
What Is Mathematics and What Should It Be? [pdf] - santaclaus
http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/hersh90.pdf
======
WiltedSpinach
Dr. Zeilberger writes a rather pleasant rant. He appears frustrated with the
lack of imagination in mathematicians which occurs when they focus too
strongly on the axiomatic nature of mathematics, and seems to wish that we
take a step back from the powerful axiomatic tools we have developed, so that
we may search for what is 'really true about the world' rather than 'playing a
mathematical game/believing in a mathematical religion'.

Three specific complaints he has involve a belief in infinity (and limits),
which he asserts do not exist in the real world; a delay in the publication of
a pair of papers on which he worked, due to a variety of circumstances; and a
"pernicious" influence of axiomatic mathematics which leads to "stupid"
questions such as Hilbert's Second Problem.

The failed publication of one paper is _particularly_ notable, as it claimed a
counterexample of Fermat's Last Theorem (according to Dr. Zeilberger's
"Opinion 123", on his Rutgers website), and (ibid) was recognized by Andrew
Wiles himself as one of three _possible_ counterexamples: the reason given for
this oversight was in fact the acceptance of the decidedly non-rigourous
statement "it is easily seen ..." (ibid). This particular instance seems to
contradict the main thrust of Dr. Zeilberger's rant against a mathematics
overburdened by rules.

Unfortunately, little example of what mathematics SHOULD look like is offered
- beyond a statement that 'obvious things should be treated as such' and an
assertion that infinities and continuities do not occur in real
life/nature/the universe. While such a statement may be understandable coming
from a respected (and clearly accomplished) combinatorist such as Dr.
Zeilberger, physics has yet to demonstrate conclusively that space and time
are discrete - undisproven interpretations of quantum mechanics exist which
allow for continuities, and 'the size of the universe' is defined as that
which we can see (ie, within ~13.8 billion light-years of Earth/Sol) so that
actual infinities are ruled out only because of our inability to perceive
them.

Perhaps Dr. Zeilberger needs to think outside his own discrete box.

~~~
hvidgaard
Re: infinities and continuities, to my understanding, it has been proved than
interaction with the world is inheriently discrete determined by Planks
constant - I am not a physicist, so may be wrong.

If we cannot interact with arbitrarily precision and can only ever observe a
finite universe, does it make sense to even ask if the it is any other way?
Should it be figure out how to interact with the world using some new
fundamental mechanic, other than a wave bound by C and h, it opens the
question.

~~~
mtzet
Even if the world is fundamentally discrete, that does not mean that
continuous models are not valuable. They are.

We are used to doing discrete approximations to continuous problems, but the
other way around works too. For example, evaluating the sum

1 + 1/4 + ... + 1/n^2

is quite hard. On the other hand, we may approximate by the integral over x^2,
which remarkably has an easy formula.

Continuous and discrete models complements each other, they are not mutually
exlusive.

~~~
hvidgaard
I wasn't trying to say that continues models are not valuable, only that the
question "is the world continues?" does not make sense if we can only interact
with it in a discrete way.

~~~
hardlianotion
That is simply a rewording of the question: is it useful to use continuity to
model the world.

~~~
hvidgaard
This is getting quite philosophical now. It depends on the intrepetation of
"world". If we mean the universe then no, it is to our current understanding
of it, not useful to model it's entirety using continuity, if we can only
messure it discretely. If by world we mean a subset of the universe, then it
makes perfect sense, because the world might be on the macro level where this
"discretness" is far smaller than we ever want to go.

~~~
hardlianotion
I think you should concentrate on the world model in my sentence rather than
world. I have something I want to analyse. How is it useful to think about it
and what techniques do I want to use?

------
pflats
I am truly confused by this screed.

>• Stupid Question 2: Trisect an arbitrary angle only using straight-edge and
compass.

>Many, very smart people, tried in vain, to solve this problem, until it
turned out, in the 19th century, to be impossible

>• Stupid Question 3: Double the cube.

>Ditto, 2^(1/3) is a cubic-algebraic number.

>• Stupid Question 4: Square the Circle.

>Many, very smart people, tried (and some still do!) in vain, to solve this
problem, until it turned out [...] to be impossible.

>Today’s Mathematics Is a Religion

>Its central dogma is thou should prove everything rigorously.

To me, that's the entire point. You prove things rigorously so that (among
many other reasons) when someone considers trying to square the circle, you
can point them to the proof that the circle cannot be squared.

~~~
kefka
>• Stupid Question 3: Double the cube.

    
    
         (X^3) * 2  -- Done.
    

>• Stupid Question 4: Square the Circle.

    
    
         r is known. X is not.
         pi * r^2 = X^2
         sqrt(pi * r^2) = X , for positive X
    

oh, geometrically? No. Algebraically works cleaner, and for any arbitrary
positive solutions for r.

>Its central dogma is thou should prove everything rigorously.

That's not a dogma. Its a proof because anyone, no matter whom, no matter
when, or where in the universe, can duplicate these results and show they are
logically true. Or they can show the results are logically false, no matter
the inputs given.

It's not "dogma", as some high edict by a Pope or something. A rank amateur
could further the field by proving a new theorem - because the _person_
doesn't matter. The soundness of logic does.

