user-1: Buddhists say that everything is empty.
Is mathematics empty?
I thought that "empty" meant conceptually constructed. But how can the fact (not the expression) 2 + 3 = 5 not be a fact which is true regardless of how we are counting?

user4722: There is a mathematician Gödel who stated back around the 1920s using something we might imagine as "meta-math" that any axiomatic system (so any mathematical system you can create or imagine) cannot satisfy simultaneously two properties.  Which two?

That the system is complete
That the system is consistent

I shall try and state in a useful metaphor these key insights.  I like to imagine a wheel and at the center of the wheel are some basic operations like multiplication, addition, subtraction.  However, it could be anything.  We are basically trying to figure out all the infinite possible combinations of our basic fundamental rules and then see if we can make this infinite variety of conclusions both internally consistent and completely exhaustive of every possible outcome.
This is like trying to build a house with some basic pieces: a roof, a chimney, a floor, a basement, a fridge, a garage, a kitchen.  Now say you try every combination of possibilities such that the fridge ends up on top of the place and the roof ends up on the ground.  This is us trying to make all the exhaustive combinations possible, but in doing so we contradict the terms we have used: fridge is now roof and roof is now floor.  
In short: if we tried to make our system demonstrably "complete" then we would invariably end up with a plethora of contradictions.  That is, we could say 1 = 1 and 2 = 2 but once we start saying 2 = 1 and 1  = 2 to get "completeness" then we have contradiction (non-consistency)
Likewise, if we want to have consistency (1 = 1 and 2 = 2 and 5 = 5 and 4-3 = 1) we cannot have completeness guaranteed, because we omit terms and therefore don't know if there is a true term we have not yet discovered.
So the basic idea of Gödel's incompleteness and completeness theorems are that mathematics cannot be both consistent (or coherent) and complete which means that mathematics, is in fact, empty.
Any postulate is empty of its own being because it is necessarily dependent on other terms to relatively define its value/position/significance.  
Any postulate is also empty of an absolutely true nature because there is no way to have absolutely true consistency, for that would require completeness, and Gödel showed that you can only have one or the other, like a scrambled egg or an unbroken egg, you cannot have both unless you sacrifice consistency (the egg is both unbroken and scrambled?) or completeness (just one or the other).
So in short, all mathematical systems must abide as either incomplete or inconsistent which is [another reason] why mathematics in itself is not graspable and not inherent.
This, however, is a long way to arrive at the basic truth that all phenomena are empty of their own being (they have inter-being as Thich Naht Hanh says).

user254: In 'pure' mathematics, the truth of an expression like 2+3=5 depends on the definition of the addition operator and the definitions of the numbers such as 1 and 2 and 3 and so on. I think these definitions are 'axiomatic'.
In 'applied' mathematics e.g.when you're adding physical objects (such as apples), then the truth depends on your ability to identity "one apple" (which isn't obvious: in the real world, some apples are smaller, some are attached to the tree, some are rotting, etc.).
Furthermore mathematics is something which you do (sometimes) and which you stop doing: therefore IMO it's conditioned and impermanent.

More generally I think it's a category of knowledge. Here's a (slightly rude or brusque or plainspoken) quote from Zen and the Art of Motorcycle Maintenance -- this quote is talking about Aristotle's "dialectic" but it could IMO equally be talking about "mathematics":

"As best I know, Aristotle's opinion is that dialectic comes before
  everything else."
"And from the dialectic come the forms," Phædrus continues, "and from. -
  - " But the Chairman cuts it off. He sees it cannot go his way and dismisses
  it.
He shouldn't have cut it off, Phædrus thinks to himself. Were he a real
  Truth-seeker and not a propagandist for a particular point of view he would
  not. He might learn something. Once it's stated that "the dialectic comes
  before anything else," this statement itself becomes a dialectical entity,
  subject to dialectical question.
Phædrus would have asked, What evidence do we have that the dialectical
  question-and-answer method of arriving at truth comes before anything
  else? We have none whatsoever. And when the statement is isolated and
  itself subject to scrutiny it becomes patently ridiculous. Here is this dialectic,
  like Newton's law of gravity, just sitting by itself in the middle of nowhere,
  giving birth to the universe, hey? It's asinine.
Dialectic, which is the parent of logic, came itself from rhetoric. Rhetoric is
  in turn the child of the myths and poetry of ancient Greece. That is so
  historically, and that is so by any application of common sense. The poetry
  and the myths are the response of a prehistoric people to the universe
  around them made on the basis of [etc.]


