Zero -- Even or Not

[Algot Runeman]

http://www.bbc.co.uk/news/magazine-20559052

Math logic takes time to develop, even historically...

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Algot Runeman
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[Tanton, James]

I found this irresistable. I made it the theme of my December curriculum letter. http://www.jamestanton.com/?p=1072. (Top of the list.) Zero is just troublesome!

- J
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From: mathf...@googlegroups.com [mathf...@googlegroups.com] On Behalf Of Algot Runeman [algot....@verizon.net]
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To: mathf...@googlegroups.com
Subject: [Math 2.0] Zero -- Even or Not

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[Bradford Hansen-Smith]

James, you stated; "There’s something worthwhile in returning to and questioning the basics. After all, true business genius often comes from taking an old and simple idea and pushing it in a completely new direction.We can work to foster that thinking"

My observations suggest the circle is a symbol for everything and nothing is the negation of everything, which conveniently leaves us with parts and without any context, which is an impossibility. Conveniently because we can then decide what things are and are not as we wish, as we love to argue about these abstract concepts.
Brad

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[Joshua Zucker]

There's also a lovely blog (mostly about cosmology) that mentions recent widely-heard confusions about the evenness of 0.

  http://blog.richmond.edu/physicsbunn/2012/12/04/a-missed-opportunity-to-teach-some-mathematics/

Enjoy,

--Joshua

[Bradford Hansen-Smith]

Zero is not a number, it is a symbol for origin of both positive and negative numbers. If we could see a way to consider the circle zero as nothing and everything at the same time it would clear up a lot of confusion. There is something about the circle and zero that is so ubiquitous that we dare not question what we have been taught about circles and that we have for centuries been drawing pictures of them.  
Brad

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[Joshua Zucker]

James, thanks for the excellent newsletter.  You left off one of my favorite conundrums about zero, though.  It follows nicely from your question about zero giraffes.

Algebraically, is 0x the same as 0y?  In that case, is zero apples the same as zero airplanes?  And if so, then what about the sentence "Oh, darn it, I'm really hungry and I have zero airplanes!"

I think the answer is that when I say "I have (stuff)" I'm giving one component of an infinite-dimensional vector ... maybe?

Or maybe the answer is just that, as mathematicians, we don't use or think about numbers with units (like apples or feet) enough.  I mean, we say "you can't add apples and oranges" and then we write x^2 + x like there's no problem with that.

On Thu, Dec 6, 2012 at 4:54 PM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

Zero is not a number, it is a symbol for origin of both positive and negative numbers. If we could see a way to consider the circle zero as nothing and everything at the same time it would clear up a lot of confusion. There is something about the circle and zero that is so ubiquitous that we dare not question what we have been taught about circles and that we have for centuries been drawing pictures of them.  
Brad

There are certainly lots of possible definitions of number, but I don't know of any in which there are positive and negative numbers but no number zero.  Is it not true that if you add two numbers, the result is a number?

Also, what do you think about sets?  Is the cardinality of a set a number?  What is the cardinality of the empty set?

--Joshua

[David Wees]

Joshua,

I think those are a problem of units though, not a problem with zero itself. If you say you have zero of something, you've indicated the units of the zero. Perhaps that is all that is necessary to avoid some interesting contradictions?

David

--

[Joshua Zucker]

Yeah, I mentioned units in the paragraph at the end of that part of my reply.  But I don't think we understand units well enough, or at least we don't teach it well enough in our math classes.  Everyone would agree that 0 inches = 0 feet, so why not 0 apples = 0 airplanes?  There's something deeper going on there.  And a lot of people think of units as a sort of multiplication: 0 miles means 0 * (1 mile), which is of course just 0, because 0 times anything is 0.  So isn't 0*(1 apple) = 0*(1 airplane) = 0?  I think this shows that there's more to units than we commonly appreciate.  They're not just extra factors to multiply together.  But what are they?

I'd love a pointer to somewhere to think about units (or, ideally, teach them to 8th graders) in a way that would make sense of these kinds of questions.

