physics requires simply no counter-example yet to have
been
> found.
Some mathematical constructs have no proof, yet are still useful. In
mathematics a single-counter example nullifies a general principle just like
in physics.
>
> So differences as to Mathematics and a physical science are easy to
> recognise. However, the definitions of science are retro-fitted concepts -
> someone observed a thing (e.g. Physics) and tried to define it. Someone
else
> probably did the same thing in a different way. No doubt it would be easy
> for one of the clever people on this list to define science in such a way
> that Music could be regarded as such.
>
> I think (while this might be a heretic thing to say) that it really
doesn't
> matter whether you regard Mathematics as science, an art, or a language.
This is certainly true. Not opinion on what mathematics is can hurt the
multiplication table. But this is not the point. I am indeed surprised by
the views expressed here. None of the arguments for mathematics not being a
science are persuasive. The Scientific Method can be applied to
mathematical research; there are mathematical experiments; mathematics
reflect many of the underlying relationships in natural phenomena, thus
being tied to Nature, and Faculties of Mathematics are usualy under the
Science offerings at universities, another clue. A scientist with no maths
would cease to be a scientist. What else is it needed to make mathematics
a scientific pursuit?
We
> all know what we mean by Mathematics and Physics, so what else is
required?
>
> Huw
>
> huw.m...@logical.com
>
> "Don't ask what a word means, but how it is used" (Can't remember who said
> it)
>
>
Daryl McCullough
(George Greene) writes:
>>Every time you make a conjecture that you can't YET
>>prove, you are beginning an experiment.
>
>No, because you're not dealing with anything external to you.
I can see where George is coming from. In physics, you have a
theory that has a testable consequence: Conduct an experiment
according to such and such specification, and you will see
this or that.
Mathematical conjectures can also have testable
consequences: The claim that every natural number can be written
as the sum of two primes has the following consequence:
Take any positive number N. Write down every pair of
numbers (N1,N2) such that N1 + N2 = N. At least one
pair will have the property that N1 and N2 are both
prime.
You could say that the difference between the physical consequence
and the mathematical consequence is that the physical consequence
is about physical objects, while the mathematical consequence is
about abstract objects. But that need not be the case. You can
cast the mathematical claim as a claim about pennies, or about
marks on a paper, or any number of different physical situations.
So mathematics has testable consequences for physical objects.
I think that it's hard to put your finger on what is the essential
difference between mathematics and physical science.
Daryl McCullough
CoGenTex, Inc.
Ithaca, NY
George Greene
>prove, you are beginning an experiment.
In article <FHC6z...@midway.uchicago.edu> me...@cars3.uchicago.edu writes:
: No, because you're not dealing with anything external to you.
Wrong.
The natural numbers are external to you.
Calculus is external to you.
Galois Theory is external to you.
Regular polyhedra and relevant theories relating
to them are external to you.
>In both cases you are trying to find some axioms that correctly describe
>an intended model.
: Nope. In math you're free to decide on your axioms, as long as
: they're not contradictory.
: In science you don't have this freedom.
Yes, you do.
In science, you are free to decide on your axioms EVEN if they
are contradictory.
And in Math, too.
The point being that you can't PROVE your axioms aren't contradictory,
in the most interesting cases. In the slightly less interesting
cases, your axioms are contradictory, but you don't KNOW it yet.
It takes a WHOLE LOT OF TIME, EFFORT, and EXPERIMENTATION to uncover
the contradiction.
For example, we could add to Peano's axioms an axiom asserting
the existence of an even number that was not the sum of two primes.
That's probably inconsistent, but proving it will be even harder
than it was for Michelson and Morley to prove that "there is an ether"
was inconsistent with other relevant axioms & observables.
George Greene
: Sorry to repeat myself ad nauseam here, but this ought to be part of
: the FAQ.
I am reading this in sci.logic.
Do We Even Have a FAQ?
George Greene
: It's the scientific method, but it's not natural science, due to
: the fact that nature is not being studied, but rather a particular
: logical system or language, and its implications.
"Particular" is wrong.
Math includes ALL logical systems and languages, by definition.
Science is what's particular; it only gets to be about
THIS universe.
: One just as well be
: studying painting, ballet, or German in this hypothetico-deductive
: fashion (all quite possible to do, I assure you).
Please, you're quite wrong.
Painting and ballet are arts. They don't have laws indepenent
of artists. The scientific method is FRUITFULLY applied to
arenas that are LAW-governed, that HAVE enough pattern and regularity
to make things like "axioms", LIKE general laws, relevant, meaningful,
discoverable, applicable, and true. There cannot be laws of painting.
There can be schools of painting but they don't become laws until
we know a lot more about the biophysics of aesthetics than we know today.
: This does not make any of them natural sciences.
Nobody cares.
: When we say "science" in standard American English we *mean* it as
: a shorthand for natural science.
Except when we are talking about Math.
In the case of Math we typically say that it is "the
Queen of the Sciences". Everybody who's opinion counts can see
that it is like physical sciences in some ways and different
in others; what matters is whether the particular constellation
of similiarities and differences JUSTIFIES classifying math
as the ABSTRACT (as OPPOSED to "natural") science. Nature is
not the only thing that can be studied via the scientific method.
Unless you think that transfinite ordinals or arbitrarily large
finite numbers, or pi, or e, or i, are found in nature.
Given that Maxwell's equations describe nature as we currently
think we know it, and these are indispensable to those,
we might be tempted to say so, but in either case, science and
math wind up closer together, not farther apart.
: One can see the "scientific method" used in, say, political
: science--
No, really, one can't. You cannot repeat controlled experiments.
You can only statistically generalize after the fact about situations
that were similar, but, unfortunately, never quite similar enough.
: but "science", as we use the word as shorthand for waht
: was historically "natural philosophy," is more than
: the method. It's also the target of study, and the body of results.
That's equally true of "math", which is also more the body of results
than the method. Unless you belong to the "applied cryptography is not
math" school.
me...@cars3.uchicago.edu
>In article <FHC4n...@midway.uchicago.edu>,
> me...@cars3.uchicago.edu wrote:
>
>> Natural numbers are whatever we define them to be. We choose a systme
>> of axioms, then we study what follows from them. We can choose a
>> different system of axioms and we get entities with different
>> properties and relationships. And that's what mathematics is, the
>> study of properties and entities which we create, by the defining the
>> governing axioms.
>>
>
>This is a myth.
>Look at history. Calculus was being done for hundreds of years before
>there was even an attempt at axiomization.
>
mathematics with its applications.
Kyle R. Hofmann
> In article <7qfdnj$r40$1...@news-01.meganews.com>
> "D.E.J.A." <em...@address.com> writes:
> >
> >Is this in the FAQ?