user37: Emptiness in eastern philosophies usually have a more subtle meaning than we use in the west. Check this wikipedia article on Emptiness. And there are many answers regarding emptiness (sunyata) like this one. There is also an article in wikipedia on Śūnyatā.
The most common understanding on emptiness (on Buddhism) is that it is dependently originated. It means that it does not have inherent meaning or value, but it relies on other things. In mathematics the numbers two, three, five and so on just has meaning because of the number one. And also, the number one only has meaning relying on the other numbers.
In a more practical sense, numbers only has meaning because of the objects, like apples, rocks or sheep. Now, emotions, that have no clear bounds, can't be numerically described. You could say that you felt happy twice today, but you can't express its intensity and form numerically. Then math is true not because it has inherent meaning, but because it works for what it is used.
There is another interpretation I read on this QA (and also in some wikipedia article but I couldn't find it):

[Emptiness is] a flashy way of reminding us that we only define things in relation to what they are not. 

In math words, the number one only exist because there is the zero. And there is zero only because of the one. (Although the zero was "created" much after the integer numbers). If we couldn't distinguish between zero and one, no numbers would make sense. There is only happy because there is not-happy. There is only "self" because we define our bounds, we define what is not-me.
Likewise we can only perceive suffering if we know what is not-suffering.
So, when we say that math is empty, we don't mean that it is false or is meaningless. But we mean that is not absolutely true, that it is true in relation to something. Truth is a relation, not an absolute thing.

-1: <p>Buddhists say that everything is empty.</p>

<p>Is mathematics empty?</p>

<p>I thought that "empty" meant conceptually constructed. But how can the fact (not the expression) 2 + 3 = 5 not be a fact which is true regardless of how we are counting?</p>
 4722: <p>There is a mathematician Gödel who stated back around the 1920s using something we might imagine as "meta-math" that any axiomatic system (so any mathematical system you can create or imagine) cannot satisfy simultaneously two properties.  Which two?</p>

<ul>
<li>That the system is complete</li>
<li>That the system is consistent</li>
</ul>

<p>I shall try and state in a useful metaphor these key insights.  I like to imagine a wheel and at the center of the wheel are some basic operations like multiplication, addition, subtraction.  However, it could be anything.  We are basically trying to figure out all the infinite possible combinations of our basic fundamental rules and then see if we can make this infinite variety of conclusions both internally consistent and completely exhaustive of every possible outcome.</p>

<p>This is like trying to build a house with some basic pieces: a roof, a chimney, a floor, a basement, a fridge, a garage, a kitchen.  Now say you try every combination of possibilities such that the fridge ends up on top of the place and the roof ends up on the ground.  This is us trying to make all the exhaustive combinations possible, but in doing so we contradict the terms we have used: fridge is now roof and roof is now floor.  </p>

<p>In short: if we tried to make our system demonstrably "complete" then we would invariably end up with a plethora of contradictions.  That is, we could say 1 = 1 and 2 = 2 but once we start saying 2 = 1 and 1  = 2 to get "completeness" then we have contradiction (non-consistency)</p>

<p>Likewise, if we want to have consistency (1 = 1 and 2 = 2 and 5 = 5 and 4-3 = 1) we cannot have completeness guaranteed, because we omit terms and therefore don't know if there is a true term we have not yet discovered.</p>

<p>So the basic idea of Gödel's incompleteness and completeness theorems are that mathematics cannot be both <em>consistent</em> (or coherent) and <em>complete</em> which means that mathematics, is in fact, empty.</p>

<p>Any postulate is empty of its own being because it is necessarily dependent on other terms to relatively define its value/position/significance.  </p>

<p>Any postulate is also empty of an absolutely true nature because there is no way to have absolutely true consistency, for that would require completeness, and Gödel showed that you can only have one or the other, like a scrambled egg or an unbroken egg, you cannot have both unless you sacrifice consistency (the egg is both unbroken and scrambled?) or completeness (just one or the other).</p>

<p>So in short, all mathematical systems must abide as either incomplete or inconsistent which is [another reason] why mathematics in itself is not graspable and not inherent.</p>

<p>This, however, is a long way to arrive at the basic truth that all phenomena are empty of their own being (they have inter-being as Thich Naht Hanh says).</p>
 254: <p>In 'pure' mathematics, the truth of an expression like <code>2+3=5</code> depends on the definition of the addition operator and the definitions of the numbers such as <code>1</code> and <code>2</code> and <code>3</code> and so on. I think these definitions are 'axiomatic'.</p>