--Joshua

On Thu, Dec 6, 2012 at 10:35 PM, David Wees <davi...@googlemail.com> wrote:

I think those are a problem of units though, not a problem with zero itself. If you say you have zero of something, you've indicated the units of the zero. Perhaps that is all that is necessary to avoid some interesting contradictions?

[Sue VanHattum]

Not in relation to 0, but I often say to students (in beginning algebra mostly), "You can't add apples and oranges, unless you call them fruit." My attempt to explain common denominators...

---

From: joshua...@gmail.com
Date: Thu, 6 Dec 2012 23:03:25 -0800
Subject: Re: [Math 2.0] Zero -- Even or Not
To: mathf...@googlegroups.com

[Linda Stojanovska]

Hiya Sue –

I too love the “fruit” denominator.  Linda

[kirby urner]

I think it's useful to stay conscious of the "not math meanings" of these words, like even and odd.

"Even" tends to mean "level" but also in the sense of "fair" as in "level playing field" and "even handed".

"Odd" tends to mean "peculiar" and/or "off" and/or "weird".

We're rediscovering on math-teach how we teach the number line and coordinate systems by relating "positive" to "right" and "up", "negative" to "left" and "down".

These are such relative terms though. 

My right is your left when we're facing each other (theater:  stage left vs. house left).

We think of "right" and "upright" as in "righteous" and/or "normal" -- "orthodox" and "orthogonal" go together (orthogonality is the prevailing orthodoxy).

Mathematics is riddled with moralisms, not just truisms.  One may say they're just connotations, not denotations, so not "real". 

That's whistling in the dark for sure.  These ghost-meanings are everywhere.

Kirby

[Bradford Hansen-Smith]

Yes, it is as important to be conscious of the effect common language has on math, and how math effects common language., Logic is relative to the frame, of which each is only partial truth.
Kirby, can you talk more about the morals in math, we need more discussion about this aspect of mathematics.

Isn't whistling in the dark a reaction to fear of something being there we don't know.
Brad

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[David Chandler]

In vector spaces the additive identity is not zero, but the zero vector.  In groups, the additive identity must be an element of the group.  Zero airplanes is in a set which contains sets of airplanes.  Zero apples is in a set which contains sets of apples.  The two zeros are not identical because they refer to different sets.  Someone can stand and complain that he has zero airplanes while munching on an apple.
--David Chandler

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[Bradford Hansen-Smith]

Joshua,

Sets are a way of segregating through separation by eliminating connections. An empty set is the potential of unexpressed ideas, grouping the unimaginable.
Brad

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[Jon Adie - Sky]

Dystopia - Utopia .... I am a bit lost.. in this  void! - found Indian .. Devanagari - Sanskrit and Arabic concepts.. Mahavira  ..dividing Zero   SHUNYA /SIFRE ... CYPHERS AND CHIFFRE'S in Romance languages

I see (intuit) approximate ... not like Maths , which demands proof..

Is this a case of where two fields meet  two sets merge creating a sub set  Art & Maths

 

I am a Visual artist , liked your leads on paper plate folding ... 'the circle' ..  Buckminster Fuller's Geodesics C60 ...  Space time and Ideas of the sacred 'temenos'... or templum..   Asylum ..Safe Inside.. a place  Out of time and Space .. read the mail re-Zero visual artists Using (Mandala/Icons) for Centuries.. and yet the enigma .. of Zero is huge ..  as are it's  concepts and products.

 

Is this fuzzy logic .. or just a blurred idsea badly presented ? Jon A  

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of David Chandler
Sent: 08 December 2012 03:25
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Zero -- Even or Not

 

In vector spaces the additive identity is not zero, but the zero vector.  In groups, the additive identity must be an element of the group.  Zero airplanes is in a set which contains sets of airplanes.  Zero apples is in a set which contains sets of apples.  The two zeros are not identical because they refer to different sets.  Someone can stand and complain that he has zero airplanes while munching on an apple.
--David Chandler

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[michel paul]

On Dec 7, 2012 7:25 PM, "David Chandler" <david...@gmail.com> wrote:

> In vector spaces the additive identity is not zero, but the zero vector. 