>
> Is there a sci.logic FAQ on the web?
>
> Neither it nor sci.math has a web version linked from the sci.physics FAQ.
There is an excellent sci.math FAQ; it's at
http://www.cs.unb.ca/~alopez-o/math-faq/math-faq.html
--
Kyle R. Hofmann <rhof...@crl.com> | "...during the years between 960 and
1000 there was great activity in the production of homilies ... [ The
Blickling Homilies ] voice the almost universal belief that the world would
end in the year 1000." -- The Concise Cambridge History of English Literature
George Greene
: Natural numbers are whatever we define them to be.
No, they are NOT!
"Natural number" is a term with a history!
It is Much OLDER than the axiomatic method!
You figure out how to count LONG before you figure
out what logic is.
: We choose a system of axioms,
: then we study what follows from them.
No, actually, we don't. We CAN, but we don't.
In REAL life, we have an INTENDED model in mind and
are looking for axioms that actually DESCRIBE the REAL
behavior of this model, JUST LIKE IN PHYSICS.
First, we can define matrices.
THEN, we can define matrix multiplication.
THEN, we can ask whether matrices under multiplication form a group.
When it turns out they don't, we go looking for some
other axioms (other than the group axioms) to describe
the kind of thing they DO form. Similarly for natural numbers.
Natural numbers were not originally defined by Peano's
axioms. We already KNEW what natural numbers were.
Peano's axioms are just a theory that approximates their
"real" behavior. In fact, THERE IS NO (first-order)
axiomatic theory of the natural numbers. But that doesn't
stop them from existing and it doesn't stop us from using
math to describe some aspects of them.
: We can choose a different system of axioms and we
: get entities with different properties and relationships.
Only when we're on NEW ground. Only when we're actually OPEN
to the possibility of having NEW entities. And in the case of
natural numbers, we are NOT open: we already KNOW what natural
numbers are. Just like we already knew what SETS were.
: And that's what mathematics is, the study of properties
: and entities which we create, by the defining the
: governing axioms.
Have you ever met a mathematician?
You don't create 0 and 1 by defining axioms about them.
They are already out there. Ditto for regular sets, complex
numbers, ad nauseam.
: Science is different. It deals with things which exist independent of
: us. We're not free to define their relationships any way we wish.
That's absolutel false.
Science does NOT deal with things independent of us.
It deals with OUR measurements of experiments designed according
to OUR rules and observed according to OUR biology.
Moreover, scientific theories are OUR THEORIES about the
real world, and that's what we deal with when we deal with
science. We ARE fre to re-define the relationships of terms
in these theories and we in fact DO DO that EVERY time the
"current" theory gets falsified and we have to adopt a new one.
We did it after the Michelson-Morley experiment. We did it after
the photoelectric effect was discovered. We did it after quarks
were discovered in 1968. Scientific theories are JUST as fanciful
and solpsistic as mathematical ones, and even LESS grounded in certainty,
because the requirement that we discard them in the face of experimental
outcomes means they are NEVER confirmed. At least we don't have to throw
out "old" math for anythign less than a logical contradiction, in order
to accomodate "new" math.
: We tend to believe that they too can be described using axiomatic systems
: but first, it is only a belief (not provable) and second, even if
: true, we do not know what the governing axioms are (and we'll never
: know for sure).
You'll never know that for the natural numbers, EITHER.
: All we can do is to come with educated guesses and
: check whether their consequences seem to much what we observe.
And that is all you can do in arithmetic, too.
: None of this, of course, negates the value of mathematics as tool for
: science. But, don't confuse the tool with the activity. When you use
: mathematics in science, you are not doing mathematics, just using it.
: When you do mathematics, you're not doing science.
Define "doing math". When you are calculating, you pretty much
are conducting an experiment. You might get it wrong. When you
are looking for axiomatic regularities governing some class of
mathematically-defined objects, that is a lot like looking for
regularities among experimentally-defined objects in physics.
George Greene
> firms. Have this purely mathematical products have been through a process
> any less scientifc than producing a drug, or a new plastic?
In article <7qgio1$vvu$1...@thorn.cc.usm.edu> Larry Mead <lrm...@orca.st.usm.edu> writes:
: Good mathematics, yes. Science, no. The experiments must be
: with real-world objects, not abstract numbers.
Why?
Support that!
You have made the opposite argument from the one you intended.
The point is that the fact that the modeled objects are abstract
in one case and concrete in the other is the ONLY difference.
EVERYthing else is THE SAME.
That is why math IS a science.
bo...@my-deja.com
me...@cars3.uchicago.edu wrote:
> >This is a myth.
> >Look at history. Calculus was being done for hundreds of years before
> >there was even an attempt at axiomization.
> >
> And same is true about geometry. This doesn't matter. Do not confuse
> mathematics with its applications.
It "doesn't matter" if you define mathematics as "chasing the logical
consequences of arbitrarily chosen axioms" rather than as what people
actually do when they do mathematics.
Do not reduce mathematics to formalism.
Arthur Fischer
> The Professor claims that:
>
> No, mathematics is not science. It's used a lot in science, but the
> processes are different. Basically, mathematics is applied logic. It lacks
> anything comparable to experiment. It also lacks the step of abduction,
> which is part of theory-formation, an essential part of science.
I agree with all of the people who say that we need to define the terms that we
are using. So I'll give you my definitions:
SCIENCE: an intellectual endevour whose aim is to understand the processes
that control nature, meaning the physical aspect of nature (animals, atoms,
chemical reactions, etc. etc. etc.)
MATHEMATICS: an intellectual endevour whose aim is to understand the
consequences of a collection of assumed propositions (axioms).
At least this is my idea on what these two are TODAY. Mathematics may not have
started as such an endevour, but this is what I believe is the current state of
affairs. According to my definitions, I guess, mathematics is not a science
because it does not deal with the physical world as its object, as I believe
sciences do. But notice that my definitions do not take experimentation out of
mathematics. I do believe that some experiementation is necessary. Especially
as I am a student of mathematics, and in order to learn and understand certain
theorems I have found myself going back to what I feel are more concrete
ideas. The proof of the four-colour theorem was an experiement, done for
reasons that I believe are similar: an imperfect understanding of the
situation at hand.
I don't think that it is fair to bring up the state of the world 500 years ago
in order to state that math is a science, since many mathematicians then were
astrologers, who used their understanding of both nature and math in order to
give out horoscopes. If you bring the history of the subjects into play, you
may have to admit that all of science and mathematics is nothing but mysticism.
