Instigator / Pro
0
1500
rating
8
debates
50.0%
won
Topic
#4381

# Math is objective.

Status
Finished

The debate is finished. The distribution of the voting points and the winner are presented below.

Winner & statistics
Winner
0
2

ComputerNerd
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Last updated date
Type
Standard
Number of rounds
5
Time for argument
Two days
Max argument characters
10,000
Voting period
Two weeks
Point system
Winner selection
Voting system
Open
Contender / Con
2
1518
rating
15
debates
40.0%
won
Description

The nature of mathematics is a long debated question. What is math? What is mathematical truth? Is math invented or discovered? The questions are endless. This debate will be on one particular such question: Is math objective, existing independently from the physical world, or is it purely a human construct, helpful for solving real-world problems, but meaningless beyond that? Naturally, as pro, I argue that math is indeed objective. You, as con, will argue that it is not objective.

Round 1
Pro
#1
Modern mathematics rests upon the foundation of set theory. Everything from numbers to shapes is defined using sets. This is backed up by these sources: https://brilliant.org/wiki/set-theory/https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/set-theory. With that in mind, let's focus on set theory. Modern set theory is based upon the ZFC axioms. These nine simple axioms are the foundation for all of modern mathematics. With this, we see that every imaginable mathematical statement falls under one of three categories:

1. True
2. False
3. Independent of ZFC

It is true if it is provable from the axioms of ZFC. It is false if it is refutable from the axioms of ZFC. Lastly, it is independent of ZFC if it can neither be proved nor refuted, and thus there isn't a meaningful way of assigning a truth value to it. An example of this third and stranger category is the continuum hypothesis, which is a statement regarding the existence of a particular kind of set. (I want to keep this friendly for people who aren't familiar with topics in advanced mathematics, so I am omitting the specific statement of the continuum hypothesis. For anyone who is curious, go to this link: https://plato.stanford.edu/entries/continuum-hypothesis/) On the one hand, there is nothing stopping such a set from existing, but on the other, none can be explicitly constructed, so it is not meaningful to say that it is true or false, but rather it is independent of ZFC. Now where am I going with all of this? Well, consider the following:

If something is merely a human concept, it does not exist until it is created by humans.

The truth value of the Riemann zeta hypothesis is part of math.

The Riemann zeta hypothesis has an existing truth value that is "true," "false," or "independent of ZFC."

The truth value of the Riemann zeta hypothesis has not been created by humans.

Math, therefore, cannot be merely a human construct, as if it was, the truth value of the Riemann zeta hypothesis would have yet to exist.

Moreover, to demonstrate fully that math is objective, consider this:

Every mathematical statement has a fixed truth value (true, false, or independent of ZFC).

This truth value is not determined by humans, or any other subjective determining factor.

Math is therefore itself objective.
Con
#2
Thank you Math_Enthusiaist,

TOPIC: Math is objective.

As PRO has failed to do as such, I will be defining the definitions in play:

Math: the abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics)
Objective: (of a person or their judgement) not influenced by personal feelings or opinions in considering and representing facts.

CON's goal in this debate is not to prove that Math is subjective, but to disprove that Math is objective. Therefore, a constructive is not necessary.

PRO's first assumption: A fallacy of composition

Fairly simple mistake, but a big one. PRO assumes that if the Riemann Zeta Hypothesis had an objectively correct answer, all of math must be objective. However, this does not follow. If I stated that one thing on Earth is a car, that does not mean Earth is a car. It means part of Earth is a car. Therefore, one cannot make assumptions on a whole system based on one single example.

PRO's second assumption: Math is objective

The argument already assumes that math is objective to prove that math is objective.

In his argument, PRO states:

The truth value of the Riemann zeta hypothesis is part of math.
and then proceeds to use the existence of this truth value as an argument for the objectivity of math. This is an example of circular reasoning, where one uses a conclusion as the premise to support itself.

PRO is assuming that the truth value of the Riemann zeta hypothesis exists independently of human knowledge or discovery, which is exactly what he is trying to prove. PRO is taking for granted that math is objective, and then using that assumption to argue that math is objective. This does not show why or how math is objective, but only repeats the claim without any support.

PRO's third assumption: Every statement has a fixed truth value.

Within the set theory and classical logic PRO employs throughout his argument, there exists certain statements and contradictions which arise, where there aren't clearly defined truth values.

For example, consider the following statement:

This statement is false.

This is a well-known paradox, and it does not have a clearly defined truth value. If it was true, the statement must be false. If it was false, it would be inherently true. Since it can't be measured in terms of truth or falsity, it falls outside the scope of ZFC. Therefore, it does not have a fixed truth value.

PRO's fourth assumption: Set theory is the only valid set of axioms.

All of PRO's constructive is based off of set theory, and the ZFC axioms. However, there are alternate sets which can be used, which may lead to different answers depending on which set you use. For example, Euclidean geometry is the most well-known system of geometry, based on five axioms that describe the properties of points, lines, and angles. However, non-Euclidean geometry,  rejects the parallel postulate and allows for curved spaces. These systems are not less valid or less mathematical than Euclidean geometry; they simply explore different aspects and consequences of different assumptions.