This is a beautiful example of why CS should be integrated with math education. It makes reasoning about things like groups and sets a practical matter. These are classes. If you have a Vector class, then it's clear that the additive identity will have to be a Vector object. If you try to add the integer 0 to a Vector object, you'll get an error.

On Thu, Dec 6, 2012 at 10:32 PM, Joshua Zucker <joshua...@gmail.com> wrote:

> Algebraically, is 0x the same as 0y?

It depends on what we mean by 'same'. They are equivalent values, so 0x = 0y, but they are definitely different ideas.

> maybe the answer is just that, as mathematicians, we don't use or think about numbers with units (like apples or feet) enough.

I think that's definitely true. I've always loved contemplating the old question, "What is a number?" Then one day I shifted to, "What is a quantity?" I found that very useful, as it's an easier question to answer, and it immediately sheds light on how we think about numbers. We can say that a 'quantity' is a 'number of units'. Now, the nature of the units we use can vary tremendously. Some can be subdivided, and others cannot be.

-- Michel

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
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[Juan]

Bradford,

With all due respect, I do not know what you mean by "number," or "a number," or "being a number," or by the word "is." You write "Zero is not a number" as if everybody understood and agreed with you on the meaning (whatever it is) you assign to those words. I just want to point out the fact that many mathematicians in the last two centuries have agreed on the notion that Zero is a real number, a complex number, a rational number, and an integer as well. So I would say "Zero is a number" is kind of the currently accepted wisdom. If you say otherwise, it would be nice of you to provide some context as to what you mean by that. Why is Zero not a number, in your opinion? The debate on whether Zero is a number or not is an old one. It was very meaningful during the classic Greek period. I like to believe we have moved on since then, and at least mathematicians have gotten over that debate, and left it behind for good, having collectively decided, a long time ago, that Zero is, indeed, a number, with all equal membership rights to "numbership" as Pi, e, One, Negative One, One-Half, Square-Root-of-Two, the Imaginary Unit, and at least all other complex numbers. No caveats, no exceptions, no exclusionary clauses, no nothing. Zero is as number as a number can be, at least in my opinion, and I believe many mathematicians share that opinion.

Juan

[Algot Runeman]

On 12/07/2012 02:03 AM, Joshua Zucker wrote:

Yeah, I mentioned units in the paragraph at the end of that part of my reply.  But I don't think we understand units well enough, or at least we don't teach it well enough in our math classes.  Everyone would agree that 0 inches = 0 feet, so why not 0 apples = 0 airplanes?  There's something deeper going on there.  And a lot of people think of units as a sort of multiplication: 0 miles means 0 * (1 mile), which is of course just 0, because 0 times anything is 0.  So isn't 0*(1 apple) = 0*(1 airplane) = 0?  I think this shows that there's more to units than we commonly appreciate.  They're not just extra factors to multiply together.  But what are they?

I'd love a pointer to somewhere to think about units (or, ideally, teach them to 8th graders) in a way that would make sense of these kinds of questions.

--Joshua

Math is, I think, an abstraction of our commonplace reality.
(Red Rome apple image from: statesymbolsusa.org)
plus equals two apples.

But how equally true, in terms of commonplace reality is this?
(crispin apple image from: nyapplecountry.com)
plus equals two apples.

I think I'm okay with this equation.

Math abstracts away the issue of units from our commonplace reality.
I can accept both of the apple math abstraction statements to be "true", but it would stretch my understanding of the world to think that a Red Rome tasted like a crispin. Often two red delicious apples don't taste much alike, especially if one is a week or more older and sitting on a kitchen counter.

1 apple + one orange = two fruit. (Unit conversion isn't much challenge in this case.)
1 apple + one airplane = two ... objects? Commonplace reality makes these units seem 'silly' to combine. Depending on the degree of math abstraction, though, the units can be mixed.

However, my ability to extract meaning from the math breaks down if I try to do some unit combinations.
1 gram of water + 1 degree of Celsius temperature = 2 ... what?
Can such a level of abstraction from reality work?