__
Arthur
bo...@my-deja.com
Arthur Fischer <afis...@ucalgary.ca> wrote:
> I don't think that it is fair to bring up the state of the world 500
years ago
> in order to state that math is a science, since many mathematicians
then were
> astrologers, who used their understanding of both nature and math in
order to
> give out horoscopes. If you bring the history of the subjects into
play, you
> may have to admit that all of science and mathematics is nothing but
mysticism.
>
The axioms for set theory are less than 100 years old. Those for the
real numbers, less than 150. Hardly back to the dark ages.
Nobody noticed until Pasch in 1882
that Euclid needed another axiom, which is now known as
Pasch's Axiom: A straight line cutting one side of a triangle
necessarily cuts one, and only one, of the other two sides, except
in the case when it passes through the opposite vertex.
So the "undoubtedly true" theorems of euclidean geometry which
had been "absolutely proved"... well...
John T. Lowry
always use for 'phyics':
Physics: a (mostly) mathematical model of (part of) nature.
John
--
John T. Lowry, PhD
Flight Physics; Box 20919; Billings MT 59104
Voice: 406-248-2606
Arthur Fischer wrote in message <7qhdam$d...@ds2.acs.ucalgary.ca>...
>I don't think that it is fair to bring up the state of the world 500 years
ago
>in order to state that math is a science, since many mathematicians then
were
>astrologers, who used their understanding of both nature and math in order
to
>give out horoscopes. If you bring the history of the subjects into play,
you
>may have to admit that all of science and mathematics is nothing but
mysticism.
>
>__
>Arthur
>
Jeremy Boden
<j...@ibms48.scri.fsu.edu> writes
...
>In article <7qfnbk$p$1...@autumn.news.rcn.net>
>"tiglath" <tig...@usa.net> writes:
>>
>>How about people who broke the Enigma. Didn't they do thousands of
>>experiments till the found the right set of rotors to break the cipher? Is
>>this not science?
>
> The Polish mathematicians who *broke* the Enigma did so based
> on mathematical methods, then used the result to design a machine
> that could be used to determine the daily settings. From that
> point on it ceased to be mathematics.
>
breakers (some of whom were logicians or mathematicians), with some
later input by USA. A recent programme on TV about Enigma revealed that
one of the UK government's criteria for code-cracking was the ability to
do the Times crossword speedily!
Actually different parts of the German military used different versions
of Enigma. So the code had to be cracked a number of times.
--
Jeremy Boden
bo...@my-deja.com
terms that we
> are using. So I'll give you my definitions:
>
> SCIENCE: an intellectual endevour whose aim is to understand the
processes
> that control nature, meaning the physical aspect of nature (animals,
atoms,
> chemical reactions, etc. etc. etc.)
>
> MATHEMATICS: an intellectual endevour whose aim is to understand the
> consequences of a collection of assumed propositions (axioms).
>
I appreciate your trying to bring some rationality to the thread.
However, I must disagree with your characterization of MATHEMATICS.
That's just not how it happens.
Axioms are discovered/invented in order to clarify or formalize some
already known and worked-with mathematical area or problem.
The axioms for convergent series were invented/discovered after a
couple hundred years of playing around with things like
1 - 1 + 1 - 1... and 1 + x + x^2 + x^3 +...
me...@cars3.uchicago.edu
>
>I don't think that it is fair to bring up the state of the world 500 years ago
>in order to state that math is a science, since many mathematicians then were
>astrologers, who used their understanding of both nature and math in order to
>give out horoscopes.
Indeed. And even later the borderline between Math and science was
rather vague, for quite a while.. Some of the biggest names in
mathematics from the period between Newton and Maxwell are the biggest
neames in physics, as well (Euler, Lagrange, Laplace, Gauss, not to
mention Newton himself).
As far as I can see, a distinction between study of logical structures
and study of the physical world only became clear in the middle of the
last century, with the advent of non-Euclidean geometries. This
development brough home the point that logical consistancy and
consistancy with observable reality are two different things. You
should be more familiar with the history of mathematics than I'm but
I'll venture the guess that you'll hardly ever (if at all) find the
term "pure mathematician" being used before Riemann, but you'll
encounter it quite often after Riemann.
me...@cars3.uchicago.edu
>In article <FHCGF...@midway.uchicago.edu>,
> me...@cars3.uchicago.edu wrote:
>
>> >This is a myth.
>> >Look at history. Calculus was being done for hundreds of years before
>> >there was even an attempt at axiomization.
>> >
>> And same is true about geometry. This doesn't matter. Do not confuse
>> mathematics with its applications.
>
>It "doesn't matter" if you define mathematics as "chasing the logical
>consequences of arbitrarily chosen axioms" rather than as what people
>actually do when they do mathematics.
>
mathematics" or applying it?
Nicolas Bray
On 31 Aug 1999, George Greene wrote:
> In the math case, the model is abstract; in the physics case, it is
> concrete. That is the ONLY difference.
No, the crucial difference that you are missing is that in math we know
the axioms whereas in physics not only do we not know them, we cannot know
them. We can gather evidence which indicates our axioms are correct but we
can never be sure whether this is the case or whether our axioms just
provide an approximation of the actual laws that is within our current
abilities to measure.
bo...@my-deja.com
me...@cars3.uchicago.edu wrote:
> Do you think that when you solve a set of equations you are "doing
> mathematics" or applying it?
>
doing
MPW
Kind of interesting, that the more opinion and less knowledge an answer
requires, the more people answer to it. :)
Who cares whether maths is a science or not? All that matters is that it
has its applications for which we use it.
The question is just academic - a waste of brain power. :)
--
Regards, Matt Webster.
____________________________
"The important thing is not to stop
questioning." - Albert Einstein
____________________________
--
tiglath wrote in message <7qfb0s$9ne$1...@autumn.news.rcn.net>...
:
:
:I have had this surreal conversation with this guy I met first in a
:religious newsgroup, I don't have his permission to print what he said to
:me, in private email, so I will paraphrase what he said by changing a
couple
:of inessential adverbs and adjectives with synonyms. The meaning remains
:unchanged.
:
:The author of the paragraph below, The Professor, teaches physics in the
:Wake Forest University in North Carolina. I pity his students.
:
:
:The Professor claims that:
:
:No, mathematics is not science. It's used a lot in science, but the
:processes are different. Basically, mathematics is applied logic. It lacks
:anything comparable to experiment. It also lacks the step of abduction,
:which is part of theory-formation, an essential part of science.
:
:
:He defines abduction as "borrowing ideas from a known situation that is
:similar or analogous to one the you're wondering about, and use them as a
:source of hypotheses for the present situation."
:
:Whereas, I thought that abduction is a process that proceeds by inferences
:where the major premise is certain and the minor one only probable.