~~~
j2kun
> oh, geometrically? No. Algebraically works cleaner, and for any arbitrary
> positive solutions for r.

I hope you're not being serious. Just in case you are, your algebra is wrong.
I'm quite certain you didn't look up what "doubling the cube" means, since the
(faux) algebraic solution is y = cube_root(2 * x^3). It undercuts the rest of
your comment.

~~~
kefka
Doubling the cube was being ridiculous to prove the bad language to explain
the problem.

If it really meant doubling the volume of a cube of X unit size, then
absolutely it's (2 * x^3)^(1/3)

~~~
j2kun
I don't think the author was trying to give a precise description of the
problem. "Doubling the cube" is a term of art. It's like if he used the word
"derivative" and you thought it meant a cheap copy of something, and then went
further to prove how silly calculus was because of your misunderstanding of
the term.

You're also selling the problem short. Doubling the cube is about producing a
finite algorithm (given a limited set of operations) that realizes the value
of (2 * x^3)^(1/3) concretely. An algebraic solution does not do this, because
it stops at the inability to realize, say, the cube root of 2 explicitly.

------
fmap
> In particular, one should abandon the dichotomy between conjecture and
> theorem.

Wasn't that the status quo before the 20th century? It's strange to suggest
that working with infinite (or rather, ideal) objects is stupid. The sheer
amount of progress in 20th century mathematics provides incontrovertible
evidence that ideal objects are a useful reasoning tool. This is even true in
combinatorics: working with generating functions is working with an algebraic
structure on infinite streams...

This essay is written to incite... there is so much in there that invites
comment from everyone who has ever spent five minutes thinking about these
things. At the same time it is lacking in examples for ways in which a
mathematical world without rigor would be better than what we have today. So
instead of getting worked up about the essay itself, does anybody here have
concrete examples where a lack of rigor lead to faster progress?

~~~
OJFord
It seems strange to me for the author to write that, and then to end with:

> _But I do believe that it is time to make [mathematics] a true science._

Surely any 'true science' has a dichotomy between conjecture or hypothesis,
and theorem or accepted fact?

------
110011
Nothing is preventing you from using computers cleverly to gain a better
understanding of mathematical problems you are facing. But abandoning or
discounting the importance of rigor is down right stupid. The great
achievement of mathematics is the vast repository of irrefutable statements
which will stand for eternity. And this comes from rigor, which helps and does
not hinder.

Going back to his example, how does it help if we all collectively agree that,
yes, there must be infinitely many twin primes because of computational
evidence X, Y and Z. Nothing more than agreement has been achieved in such a
case. And we are none the wiser in the way of insight or explanation besides
the computations we already had...

~~~
trentmb
> But abandoning or discounting the importance of rigor is down right stupid.

[https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...](https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry#Collapse_of_the_school)

------
ldp01
From an engineering perspective... Mathematics looks like an add-on to natural
language designed to express precise concepts in the simplest possible way.
(Simple != easy, of course).

In a scientific or engineering context mathematics evolves naturally whenever
the requirement for precision exceeds the capability of the available language
and you have the right folks around to develop it.

~~~
majewsky
As evidence, look at how math looked before we formalized notation. For
example, the algebra book by al-Khwarizmi (the guy whom algorithms are named
after). Wikipedia has a commented excerpt:
[https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-
Khwarizmi...](https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-
Khwarizmi#Algebra)

------
0x264
Mathematics is the art of highlighting the necessary consequences of a
situation. That's all.

The situations that we mathematicians mostly consider are axiomatic
constructs, because it's easier to then unambiguously establish necessity.

That this art, its methods, techniques and tools, happens to be so useful to
other sciences (and in fact most of human knowledge) is an interesting
phenomenon...

~~~
amelius
Mathematics is also used for purely abstract entertainment.