<p>In 'applied' mathematics e.g.when you're adding physical objects (such as apples), then the truth depends on your ability to identity "one apple" (which isn't obvious: in the real world, some apples are smaller, some are attached to the tree, some are rotting, etc.).</p>

<p>Furthermore mathematics is something which you do (sometimes) and which you stop doing: therefore IMO it's conditioned and impermanent.</p>

<hr>

<p>More generally I think it's a category of knowledge. Here's a (slightly rude or brusque or plainspoken) quote from <em>Zen and the Art of Motorcycle Maintenance</em> -- this quote is talking about Aristotle's "dialectic" but it could IMO equally be talking about "mathematics":</p>

<blockquote>
  <p>"As best I know, Aristotle's opinion is that dialectic comes before
  everything else."</p>
  
  <p>"And from the dialectic come the forms," Phædrus continues, "and from. -
  - " But the Chairman cuts it off. He sees it cannot go his way and dismisses
  it.</p>
  
  <p>He shouldn't have cut it off, Phædrus thinks to himself. Were he a real
  Truth-seeker and not a propagandist for a particular point of view he would
  not. He might learn something. Once it's stated that "the dialectic comes
  before anything else," this statement itself becomes a dialectical entity,
  subject to dialectical question.</p>
  
  <p>Phædrus would have asked, What evidence do we have that the dialectical
  question-and-answer method of arriving at truth comes before anything
  else? We have none whatsoever. And when the statement is isolated and
  itself subject to scrutiny it becomes patently ridiculous. Here is this dialectic,
  like Newton's law of gravity, just sitting by itself in the middle of nowhere,
  giving birth to the universe, hey? It's asinine.</p>
  
  <p>Dialectic, which is the parent of logic, came itself from rhetoric. Rhetoric is
  in turn the child of the myths and poetry of ancient Greece. That is so
  historically, and that is so by any application of common sense. The poetry
  and the myths are the response of a prehistoric people to the universe
  around them made on the basis of [etc.]</p>
</blockquote>
 37: <p>Emptiness in eastern philosophies usually have a more subtle meaning than we use in the west. Check this <a href="http://en.wikipedia.org/wiki/Emptiness" rel="nofollow noreferrer">wikipedia article on Emptiness</a>. And there are many answers regarding emptiness (<em>sunyata</em>) like <a href="https://buddhism.stackexchange.com/questions/3836/are-there-different-types-of-emptiness-%C5%9A%C5%ABnyat%C4%81">this one</a>. There is also an article in wikipedia on <a href="http://en.wikipedia.org/wiki/%C5%9A%C5%ABnyat%C4%81" rel="nofollow noreferrer">Śūnyatā</a>.</p>

<p>The most common understanding on emptiness (on Buddhism) is that it is <a href="http://en.wikipedia.org/wiki/Prat%C4%ABtyasamutp%C4%81da" rel="nofollow noreferrer">dependently originated</a>. It means that it does not have inherent meaning or value, but it relies on other things. In mathematics the numbers two, three, five and so on just has meaning because of the number one. And also, the number one only has meaning relying on the other numbers.</p>

<p>In a more practical sense, numbers only has meaning because of the objects, like apples, rocks or sheep. Now, emotions, that have no clear bounds, can't be numerically described. You could say that you felt happy twice today, but you can't express its intensity and form numerically. Then math is true not because it has inherent meaning, but because it works for what it is used.</p>

<p>There is another interpretation I read on <a href="https://buddhism.stackexchange.com/questions/7119/how-does-buddhism-refute-the-inherent-existence-of-sub-atomic-particles">this QA</a> (and also in some wikipedia article but I couldn't find it):</p>

<blockquote>
  <p>[Emptiness is] a flashy way of reminding us that we only define things in relation to what they are not. </p>
</blockquote>

<p>In math words, the number one only exist because there is the zero. And there is zero only because of the one. (Although the zero was "created" much after the integer numbers). If we couldn't distinguish between zero and one, no numbers would make sense. There is only happy because there is not-happy. There is only "self" because we define our bounds, we define what is not-me.</p>

<p>Likewise we can only perceive suffering if we know what is not-suffering.</p>

<p>So, when we say that math is empty, we don't mean that it is false or is meaningless. But we mean that is not absolutely true, that it is true in relation to something. Truth is a relation, not an absolute thing.</p>