--Algot

[Sue Hellman]

Algo -- thanks very much for this your metaphor I'm working on a workshop in active learning strategies for math educators and would like to use it if I may. David Tall has had a lot to say about difficulties in math and the development of mathematical thinking (see for example: http://digilander.libero.it/leo723/materiali/algebra/dot1995b-pme-plenary.pdf). As I understand what he says, struggling students' difficulties come not as much from difficulty with abstraction, but from a reluctance or inability to let go of primitive processes or to reconceptualize them as objects which can be manipulated in their own right. He explains jump more as a 'compression' than an abstraction -- i.e. critical in order to reduce the cognitive load on working memory. Unfortunately he doesn't offer much to help educators to get the students who cling desperately to reproducing processes when solving more complex problems (i.e. counting to multiply large numbers instead of compressing that process into a known fact which has a use of its own independent of the original meaning or concrete situation from which it sprang).
 
BTW: Here's my variation on your 2 apples problem:
 
  plus    =  
 
(My additions are adapted from images in wikimedia commons.)
 
Sue 

--

[kirby urner]

On Fri, Dec 7, 2012 at 7:19 AM, Bradford Hansen-Smith
<wholem...@gmail.com> wrote:
> Yes, it is as important to be conscious of the effect common language has on
> math, and how math effects common language., Logic is relative to the frame,
> of which each is only partial truth.
> Kirby, can you talk more about the morals in math, we need more discussion
> about this aspect of mathematics.
>

Probably it would help strengthen the mathematics curriculum if we
spent more time investing in off-the-beaten-track approaches that take
us in another direction.

On the issue of positive and negative, I think it'd make sense to
bring in more discussion of particles and anti-particles e.g.
electrons and positrons, to to have some discussion of why anti-matter
is not equally prevalent. There's asymmetry we might want to discuss.

What I do more, in getting off the beaten track, is I stop showing
multiplication as exclusively a right angles affair. A triangle with
all edges n and an area of n makes enough sense to support an
internally consistent logic. A tetrahedron of edges n has volume n^3.
So we don't say 'squared' and 'cubed' for 2nd and 3rd powering
respectively. That's not the knee-jerk reflex it becomes for your
standardized math-head.

Rather that just say "live and let live" and "we should tolerate
differences" it's more eye-opening to push back a little and suggest
the way we do it now is the less intelligent way i.e. not only are
there other ways to design maths, but some of these maths lead to
better thinking than we currently access, thanks to our collective
closed-mindedness.

I do this with my unit volume tetrahedrons, dividing them evenly into
other shapes with easy relationships. So many more wholesome whole
numbers that the unit volume cubes give us. The connotation that
cubes are "relatively awkward, stupid, slow, overbuilt" (relative to
tetrahedrons) is allowed to seep through the lines.

Such a math is "not normal" i..e "not orthodox" and so doesn't get
much discussion, even though the literature is well developed (not all
my invention by a long shot).

My intention is to show how mathematics is laced with conventions and
agreements that are anything but "proved". A way to show this is to
lace a mathematics with different conventions and agreements.

> Isn't whistling in the dark a reaction to fear of something being there we
> don't know.

Yes. Seeing a rather different math growing up, based on different
conventions and assumptions, can be eerie to witness, perhaps somewhat
scary, like witnessing the growth of that alien plant (Audrey 2) in
'Little Shop of Horrors' maybe.

Kirby

> Brad

[michel paul]

On Sun, Dec 9, 2012 at 10:54 AM, kirby urner <kirby...@gmail.com> wrote:

Probably it would help strengthen the mathematics curriculum if we
spent more time investing in off-the-beaten-track approaches that take
us in another direction.

Absolutely. I'd say the biggest obstacle to this getting to happen is high stakes testing. It's deadly. People are very hesitant to veer off the beaten track, at least in US public ed, since everything is so heavily tied to test scores. Why don't people teach Euclid's Algorithm? Well, because it isn't tested. The fact that it's a beautiful piece of reasoning that is a doorway into number theory and computational thinking isn't as important as the fact that it's not tested.  