Also, your definition applies to physics as well.

~~~
0x264
> Also, your definition applies to physics as well.

Um.... I think that it looks like that a lot of what maths does is shared with
physics, but I think (I could be wrong) that it's mostly because physics has
adopted the language of mathematics.

That having been said, there is an aspect where they are definitively
different. Physics is motivated by the understanding of the physical world.
For instance a physics theory gets dropped when we discover that we mis-
observed whatever it was meant to explain. This doesn't happen in maths. The
theories in maths (here defined as axiomatic structures) do not need to align
with the natural world and get studied for other reasons than because they
would increase the understanding of the natural world. Such understanding may
eventually happen, but was not the motivation factor.

What do you think ? :)

~~~
amelius
In my opinion, math is:

\- an extension of pure logic

\- not a science (since it cannot be falsified)

\- only indirectly concerned with observations (because we don't know the
universe)

\- used to predict, but also to reason about the past (big bang theories)

\- a tool

\- a game (puzzle)

------
tpeo
Reading the word "pseudo-problems" sets me off to a rough start. But calling
problems stupid merely because they're unsolvable is a mortal sin, specially
if there's a non-trivial reason _why_ we cannot mathematically go from this to
that. That the impossibility of squaring the circle is due to it's equivalence
with the transcendentality of π is in itself an interesting piece of
mathematics.

~~~
marcosdumay
He didn't call them stupid because they were unsolvable. He called them stupid
because they didn't exist on any practice.

And, of course, the knowledge that solved them also become essential to other
areas of Math, so I also don't like his framing.

------
roenxi
This gentleman needs to watch Feynman talking about the difference between
maths and physics.

Science is all about gathering evidence and the scientific method focuses very
heavily on _observation_ and _testing_. Basically, it is impossible to conduct
science without data.

Maths is data independent. More data or less data doesn't influence what maths
is. Maths links the axiom and the result - once a result has been prooven to
follow from an axiom, the data is irrelevant. No amount of observation or
testing will change the value of Pi.

[https://www.youtube.com/watch?v=obCjODeoLVw](https://www.youtube.com/watch?v=obCjODeoLVw)

~~~
empath75
I wouldn't say it's data independent. Look at goldbach's conjecture. While
there is no rigorous proof, it seems almost impossible to be the case that it
is not true. A lot of conjectures like that are based on data gathered first.

~~~
roenxi
You are referring to an as yet unsolved problem. The maths is as yet un-done!

Sure, there is data that suggests it is probably true. The scientific can say
there is enough data to be almost certain it is true and move on. The
mathematical community cannot say for certain it is true because it is not yet
prooven. Which is why they are still working on it.

~~~
empath75
I would suggest that formulating conjectures based on data is part of the
mathematical process, would you not?

~~~
roenxi
I would suggest that formulating conjectures based on data is specifically not
maths.

Some mathematicians start with data, and no doubt about it data is effective.
But I'm specifically saying that that is employing a tool of the scientific
community (see [1]) as a starting point before then proceeding to do some
actual mathematical work. This distinction is why maths is often classified in
the Arts rather than the sciences.

But if you can point to a maths textbook that teaches someone "general
theories" by printing 10,000 data points followed by a QED then I'd suggest it
is a pretty extraordinary theory.

[1]
[https://en.wikipedia.org/wiki/Scientific_method](https://en.wikipedia.org/wiki/Scientific_method)

 _EDIT_ I'll throw in an example; a software consultant might be involved in
invoicing for a project. The invoicing is still accounting work, even if it is
being done for a software project.

~~~
Jtsummers
Formulating a conjecture and proving a conjecture are two different
activities. A conjecture is based on incomplete information (data gathered so
far), and then we try to prove or disprove it using the mathematical tools
available (and sometimes developing new tools).

Consider making the observation (shown as a table):

    
    
      +-----+-------------+
      |  n  | sum(1 to n) |
      +-----+-------------+
      |  1  |      1      |
      +-----+-------------+
      |  2  |      3      |
      +-----+-------------+
      |  3  |      6      |
      +-----+-------------+
      |  4  |      10     |
      +-----+-------------+
      |  5  |      15     |
      +-----+-------------+
      |  6  |      21     |
      +-----+-------------+
    

We can come up with the formulation: sum(1 to n) = n(n+1)/2 by several
methods, but from the data given it's only a conjecture. Depending on how we
came up with that formulation we may already have proven the conjecture. Or if
we constructed it using the data only, we can prove it via induction or other
methods.

The same is how many other conjectures begin on less trivial examples. The
four color theorem, for instance, was notably hard to prove (and there was a
lot of controversy over its method of proof). But it was still just a
conjecture until they laid out their proof, though no one had ever found a
counterexample.

------
danielam
Mathematics is a formal science, not an empirical science. In other words,
both Zeilberger and those he criticizes (as he characterizes them) are wrong.
Some interesting books on the subject...

"An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science
of Quantity and Structure" by James Franklin

"German Science" by Pierre Duhem

------
jhanschoo
It is unclear to me why this rant was necessary. The field of applied
mathematics and other mathematical modeling sub-fields seems to be what the
author means when he talks of mathematics as science and of mathematical
truths. His criticism would have been much more salient if applied mathematics
weren't thriving today more than ever--with the explosion of computational
power every field now seeks to make evermore complex mathematical models of
the problems they study and solve.