:
:I
:How can a Physics teacher write that Mathematics is not science? Is this
:another Professor On Acid in the Silly Season at American Universities?
:Comments welcome.
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
me...@cars3.uchicago.edu
>In article <FHCn2...@midway.uchicago.edu>,
> me...@cars3.uchicago.edu wrote:
>
>> Do you think that when you solve a set of equations you are "doing
>> mathematics" or applying it?
>>
>doing
>
Steven B. Harris
writes:
>
>Virgil <vm...@frii.com> wrote in message
>news:vmhjr-30089...@ftc-0409.dialup.frii.com...
>}
>} Mathematics is a large part of the language of science, but
mathematics is
>} no more science than grammar is literature.
>
>In article <7qfnbk$p$1...@autumn.news.rcn.net>
>"tiglath" <tig...@usa.net> writes:
>>
>>How about people who broke the Enigma. Didn't they do thousands of
>>experiments till the found the right set of rotors to break the
cipher? Is
>>this not science?
>
> The Polish mathematicians who *broke* the Enigma did so based
> on mathematical methods, then used the result to design a machine
> that could be used to determine the daily settings. From that
> point on it ceased to be mathematics.
It wasn't even all math to begin with, but relied on some insight
into human nature. To wit, that the Germans would be lazy, and not
change the code wheels randomly, but leave some alone, though we didn't
know which. That they did in fact do so, made the task within
computational power of the time. Had the Enigma worked (or rather,
been used in the field) as designed, we'd never have broken it before
war's end.
Steven B. Harris
writes:
>>>
>>Abduction, not induction. Look it up.
>
> It is not in the biggest dictionary I have in the office, so
> either I need the OED or a more recent one. The only role of
> 'abduction' in math that I know of is the one described by
> Tom Lehrer: "I asked a friend in Minsk, who had a friend ...."
> and "Let no one's work evade your eyes ..."
Plagerize, plagerize! Only be sure, always to call it... research.
>
> How would you distinquish abduction from induction? I would say
> the latter would be an example where some set of cases is used
> to guess at the major premise.
Abduction is where there is a minor (inductive) premise which is
hidden so well that it isn't noticed, so it can look a lot like
deduction. You've seen a lot of people on there "prove" logically that
Einstein's theories are paradoxical and wrong, because they require
people passing each other to see that each of the other's clocks run
slow. The minor premise, not examined, is that this really is a
paradox. A paradox, however, is when something is physically
impossible-- it both happens and doesn't happen. If two guys passing
each other see the other's clocks run slow, there's no paradox there--
it's just weird. Not so with events. If two guys pass an atom bomb
and one guy sees it go off and incinerate him, and the guy going by
just inches away sees nothing, that's a paradox. SR does not permit
this kind.
G. A. Edgar
> :The Professor claims that:
> :
> :No, mathematics is not science. It's used a lot in science, but the
Here's a book listing...
Mathematics : The Science of Patterns : The Search for Order in
Life, Mind and the Universe (Scientific
American Paperback Library) -- Keith J. Devlin
--
Gerald A. Edgar ed...@math.ohio-state.edu
Department of Mathematics telephone: 614-292-0395 (Office)
The Ohio State University 614-292-4975 (Math. Dept.)
Columbus, OH 43210 614-292-1479 (Dept. Fax)
Lapidary
"tiglath" <tig...@usa.net> wrote:
>
> Virgil <vm...@frii.com> wrote in message
> news:vmhjr-30089...@ftc-0409.dialup.frii.com...
> > Mathematics is a large part of the language of science, but
mathematics is
> > no more science than grammar is literature.
>
> experiments till the found the right set of rotors to break the
cipher? Is
> this not science?
>
factor
> bigger and bigger numbers into large prime factors, are this peoples
> "grammarians."
>
These experiments aren't telling us anything about the
world, just about maths. Science is about the world,
maths is about maths. Therefore maths is not science.
> What about the cryptographic products produced by the NSA and
commercial
> firms. Have this purely mathematical products have been through a
> process
> any less scientifc than producing a drug, or a new plastic?
>
>
Product testing is business/engineering, not science.
--
Regards, Peter D Jones .
"I had a million pounds in the bank. I spent most
of it on booze, women and fast cars. The rest I
wasted" -- George Best.
George Greene
: SCIENCE: an intellectual endevour whose aim is to understand the processes
: that control nature, meaning the physical aspect of nature (animals, atoms,
: chemical reactions, etc. etc. etc.)
:
: MATHEMATICS: an intellectual endevour whose aim is to understand the
: consequences of a collection of assumed propositions (axioms).
To understand why this is a similarity and not a difference,
you need do only ONE thing: define UNDERSTAND.
What it actually MEANS to "UNDERSTAND the processes that
control [the physical aspect of] nature" IS
to have a collection of assumed propositions (axioms) that
your natural expirements don't violate. Like Maxwell's laws.
General relativity is called a "theory" BECAUSE it has axioms.
Larry Mead
: In article <7qgija$vvu$9...@thorn.cc.usm.edu> Larry Mead <lrm...@orca.st.usm.edu> writes:
: : Because it is not. Science depends on the process of "induction":
: : generalization from specific to the general case. But most importantly,
: : science depends on *experimentation* . Clearly mathematics lacks that.
: That is patently ridiculous.
It is patently true. When a physicist, for example, verifies a Law like
Newtons second law he uses real objects and takes data. When doing
mathematics, the mathematician may "experiement" with abstract concepts
while attempting to formulate a theorem, then uses deductive reasoning
(even now computer aided) to prove the theorem. Nowhere in this process
does (s)he do what the physicist does.
: Every time you make a conjecture that you can't YET
: prove, you are beginning an experiment.
No. You are imagining, not experimenting. Even the very objects on which
you ponder have no measurement errors; they may not even be numbers.
: You can either add it as an axiom and see if any
: proofs of contradictions result, or you can go
: actively in pursuit of counterexamples. BOTH of these
: searches are experiments because there are well-defined
: rules for conducting them and measuring/recording the outcome.
: And you DON'T usually know until AFTER you perform the calculation/
: derivation/EXPERIMENT what the OUTCOME is going to be.
As I said, this is not what is meant by experimentation in science.
: In physical science, an axiom of your proposed theory can
: be contradicted by predicting an experimental outcome that
: (to within error of execution) seems not to be occurring.
Yes, indeed, that's why we call it doing science not math. That is the
only distinction.
--
Lawrence R. Mead Ph.D. (Lawren...@usm.edu)
Eschew Obfuscation! Espouse Elucidation!
www-dept.usm.edu/~physics/mead.html
George Greene
: : generalization from specific to the general case. But most importantly,
: : science depends on *experimentation* . Clearly mathematics lacks that.