Similarly, there are other systems of logic and set theory that use different axioms than ZFC, such as intuitionistic logic, which rejects the law of excluded middle and only accepts constructive proofs; or ZF without the axiom of choice, which allows for the existence of non-well-orderable sets; or NFU, which is a variant of Quine’s New Foundations that allows for a universal set. These systems are not less valid or less mathematical than ZFC; they simply explore different aspects and consequences of different assumptions.

Conclusion:
PRO's case was heavily fallacious and ignored other possible sets of axioms. Therefore, PRO has failed to decisively prove that Math is objective.

Thank you, I await your response.

ComputerNerd
Round 2
Pro
#3
PRO's first assumption: A fallacy of composition

Fairly simple mistake, but a big one. PRO assumes that if the Riemann Zeta Hypothesis had an objectively correct answer, all of math must be objective. However, this does not follow. If I stated that one thing on Earth is a car, that does not mean Earth is a car. It means part of Earth is a car. Therefore, one cannot make assumptions on a whole system based on one single example.
This is somewhat of a moot point, since I then went on to say this:
Moreover, to demonstrate fully that math is objective, consider this:

Every mathematical statement has a fixed truth value (true, false, or independent of ZFC).

This truth value is not determined by humans, or any other subjective determining factor.

Math is therefore itself objective.
I will also comment that every mathematical fact is an assertion of mathematical truth, so if I can show that mathematical truth is objective, then I will have in fact shown that math is objective.

PRO's second assumption: Math is objective

The argument already assumes that math is objective to prove that math is objective.

In his argument, PRO states:

The truth value of the Riemann zeta hypothesis is part of math.
and then proceeds to use the existence of this truth value as an argument for the objectivity of math. This is an example of circular reasoning, where one uses a conclusion as the premise to support itself.

PRO is assuming that the truth value of the Riemann zeta hypothesis exists independently of human knowledge or discovery, which is exactly what he is trying to prove. PRO is taking for granted that math is objective, and then using that assumption to argue that math is objective. This does not show why or how math is objective, but only repeats the claim without any support.
The axioms of ZFC are subject to basic rules of deduction. (Those of propositional logic.)

For any given mathematical statement, one of a few things are true:

1. It is possible to deduce this statement from ZFC.
2. It is possible to deduce the negation of this statement from ZFC.
3. It is not possible to do either of these things.

This is an objective question. To demonstrate this, let me demonstrate that it is not up to human interpretation what counts as a proof of a statement, and thus that there either objectively is one, or objectively isn't one:

For a given supposed proof, either every step follows from the last or from one of the axioms by direct application of one of the axioms of propositional logic, or this is not the case.  Commonly, mathematical proofs neglect to do this precisely, but any valid proof should be reducible to such a thing. Now, there either is such a proof or there isn't one. When there is one, then it is said that it is possible to deduce this statement from ZFC.

With that out of the way the three possibilities listed above can assign a truth value to any statement out of these three possibilities:

1. True.
2. False.
3. Independent of ZFC.
PRO's third assumption: Every statement has a fixed truth value.

Within the set theory and classical logic PRO employs throughout his argument, there exists certain statements and contradictions which arise, where there aren't clearly defined truth values.

For example, consider the following statement:

This statement is false.

This is a well-known paradox, and it does not have a clearly defined truth value. If it was true, the statement must be false. If it was false, it would be inherently true. Since it can't be measured in terms of truth or falsity, it falls outside the scope of ZFC. Therefore, it does not have a fixed truth value.
It does indeed fall outside of the scope of ZFC, but it is also not within set theory as con claims. Nonetheless, if it did fall within the scope of set theory, it would mean math was inconsistent. This wouldn't make it any less objective, it would simply mean that every statement has two truth values: True and false. The reason for this is that if there is indeed a contradiction lurking within our current model of math, it would allow every statement to be proven via argumentum ad absurdum, which includes every statement's negation.
PRO's fourth assumption: Set theory is the only valid set of axioms.

All of PRO's constructive is based off of set theory, and the ZFC axioms. However, there are alternate sets which can be used, which may lead to different answers depending on which set you use. For example, Euclidean geometry is the most well-known system of geometry, based on five axioms that describe the properties of points, lines, and angles. However, non-Euclidean geometry,  rejects the parallel postulate and allows for curved spaces. These systems are not less valid or less mathematical than Euclidean geometry; they simply explore different aspects and consequences of different assumptions.