-- Michel

[michel paul]

Very nice - this discussion of units turns out to be important as well for discussing the difference between addition and multiplication! I totally agree that thinking about what units are should be done more in math. They tend to get neglected if the focus is on the bottom line computation. 

A sum only makes sense when we add units of the same type, and a sum will be composed of the same units as its addends, but a product will always involve units of different types, as in square units from linear units, or as in rate*time. In one way or another a product involves a combination of units, but a sum always involves units of the same type.

A repeated addition model might think in this fashion: a * (b units) = (b units) + (b units) + (b units) ... a times. Here the a apparently has no units. It is a scalar. Someone might think that here we have an example of a product where there is only one type of unit involved. However, it is still the case that a is a different kind of mathematical object than the b units.

We could also express the product as (a groups) * (b units/group). And again, ratio appears as an essential part of what multiplication is.

- Michel

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[kirby urner]

On Sun, Dec 9, 2012 at 11:18 AM, michel paul <python...@gmail.com> wrote:
> On Sun, Dec 9, 2012 at 10:54 AM, kirby urner <kirby...@gmail.com> wrote:
>>
>>
>> Probably it would help strengthen the mathematics curriculum if we
>> spent more time investing in off-the-beaten-track approaches that take
>> us in another direction.
>
>
> Absolutely. I'd say the biggest obstacle to this getting to happen is high
> stakes testing. It's deadly. People are very hesitant to veer off the beaten
> track, at least in US public ed, since everything is so heavily tied to test
> scores. Why don't people teach Euclid's Algorithm? Well, because it isn't
> tested. The fact that it's a beautiful piece of reasoning that is a doorway
> into number theory and computational thinking isn't as important as the fact
> that it's not tested.
>
> -- Michel
>

Yes, high stakes testing has a huge impact.

I wonder if the response should be alternative high stakes testing.

On the other side of the tests, you need a network of academies that
offer worthwhile training and experience.

One need not compete for huge numbers of student in a high profile
way. It's not like ETS (SAT, AP...) will go out of business.

Sure we need to cultivate some sense of "recruiters" or "talent
scouts" but when it comes to underdog subcultures, smaller is better.

You get to be elitist more quickly.

If your biggest challenge is school pride and snobbery, that's likely
not a bad place to be.

Kirby

[Bradford Hansen-Smith]

Jaun, thank you for asking.

You have asked; "If you say otherwise, it would be nice of you to provide some context as to what you mean by that. Why is Zero not a number, in your opinion? "

Zero is a symbol, and yes, there have been many very bright and respected people over centuries that have worked to clarify mathematical issues. I do not discount any of them, but must go by my own observations and understanding as well. Using the circle as a symbol for zero, nothing, does not make the circle a zero. The zero is not a number; what we call numbers are different symbols with different values. The zero and numbers are only by common agreement. I see the circle symbol as an image that represent a circle disk in space, as well meaning zero,

I function in s 3-D world as well as in mind. There must be a balance between the physical, mental, and yes the spiritual. For understanding I need a larger context than what has been agreed upon. We have left out too much. It is important to establish context since everything is in fact multifunctional with more than one meaning. The generalization about abstract objects does not take into consideration the context, rather we talk about isolated objects, units in separation; this does not hold up in understanding the complexities of interrelationships withing a larger context. 

I must side with Kirby, for there are other math and logic systems equally important as what we have agreed upon. One is not better, they all hold some degree of understanding truth, which is the reason to put out all this mental energy in the first place. The symbol of the circle represents a 3-D circle as well abstractions of the zero concept. The properties of this circle disk object has five congruent circles, four more than a 2-D circle. When a circle is folded in half a tetrahedron pattern of movement is generated showing the traditional properties of a tetrahedron. This is observable but not obvious until pointed out "how" we fold the circle. This is another understanding that while very different in approach does not deny what else we have discovered and accepted about circles, but it gives a greater context of understanding making clear some of the inconsistencies that we discuss and protect.