> That is patently ridiculous.
: It is patently true. When a physicist, for example, verifies a Law like
: Newtons second law
Bzzt.
Newton's second law is false. It has never been verified.
Nothing has ever happened "when x verifies a law like Newton's second law".
Scientific laws DON'T get verified. They ONLY get falsified.
: he uses real objects and takes data. When doing
: mathematics, the mathematician may "experiement" with abstract concepts
: while attempting to formulate a theorem, then uses deductive reasoning
: (even now computer aided) to prove the theorem.
True.
: Nowhere in this process
: does (s)he do what the physicist does.
The above IS doing what the physicist does.
The fact that one investigated arena is abstract and the
other is concrete is IRRELEVANT. The point is that both
arenas behave in ways that your will doesn't controlled and
that have regularities and patterns that you can learn about
by experiment, upon which you can use induction.
: Every time you make a conjecture that you can't YET
: prove, you are beginning an experiment.
: No. You are imagining, not experimenting.
No, you are calculating, which is experimenting whenever
you have a prior inexact characterization of the answer,
which the exact answer could (subjectively) possibly fail to match.
: Even the very objects on which you ponder have no measurement
: errors; they may not even be numbers.
That is not really relevant.
Measurement error is problematic in physics but the
real progress comes precisely when you can prove that
your measurement error has been made small enough not to
matter after all. Getting to that point is
more engineering than physics.
And there are mathematical analogues. Mathematical
conjectures and intuitions can also be initially imprecise.
: You can either add it as an axiom and see if any
: proofs of contradictions result, or you can go
: actively in pursuit of counterexamples. BOTH of these
: searches are experiments because there are well-defined
: rules for conducting them and measuring/recording the outcome.
: And you DON'T usually know until AFTER you perform the calculation/
: derivation/EXPERIMENT what the OUTCOME is going to be.
> As I said, this is not what is meant by experimentation in science.
This is analogous enough to what goes on in science that it IS
experimentation, PERIOD. To say it is not 'in science' is begging
the question. Precisely BECAUSE this goes on it, it IS science,
no matter what Else (including Math) "it" may also be.
: In physical science, an axiom of your proposed theory can
: be contradicted by predicting an experimental outcome that
: (to within error of execution) seems not to be occurring.
> Yes, indeed, that's why we call it doing science not math. That is the
> only distinction.
Any time you're arguing that there is an "only" distinction, you're
arguing my side (the similarity side); if these things were as
qualitatively different as you are trying to insist, so much so
that math is not a science, then there would be a lot more
differences than one "only".
Huw Morgan
some time trying to define it, but I'm not up on his conclusions.
On your other point, a mathematical construct might be useful in some
circumstances, but not in defining something that is part of mathematics -
you can't have a Theorem that is based on a conjecture, but someone might be
able to use that conjecture to help them think in the right directions. Or
some applied scientist might use it to design a new spacecraft or something.
A Physicist might well accept something that has no counter-example, with no
rigorous proof, just on the basis of a few observations.
Huw
huw.m...@logical.com
"Don't ask what a word means, but how it is used" (Can't remember who said
it)
oh yes, it was Wittgenstein
Pertti Lounesto
> you can't have a Theorem that is based on a conjecture, but someone might be
> able to use that conjecture to help them think in the right directions. Or
> some applied scientist might use it to design a new spacecraft or something.
> A physicist might well accept something that has no counter-example, with no
> rigorous proof, just on the basis of a few observations.
Many mathematical theorems, which have rigorous proofs, also
have counterexamples, which satisfy all the assumptions of the
the theorem without the conclusion being valid. See my www-
page, http://www.hit.fi/~lounesto/counterexamples.htm, where
I falsify theorems of renowned mathematicians. Mathematicians
are not much better than physicists. Mathematicians do accept
theorems, even though the theorems have counterexamples,
but only encrust their false theorems with so-called proofs.
Huw Morgan
Pertti Lounesto wrote:
>Many mathematical theorems, which have rigorous proofs, also
>have counterexamples, which satisfy all the assumptions of the
>the theorem without the conclusion being valid. See my www-
>page, http://www.hit.fi/~lounesto/counterexamples.htm, where
>I falsify theorems of renowned mathematicians. Mathematicians
>are not much better than physicists. Mathematicians do accept
>theorems, even though the theorems have counterexamples,
>but only encrust their false theorems with so-called proofs.
You may have provided counter-examples of these theorems - Clifford isn't
one of my areas of knowledge, I couldn't say. But if your work is valid,
then those so-called theorems are not valid and not part of mathematics. So
you haven't contradicted anything I've said.
Huw
Pertti Lounesto
You cannot decide whether my counterexamples are valid
or not, and you cannot decide whether the theorems were
falsified or not. Yet, you speak about a procedure to decide
whether something is a part of mathematics or not. Is your
procedure practical or not?
Huw Morgan
Pertti Lounesto wrote:
>You cannot decide whether my counterexamples are valid
>or not, and you cannot decide whether the theorems were
>falsified or not. Yet, you speak about a procedure to decide
>whether something is a part of mathematics or not. Is your
>procedure practical or not?
>
I don't think it is a difficult concept to understand. What I am saying is
that a rigorously-proven theorem is the only aspect of mathematics that is
acceptable beyond the original assumptions of the mathematical system.
You claim to have provided counter-examples to some hitherto accepted
theorems. If these are valid (and I do not have the knowledge of that area
of mathematics to say) then the proofs which were originally provided for
these theorems are flawed and not rigorous in some way.
I don't see what you're arguing about - or are you just trying to get more
publicity for your site ;-)
Huw
oho...@my-deja.com
Pertti Lounesto <Pertti....@hit.fi> wrote:
> Huw Morgan wrote:
>
> > Pertti Lounesto wrote:
> > >Many mathematical theorems, which have rigorous proofs, also
> > >have counterexamples, which satisfy all the assumptions of the
> > >the theorem without the conclusion being valid. See my www-
> > >page, http://www.hit.fi/~lounesto/counterexamples.htm, where
> > >I falsify theorems of renowned mathematicians. Mathematicians
> > >are not much better than physicists. Mathematicians do accept
> > >theorems, even though the theorems have counterexamples,
> > >but only encrust their false theorems with so-called proofs.
> >
> > You may have provided counter-examples of these theorems - Clifford
isn't
> > one of my areas of knowledge, I couldn't say. But if your work is
valid,
> > then those so-called theorems are not valid and not part of
mathematics. So
> > you haven't contradicted anything I've said.
>
> or not, and you cannot decide whether the theorems were
> falsified or not. Yet, you speak about a procedure to decide
> whether something is a part of mathematics or not. Is your
> procedure practical or not?