------
ithinkso
That was very unpleasant to read but what he basically proposes is to change
"don't know if true until proven true" to "true until proven false" which is
ridiculous.

------
hardlianotion
I can't say I like the essentially manufactured grievance against Hardy and
his young man's game quote, completely out of context. It is a cheap shot and
has nothing to do with his argument.

------
aqsalose
>[..] Since slaves did all the manual labor, the rich folks had plenty of time
to contemplate their navels, and to ponder about the meaning of life. Hence
Western Philosophy, with its many pseudo-questions was developed in the hands
of Plato, Aristotle, and their buddies, and ‘Modern’ pure mathematics was
inaugurated in the hands of the gang of Euclid et. al.

Some call it "contemplating their navels", some other call it curiosity. I'm
reminded of this _other_ essay [1] that was discussed on HN a week ago:

>[..] throughout the whole history of science most of the really great
discoveries which had ultimately proved to be beneficial to mankind had been
made by men and women who were driven not by the desire to be useful but
merely the desire to satisfy their curiosity.

Regarding the rest of the Zeilberger's essay ... while it is intuitively
obvious that there _something_ odd going with the assumptions in problems like
the Zeno's paradox (that's why we call them paradoxes), it is certainly not
intuitively obvious what that something exactly is, because it took us a good
amount of principled application of curiosity (and quite while of time) to see
_why_ and _how_. What you'd call a mind that does not yearn to know the
details but instead yells and lavishly prefers to drop "that's stupid" in
boldface and with an exclamation point? I'm not sure, but "uninteresting"
comes to mind.

In general, it's easy to see that ancient Greeks were mistaken on some issue
or their ideas on some other ones were downright silly now that we have spent
over two millenia improving on them and developing the ideas and tools such as
"experimental science".

[1] Abraham Flexner, "Usefulness of Useless Knowledge", _Harpers_ 179, 1939,
[https://library.ias.edu/files/UsefulnessHarpers.pdf](https://library.ias.edu/files/UsefulnessHarpers.pdf)
HN discussion:
[https://news.ycombinator.com/item?id=14558775](https://news.ycombinator.com/item?id=14558775)

------
openfuture
My formulation would be something like this: Mathematics is how you discover
further truths about worlds that have objective truths. We don't know if the
world "we live in" is such a world but it contains a lot of those other worlds
=)

~~~
pdfernhout
Brilliant!

------
tempodox
Just consider that academic math refused for over 300 years to acknowledge the
existence of complex numbers, leaving the problem to the engineers. No
scientist could afford this substitution of dogma for reality.

~~~
protonfish
I don't have a problem with the pure math crowd doing their own thing. They
absolutely discover some amazing things using their techniques. What I have a
problem with is that these people write the math textbooks. Why are we
bamboozling our poor children with this impractical, esoteric,
overcomplicated, pseudo-religious, gobbledygook? Primary and secondary school
mathematics need to change focus to applied and computational math.

~~~
Jtsummers
Since I haven't been in a K-12 math classroom in about 10 years (contemplated
becoming a teacher and visited some friends' classes), what's the gobbledygook
we're teaching kids these days? I don't recall any gobbledygook from my years
in K-12 education (excepting an awful, awful long-term substitute teacher).
Arithmetic and algebra (K-8, basically), geometry, trig, and calc were all
practical and applied (the latter two in advanced science courses and work
rather than daily life).

I think more probability and statistics courses would be easily justified.
Regarding computational thinking, math in the K-12 level is computational
thinking more than it's based on proof construction. Computational thinking
should be added as a subset of the math and science courses. I think it helped
many of my college classmates who weren't in CS to understand computational
thinking when we ran simulations using matlab of experiments, or in the
reverse used it to process our results.

------
jcranberry
I hate combinatorics. Number theory and algebra are so beautiful and profound
and I feel combinatorics just sucks out all the soul.

Also strange that he cites Hersch, who, if I remember correctly became
relevant during the post-modernism vs science wars, unless this is from that
particular conflict--in which case I wasted my time reading this at all.

------
Ceezy
"Its central dogma is thou should prove everything rigorously"

I'm really worried that peoples who have knowledge in computer science doesn't
get that from a false statement you can prove ANYTHING. And yes literally
ANYTHING. Should we accpet models of computer that can compute faster than
speed of light....?

~~~
kwoff
I'm not convinced that you're really worried about that. You seem fairly
dramatic, to be honest.

~~~
Ceezy
That's some undergraduate logic and it was my first weeks in college so yes
i'm concern of what people get from expensive computer science degree.

------
nippples
We are already suffering greatly in computing because reduction of rigor.
People who can't handle rigor in mathematics, the tool that every freaking
scientist in STEM should seek a different, more enjoyable, field.

~~~
kwoff
offer some proof of this?