Similarly, there are other systems of logic and set theory that use different axioms than ZFC, such as intuitionistic logic, which rejects the law of excluded middle and only accepts constructive proofs; or ZF without the axiom of choice, which allows for the existence of non-well-orderable sets; or NFU, which is a variant of Quine’s New Foundations that allows for a universal set. These systems are not less valid or less mathematical than ZFC; they simply explore different aspects and consequences of different assumptions.
ZFC is the basis for modern mathematics, and Euclidean geometry as well as non-Euclidean geometry can all be found within ZFC, modeled by the plane (Euclidean geometry) or some other manifold (non-Euclidean geometry). Nonetheless, if con demands, that I favour no one set of axioms over the others, it is possible to replace every instance of "ZFC" thus far with "RAS," (as in Random Axiom System) which can act as a placeholder for any set of axioms. Observe that except when I have made mention of the continuum hypothesis (which was merely as an example of what the "truth value" of independence from ZFC meant) or the Riemann hypothesis (which you may now consider removed from my argument, as it wasn't technically necessary in the first place) I have made no use of results of this particular set of axioms. I will thus write "RAS" instead of "ZFC" from this point on so as to make it clear that this is not specific to ZFC. The only requirement is that the axioms of RAS are well-defined, but beyond that, there are absolutely no restrictions.
Con
#4
Thank you Math_Enthusiast,

TOPIC: Math is Objective

Arguments:

#1: Math isn't just mathematical truths.
In R2, my opponent assumed:
I will also comment that every mathematical fact is an assertion of mathematical truth, so if I can show that mathematical truth is objective, then I will have in fact shown that math is objective.
That isn't so clear cut. If we are to define Math as a collection of objective mathematical truths then, yes, Math would be objective (of course assuming you could prove that). However, PRO needs to consider the fact that Math doesn't just include these truths. Math is composed of many other things, including mathematical practices, applications, values and methods. There is also a philosophical approach to math, covering the nature and validity of mathematical concepts. Additionally, math has an extensive history of discoveries. It would be unfair to simply blanket all of this under mathematical truths, so therefore, no, even if mathematical truth was objective, math is not inherently objective.

There is a split among philosophers among this very topic, with two major sides. One is the view I suppose my opponent would concur with, known as mathematical realism, or Platonism, after the philosopher Plato. This view states that Math is the study of abstract objects that are independent of our minds, and that a mathematical statement is objectively true or false depending on if they describe these objects accurately. The other is one I myself agree with, known as anti-realism, or constructivism. This view states that math is more than this collection of truths, but is in practice a human activity, involving practices, applications, values and methods that are influenced by a variety of cultural or historical factors.

In fact, this debate has been fought for millennia, and it has become increasingly clear that there are definitely arguments for both sides. No conclusion can be drawn definitively, as it is rare for philosophers to agree on anything, such is their science. Therefore, it is practically impossible to prove that Math is objective, without assuming the falsity of constructivism, which has withstood attacks for a long time.

So, as a result, it is technically impossible for PRO to win this with only mathematical statements. It has become increasingly clear to me that my opponent is definitely sound in mathematical arguments, but you cannot win this debate without venturing into philosophy, where there is NO answer.

I should win on this alone, but I will additionally address some of my opponent's mathematical arguments.

#2: An inconsistent math is a subjective one.

PRO acknowledges:

It does indeed fall outside of the scope of ZFC, but it is also not within set theory as con claims. Nonetheless, if it did fall within the scope of set theory, it would mean math was inconsistent. This wouldn't make it any less objective, it would simply mean that every statement has two truth values: True and false. The reason for this is that if there is indeed a contradiction lurking within our current model of math, it would allow every statement to be proven via argumentum ad absurdum, which includes every statement's negation.
I apologize for the oversight. As PRO rightly mentioned, my statement did not fall under set theory. So, I will be presenting another one which DOES indeed fall under set theory and is outside of the scope of ZFC.

The continuum hypothesis (CH), states that there is no set whose cardinality is strictly between that of the natural numbers and real numbers. In Godel's constructible universe model, there is a subclass of sets called L, which satisfies ZFC and the Continuum hypothesis. However, in Cohen's forcing, there is a way to add new subsets of natural numbers to this universe of sets, preserving ZFC and nullifying the Continuum hypothesis. Depending on the choice of model, CH provides different truth values, and CH falls under set theory.

Both of these models are valid. But our decision on which one to apply influences our answer. If it was truly objective, there would be one answer, and other answers would have flaws. Therefore, it follows that, since this exists, math cannot be said to be truly objective.

I will drop the final point, as it is a mere adjustment rather than an argument.

CONCLUSION:

• Math includes philosophy, where an answer can never be truly found.
• PRO cannot decisively prove that Math is objective.
Thank you, I await your response.
ComputerNerd

Some questions to clear up confusion:

- What is the BOP?
- What is the scope of this debate?
- What version of Platonism are you defending?

Round 3
Pro
#5
I apologize, as it has become clear to me that I was not sufficiently clear on the topic and scope of this debate. For this reason I am conceding. I thank you for engaging me in this debate. Perhaps I will try this again in the future with the guidelines set up more clearly.
Con
#6
I accept and appreciate the concession.

VOTE CON
Round 4
Pro
#7
Waived. (Concession.)
Con
#8
PRO has forfeited. VOTE CON.
Round 5
Pro
#9
Waived. (Concession.)
Con
#10
PRO has forfeited. VOTE CON.