Pertti, How can it be that you have a counterexample which
is not a valid counterexample...perhaps it is possibly a
counterexample or that you believe it might be a counterexample..
I don't understand what you are saying. Surely counterexamples
of a particular theorem shows that the theorem is invalid i.e.
it shows that the expression is not a theorem. Is your procedure
logical..mathematical or not.
Owen
Pertti Lounesto
> Pertti, How can it be that you have a counterexample which
> is not a valid counterexample.
My counterexamples remain valid until being invalidated.
> Surely counterexamples
> of a particular theorem shows that the theorem is invalid i.e.
> it shows that the expression is not a theorem.
Theorems hold until being falsified by counterexamples.
Huw Morgan wrote:
> I don't see what you're arguing about - or are you just trying
> to get more publicity for your site ;-)
My agenda is this: Mathematical truth varies with time,
depends on the actions of the mathematical community,
is a result of interection of man with his environemnt,
both individually and genetically.
Greg Neill
: Huw Morgan wrote:
: > I don't see what you're arguing about - or are you just trying
: > to get more publicity for your site ;-)
: My agenda is this: Mathematical truth varies with time,
: depends on the actions of the mathematical community,
: is a result of interection of man with his environemnt,
: both individually and genetically.
Three is always greater than two, even for large values of two.
--
----------------------------------------------------------------------------
Greg Neill,
HNSX Supercomputers Inc.
http://www.capecod.net/~cfoster1/orrery.htm
----------------------------------------------------------------------------
etherman
tiglath <tig...@usa.net> wrote in message
news:7qfb0s$9ne$1...@autumn.news.rcn.net...
>
>
> I have had this surreal conversation with this guy I met first in a
> religious newsgroup, I don't have his permission to print what he said to
> me, in private email, so I will paraphrase what he said by changing a
couple
> of inessential adverbs and adjectives with synonyms. The meaning remains
> unchanged.
>
> The author of the paragraph below, The Professor, teaches physics in the
> Wake Forest University in North Carolina. I pity his students.
>
>
>
> No, mathematics is not science. It's used a lot in science, but the
> anything comparable to experiment.
That's correct. The methods of science and math are just different.
> It also lacks the step of abduction,
> which is part of theory-formation, an essential part of science.
Does he mean induction?
> He defines abduction as "borrowing ideas from a known situation that is
> similar or analogous to one the you're wondering about, and use them as a
> source of hypotheses for the present situation."
There's a couple things similar in mathematics. Sometimes you can abstract
ideas(e.g. real numbers to complex numbers). Other times you can slightly
alter
an axiom or two (to go from real numbers to p-adics).
> Whereas, I thought that abduction is a process that proceeds by inferences
> where the major premise is certain and the minor one only probable.
Sounds like approximation to me.
> I
> How can a Physics teacher write that Mathematics is not science? Is this
> another Professor On Acid in the Silly Season at American Universities?
> Comments welcome.
Boils down to how you define science I guess. In my mind there are aspects
to each which are analogous. Science has hypotheses and theories. Math
has conjectures and theorems. Science has experimentation. Math has
proof.
--
Etherman
Jeff Bolz
<37CEE802...@hit.fi>...
> oho...@my-deja.com wrote:
>
> > Pertti, How can it be that you have a counterexample which
> > is not a valid counterexample.
>
> My counterexamples remain valid until being invalidated.
>
Sorry, math doesn't work this way. By this logic, JSH has had a proof of
FLT hundreds, maybe thousand of times! Counterexamples (more specifically,
showing somebody else is wrong) must be reviewed before they are accepted
by the mathematical community and considered true, just like theorems.
> > Surely counterexamples
> > of a particular theorem shows that the theorem is invalid i.e.
> > it shows that the expression is not a theorem.
>
> Theorems hold until being falsified by counterexamples.
>
If it can be falsified by a counterexample, it never was a theorem but was
only mistakenly considered one.
>
> My agenda is this: Mathematical truth varies with time,
You don't really believe this, do you?
> depends on the actions of the mathematical community,
> is a result of interection of man with his environemnt,
> both individually and genetically.
>
>
--
Jeff Bolz
ehb...@sprynet.com
Jeremy Boden
<Pertti....@hit.fi> writes
...
>Many mathematical theorems, which have rigorous proofs, also
>have counterexamples, which satisfy all the assumptions of the
>the theorem without the conclusion being valid. See my www-
>page, http://www.hit.fi/~lounesto/counterexamples.htm, where
>I falsify theorems of renowned mathematicians. Mathematicians
>are not much better than physicists. Mathematicians do accept
>theorems, even though the theorems have counterexamples,
>but only encrust their false theorems with so-called proofs.
>
proof and also a counterexample?
Either the proof is not really a proof, or the counterexample is not
really a counterexample.
--
Jeremy Boden
oho...@my-deja.com
Pertti Lounesto <Pertti....@hit.fi> wrote:
> oho...@my-deja.com wrote:
>
> > Pertti, How can it be that you have a counterexample which
> > is not a valid counterexample.
>
> My counterexamples remain valid until being invalidated.
>
> > of a particular theorem shows that the theorem is invalid i.e.
> > it shows that the expression is not a theorem.
>
> Theorems hold until being falsified by counterexamples.
>
>
> > I don't see what you're arguing about - or are you just trying
> > to get more publicity for your site ;-)
>
> is a result of interection of man with his environemnt,
> both individually and genetically.
Could you demonstrate any mathematical truth which varies
with time, or is dependent on any environment? Do you take
mathematics to be subjective? You seem to be describing
empirical truth not logical/mathematical truth.
Pertti Lounesto
> Pertti Lounesto <Pertti....@hit.fi> wrote:
>
> > My counterexamples remain valid until being invalidated.
>
> FLT hundreds, maybe thousand of times! Counterexamples (more specifically,
> showing somebody else is wrong) must be reviewed before they are accepted
> by the mathematical community and considered true, just like theorems.
My www-page http://www.hit.fi/~lounesto/counterexamples.htm
presents 20 counterexamples to statements published as
theorems by renowned mathematicians. A larger collection
of 40 counterexamples can be found in my article P. Lounesto:
"Counterexamples in Clifford algebras", in Advances in Applied
Clifford Algebras 6 (1996), 69-104, which is a refereed journal,
see http://www.clifford.org/~clf-alg/journals/jadvclfa.html.
The editorial board of this journal has 10 members, out of
which 7 were listed as mistake makers in my article.
I assume that my counterexamples have been "accepted by the
mathematical community", and remain true until being falsified.
> > Theorems hold until being falsified by counterexamples.
>
> If it can be falsified by a counterexample, it never was a
> theorem but was only mistakenly considered one.