The circle is a symbol of nothing, zero, empty set, but is also an image of a comprehensive whole, a set of everything known and unknown, it is unity of everything. The circle functions as unit and unity simultaneously; no other object can demonstrate this concept. Why disadvantage ourselves and say it is only one thing when we are only talking about an image. It can represent what ever we want it to, but that does not make it so. Every round coin is a circle disk and each has different value depending on size and surface design because of agreement, not because of the object. There are many ways to calculate the abstraction of "money" value, as there are many ways to understand images used in mathematics. I only suggest we expand our understanding of what we know and not get stuck in what someone said about something a long time ago. I do not discount the value and importance of their contributions, but I do not overly value yesterday more that what is dynamically taking place today. We are expanding our understanding about where we are and realizing things may not be what we once thought.  
Brad

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[kirby urner]

On Mon, Dec 10, 2012 at 8:09 AM, Bradford Hansen-Smith
<wholem...@gmail.com> wrote:

<< snip >>

> I must side with Kirby, for there are other math and logic systems equally
> important as what we have agreed upon. One is not better, they all hold some
> degree of understanding truth, which is the reason to put out all this
> mental energy in the first place. The symbol of the circle represents a 3-D
> circle as well abstractions of the zero concept. The properties of this
> circle disk object has five congruent circles, four more than a 2-D circle.
> When a circle is folded in half a tetrahedron pattern of movement is
> generated showing the traditional properties of a tetrahedron. This is
> observable but not obvious until pointed out "how" we fold the circle. This
> is another understanding that while very different in approach does not deny
> what else we have discovered and accepted about circles, but it gives a
> greater context of understanding making clear some of the inconsistencies
> that we discuss and protect.
>

I take the concept of "namespaces" from computer science and mix it
with "language games" from Wittgenstein.

When designing computer languages, people have needed to work out how
to allow different "meaning" of the same word to co-exist peacefully
and get work done. People have a propensity to use the same words for
different objects.

Not that "meaning" has to be seen as "pointing to objects" but that's
the crass nominalism we inherit, and object oriented programming makes
it worse, so I go with the flow sometimes.

Hence Martian Math where I set up the namespaces on either side of a
canyon. The Martians have a D^3 unit volume tetrahedron while the
Earthlings have an R^3 cube. R+R =D in length. They both share a
common reference sphere, at the center of which we can talk about Zero
(both civs agree to some extent, though the Earthlings use XYZ and the
Martians use IVM for coordinate systems / scaffolding).

The antidote to "one and only one math" is "local variables" i.e. in
talking about "lines" or "powering", we're local to a namespace.
There's no requirement to establish the global meanings of terms, as
if "zero" could not anchor in language unless it had "one true
meaning" uber alles. We don't need for it to have that. All we need
are local constructions in which Zero has a role. There may be family
resemblance among them.

Kirby

[Jon Adie - Sky]

Non - plussed... I like your   'description'.. of the symbol, and the extended circle of meanings .. with the emphasis on context ..  I agree although I am reminded of Alan Turin and Buckminster Fuller who both looked back in time while breaking ground moving forward

Jon A

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Bradford Hansen-Smith
Sent: 10 December 2012 16:09
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Zero -- Even or Not

 

Jaun, thank you for asking.

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[Juan]

On Monday, December 10, 2012 10:18:36 AM UTC-8, kirby urner wrote:

The antidote to "one and only one math" is "local variables" i.e. in
talking about "lines" or "powering", we're local to a namespace.

Kirby,
Just to let you know, there we are at polar opposite points of the spectrum. "Antidote'? I see no need for an antidote. I do love "one and only one math." I believe in "one and only one math." I support "one and only one math." I am an enthusiast of "one and only one math." That is the reason I studied math in the first place, instead of, say, political science, or linguistics. Inside the one and only one math, there is room for an infinite number of mathematical branches and sub-branches. There are many maths inside The One Math, and that makes It more fascinating. I find math's uniqueness makes it perfect, in sharp contrast with the maddening cacophony of the real world.