Can you separate the statements, published as theorems in 1998,
to theorems and to false statements. If you cannot, your
concept of "theorem" is useless. I did make such a separation
for the past two decades in my own speciality, Clifford algebras,
and found that almost everybody in the field had made mistakes.
When I have exposed the "mathematical community" to this fact,
right here in sci.math, the usual reactions have been the
following:
0. One cannot falsify theorems. But those say this, cannot give
a list of correct theorems published in mathematical literature.
1. My counterexamples are not valid. But those who take this
position, fail to single out a single invalid counterexample.
2. The errors I detected are not significant. But those who
take this position, are not specialists in Clifford algebras.
3. The ultimate putdown:
a. Clifford algebraists are not good mathematicians,
because they make so many mistakes.
b. The purpose of mathematics is to prove theorems,
not to falsify theorems (= sweep out mistakes).
Those who have taken positions 3a or 3b, I have challenged
with http://www.hit.fi/~lounesto/Robin.Chapman, that is,
making sense of a proof they challenged me to present.
Chapman, who is one of the best mathematicians in this
forum, has failed to make sense of my proof during the
time of the challenge, the year 1999. It is already past
noon of 1999, and the "mathematical community" of sci.math
has abundantly demonstrated its incomptence in mathematics.
Torkel Franzen
>Can you separate the statements, published as theorems in 1998,
>to theorems and to false statements. If you cannot, your
>concept of "theorem" is useless.
Why is that? It would indeed be pretty useless if the results
presented in refereed journals were as often as not later found
to be incorrect. But in fact the process seems pretty reliable.
Pertti Lounesto
Where do you put the limit of reliability? If in a research topic
10% of new theorems are found to be false during three years
after publication, are the new theorems reliable? What about
20% or 30% after say 5 years? This was the situation when
I begin to sweep math mistakes in my field of speciality, see
http://www.hit.fi/~lounesto/counterexamples.htm.
There are two ways to explain away the generality of errors:
1. Chapman: Clifford algebraists are poor mathematicians,
because they make so many mistakes, see my www-page
http://www.hit.fi/~lounesto/Robin.Chapman.
2. Lounesto: Errors are as common in all parts of mathematics,
only researchers in other topics are less vigilant to detect
the errors, less honest to admit the frequency of errors, and
more greedy in securing their positions by not threatening
the authorities in their own fields.
I became a mathematician, because, as a young man, I found
mathematicians more honests than other people, more truth
loving than others.
Torkel Franzen
>Where do you put the limit of reliability?
I don't put any limits. In fact I don't know what sort of
investigations would be needed to obtain any statistically significant
results. Apart from questions concerning how to select results to
examine, how to choose the people to examine them, and how to conduct
the examination, there are questions concerning what to count as an
error, and how to grade errors. But no doubt there are quite a few
false statements presented as theorems among the thousands of
theorems published every year, and no doubt there are even more
proofs that aren't really conclusive or contain downright howlers.
But why should that worry us?
Pertti Lounesto
> Could you demonstrate any mathematical truth which varies
> with time, or is dependent on any environment?
Your question is ambiguous as long as mathematical truth
is undefined. Here a definition of mathematical truth:
A mathematical statement is a true, if all the specialists
exploring the same domain agree that its proof is correct.
This definition has drawbacks: Who decides who are
the specialists? Who decides when agreement is reached?
According to this definition, almost all theorems, published
about Clifford algebras before 1985, were true. Then I
begun my error-hunt, with CLICAL, a computer program,
see http://www.hit.fi/~lounesto/CLICAL.htm, designed for
research in Clifford algebras, and checking conjectures.
I announced results of my error-hunt to the community,
part of which accepted my counterexamples and part not.
Usually the mistake makers were among the last to admit
an error. Thus, according to my definition of math truth,
the truth value of the theorems dropped from 1, slowly
and gradually, toward 0.
> Do you take mathematics to be subjective?
No. Mathematics consists of concepts, not individually
held concepts, but socially held concepts. Part of this
concept-holding is common for small groups of specialists
and part for all mankind; the part which has proved
useful for man in his struggle for existence.
> You seem to be describing
> empirical truth not logical/mathematical truth.
No. Mathematical truth is part of collective human
conciousness. Some parts of mathematical truth
are being settled, every day, by the math community.
Pertti Lounesto
> . But no doubt there are quite a few
> false statements presented as theorems among the thousands of
> theorems published every year, and no doubt there are even more
> proofs that aren't really conclusive or contain downright howlers.
> But why should that worry us?
Your question is rhetoric, but I will anwer it. A worry about
small details, to be filled or rectified, often leads a curious
explorer to new territories, new continests of knowledge,
to be conquered.
wetboy
<snip>
: The Professor claims that:
: No, mathematics is not science. It's used a lot in science, but the
: processes are different. Basically, mathematics is applied logic. It lacks
: anything comparable to experiment. It also lacks the step of abduction,
: which is part of theory-formation, an essential part of science.
: He defines abduction as "borrowing ideas from a known situation that is
: similar or analogous to one the you're wondering about, and use them as a
: source of hypotheses for the present situation."
: Whereas, I thought that abduction is a process that proceeds by inferences
: where the major premise is certain and the minor one only probable.
: How can a Physics teacher write that Mathematics is not science?
In order do decide this (for myself), I'd have to look up
the exact definitions of "science" and "mathematics" in the OED
and reflect on it for a while to try to determine whether
mathematics is science. But then, I don't have time for such
useless pursuits -- it seems to me to be just senseless hair splitting.
-- Wetboy
Pertti Lounesto
> Three is always greater than two, even for large values of two.
Where does two exist? And three? My answer is this: Two
and three are abstractions, properties of physical objects, like
their colors, red and green. Where does red exist? Are two
and three only illusions? Or concepts possessed by some
collectives of people? Does the Mars still have two Moons
after the last man has died? Does the concept of a planet
exist after the extinction of man? Are there still two classes
of celestial bodies, stars and planets, after man has disapperead?
Is there a sky after man? Is the universe divided into the Sky
and the Earth, after man? Or are there more than two parts
of the Universe, or less?
J R Partington
>
>Where does two exist? And three? My answer is this: Two
>and three are abstractions, properties of physical objects, like
>their colors, red and green. Where does red exist? Are two
>and three only illusions? Or concepts possessed by some
>collectives of people? Does the Mars still have two Moons
>after the last man has died? Does the concept of a planet
>exist after the extinction of man? Are there still two classes
>of celestial bodies, stars and planets, after man has disapperead?
>Is there a sky after man? Is the universe divided into the Sky
>and the Earth, after man? Or are there more than two parts
>of the Universe, or less?
Candidates should not attempt more than 4 questions.
JRP
---
Poincare's Optimization Satz: Let B be a normed space whose Iwasawa
algebra is weakly separable; then it can be coloured in at most four
colours.
Torkel Franzen
> Your question is rhetoric, but I will anwer it.
It wasn't rehetorical. I was wondering why you worry.
>A worry about
> small details, to be filled or rectified, often leads a curious
> explorer to new territories, new continests of knowledge,
> to be conquered.
Sure. But worrying about the probable existence of lots of
false "theorems" and even more invalid "proofs" in the literature
surely isn't a worry about small details?
karl malbrain
Pertti Lounesto <Pertti....@hit.fi> wrote in message
news:37CFFF6A...@hit.fi...
> Greg Neill wrote:
>
> > Three is always greater than two, even for large values of two.
>
> and three are abstractions, properties of physical objects, like
> their colors, red and green.
Stop giving away ground!! Your falling into a spurious induction from an
ungrounded group -- ALL values of two are the same size. Yes, an analysis
was done on color values, and there is a finite number of DIFFERENTIATORS.
Karl M
tiglath
wetboy <wet...@shore.net> wrote in message
news:BXRz3.19$JG1....@news.shore.net...
Yet, you seem to have time to give a non-answer. What is the use of that?
Pertti Lounesto
> worrying about the probable existence of lots of
> false "theorems" and even more invalid "proofs" in the literature
> surely isn't a worry about small details?
Quite right. There are more errors in the literature than
the community is willing to admit, and that as not a small
thing. But many of the individual errors were caused by
overlooking of a small detail, innocent and harmless.
Pertti Lounesto
> Can you give an example of a mathematical theorem, which
> has a rigorous proof and also a counterexample?
>
> Either the proof is not really a proof, or the counterexample
> is not really a counterexample.
It is interesting that this scribean opinion is repeated by
non-scholars lacking own experience of actual research.
Many theorems at the forefront of explorations of
mathematicians are under debate and have, at the beginning
of their life-span, both counterexamples and proofs.
See http://www.hit.fi/~lounesto/counterexamples.htm.
Jeremy, if you would have your own, first hand experience,
of research-making, you would know that such dicothomy
does not exist, the world of explorers is not black and white,
the picture has not yet formed, the explorers are jointly
drawing a map of a jungle, or maybe a new continent, who
knows.
George Greene
: Your question is ambiguous as long as mathematical truth
: is undefined. Here a definition of mathematical truth:
:
: A mathematical statement is a true, if all the specialists
: exploring the same domain agree that its proof is correct.
Not only every mathematician except you, but
almost every philosopher of science as well, knows that
that is not a definition of mathematical truth.
It is not a definition of mathematical anything.
It is not a mathematical definition. If it is
a definition at all, it is a definition in the
applied sociology of science.
Please don't whack me for asking such a basic question,
but have you read T.S.Kuhn (The Structure of Scientific
Revolutions) ? I am 95% certain that you have; I just
need to clear the other 5% before I say anything else.
George Greene
> has a rigorous proof and also a counterexample?
>
> Either the proof is not really a proof, or the counterexample
> is not really a counterexample.
In article <37D03416...@hit.fi>
Pertti Lounesto <Pertti....@hit.fi> writes:
: It is interesting that this scribean opinion
This is not an opinion. It is a definition.
: is repeated by
: non-scholars lacking own experience of actual research.
Yes, our acquaintance with the DICTIONARY has not been
corrupted by bad habits and jargon like mathematicans'.
: Many theorems at the forefront of explorations of
: mathematicians are under debate and have, at the beginning
: of their life-span, both counterexamples and proofs.
Then fact that they are theorems is not yet *known* at this
time. All of them for which there really ARE counterexamples
are NOT theorems. BY DEFINITION. The fact that even humanity's
best experts in the field are not yet competent to tell the
difference is IRRELEVANT FOR PURPOSES OF *THIS* DISCUSSION.
If a coin falls behind my desk, I do not know whether it has
fallen heads or tails, but there is still a fact of the
matter, and it is not changed by whether experts disagree,
or by how many years it takes someone to move the desk and
deliver PERSPICUOUSLY incontrovertible proof.
: Jeremy, if you would have your own, first hand experience,
: of research-making, you would know that such dicothomy
: does not exist, the world of explorers is not black and white,
: the picture has not yet formed, the explorers are jointly
: drawing a map of a jungle, or maybe a new continent, who
: knows.
And they are therefore NOT proving THEOREMS!
Unless and until they get lucky.
In all seriousness, the problem is highlighted when you say:
: Many theorems ... have both counterexamples and proofs.
These things ARE NOT THEOREMS. They are CONJECTURES
until somebody understands a RIGHT proof and presents
a version of it in which errors are minor or of sufficiently
small significance. The fact that experts in the field
are going to CALL some things theorems when they ARE NOT
is one that we are DEEPLY grateful to you for proving.
But the fact that referees of journals have called something
a theorem, even if it DOES MAKE it a theorem IN PRACTICAL TERMS,
does NOT make it a theorem. Being a theorem (or not) is, like
a being a rock, a PURELY-EXTRA-HUMAN kind of thing.
karl malbrain
George Greene <gre...@austin.cs.unc.edu> wrote in message
news:xesvh9r...@austin.cs.unc.edu...
> If a coin falls behind my desk, I do not know whether it has
> fallen heads or tails, but there is still a fact of the
> matter, and it is not changed by whether experts disagree,
> or by how many years it takes someone to move the desk and
> deliver PERSPICUOUSLY incontrovertible proof.
>
> : Jeremy, if you would have your own, first hand experience,
> : of research-making, you would know that such dicothomy
> : does not exist, the world of explorers is not black and white,
> : the picture has not yet formed, the explorers are jointly
> : drawing a map of a jungle, or maybe a new continent, who
> : knows.
>
> And they are therefore NOT proving THEOREMS!
> Unless and until they get lucky.
Sorry, but luck has nothing to do with it. `Proof' is a process of removing
RANDOMNESS from a system. This requires WORK together with sufficient
GROUNDS. Karl M
Richard Carr
:Date: Fri, 3 Sep 1999 15:37:53 -0700
:From: karl malbrain <kar...@acm.org>
:Newsgroups: sci.logic, sci.math, sci.physics
:Subject: Re: A Most Basic Question
:
:
:George Greene <gre...@austin.cs.unc.edu> wrote in message
:news:xesvh9r...@austin.cs.unc.edu...
:> If a coin falls behind my desk, I do not know whether it has
:> fallen heads or tails, but there is still a fact of the
Couldn't you just turn your head around and look?
:> matter, and it is not changed by whether experts disagree,
:
:
:
: