Instigator / Pro
0
1500
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Topic
#4480

# Math is objective

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Finished

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

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It's a tie!
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Rated
Number of rounds
5
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Three days
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Voting period
One month
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1,450
Contender / Con
0
1493
rating
24
debates
62.5%
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Description

This is going to be a debate on modern pure mathematics. Specifically, this debate will focus on the objectivity of mathematical truth. As pro, I will be arguing that math is indeed objective. That mathematical truth is what it is independent of human interpretation and preference. You as con, are to argue that math is merely a subjective, human construct. Please put any questions on the debate topic in the comments.

Round 1
Pro
#1
Modern math is centered around formal systems. It will be important to be familiar with formal systems. While the explanation given by the website I have linked to may look daunting, only two key concepts will be necessary. The language of a formal system, and the deductive system, which consists of both deduction rules, and the set of axioms. The majority of modern mathematics is based upon the axioms of ZFC. Areas like proof theory, however, do investigate formal systems more generally, so it will be important for me to fulfill my BoP for any such formal system. Suppose we are considering a formal system, which I will call F. I intend to demonstrate that no matter what F is, even if it is inconsistent, math, as based upon F, is objective. It is important to note that the axioms of F are never themselves considered to be objective, but rather math studies what we can deduce from these axioms, under the deduction rules of F. Now, for a particular proposition in the language of F, which I will call P, there are only two possibilities:

1. P is a theorem in F. That is, using only the deduction rules of F, we can deduce P from the axioms of F.
2. P is not a theorem in F. That is, not such thing is possible.

Every proposition P in the language of F will necessarily fall into one of these to categories. Either P is a theorem in F, or it isn't. This is independent of whether or not we as humans have or ever will prove this. Even if F is inconsistent, this does not mean that what is and isn't a theorem in F becomes subjective. It just means that it will not be particularly useful. Usefulness is, however, outside of the scope of mathematical truth. F will not even necessarily contain the logical connectives we are familiar with, and if it does, they are no different then anything else in F: They are subject to the same deductive system, regardless of our human intuition about those logical connectives.
Con
#2
Reading into math has been interesting.  Thanks for the debate topic and chance to talk about it.

I want to make sure I understand what you are presenting.
1. Questions
A ) Is formal system objective?

B)  What would you say is the "language" refered to in definition/description provided for formal systems, and would this language be objective?

C) The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following (Jech 1997, p. 1),
stands for exists,  means for all,  stands for "is an element of,"  for the empty set,  for implies,  for AND,  for OR, and  for "is equivalent to."

A)  Are these symbols and meanings used through out the world? Do you know what existed or was used before these symbols were promoted by Ermelo?

B) there are and/or have been different formulas & methods used to find the same answer. A common running joke in media is a parent trying to help their kid do math but realizing the process to find the answer is different.

Ancient societies like methods used by Byzantines had similar mathmatical processes to today's standard. Would a change in formula or method to determine an answer demonstrate subjectivity in math because the  formulas/methods in doing math change?

In your given presentation for F  and P. You give only two options. These options are
i p is in f
ii p is not in f

Is this accurate depiction for your presentation?

3.  Is math objective?
Let us do math to see if it is.

2 + 2 = 4 ..  yes? Can only be 4 then. That's objective.

Here is a good one.
0 = 0
Lets add some numerical values so zero is zero .

4-4 = 10-10
We can rewrite this as a product of 2

2*2 & 2*5
Lets put it into a solving equation.

(2-2)(2+2)= 5(2-2).
Interesting enough. What if we cancel (2-2) from each side?

We get
(2+2)=5

Ah! Now we see it.
2+2 = 5
Although we have proven 2+2 is 5, I am sure we are also thinking 2+2 is only 4. Here we can see the subjectivity in math because our own understanding for 2+2 is going into conflicting ways.  We know 4 is not 5 but at the same time a formula to get 4 can get 5!

If a thing is objective then paradoxes such as this should not exist. The mere presence shows us subjectivity

Round 2
Pro
#3
Reading into math has been interesting.  Thanks for the debate topic and chance to talk about it.
Thank you for accepting. I hope we will have an interesting debate.

A ) Is formal system objective?
I'm not entirely sure how to interpret this. Is there one objectively "true" formal system? No, but math does not consider that a meaningful question, so this does not function as a valid counterargument. Is it objective which statements are and are not theorems in a formal system? Certainly.

B)  What would you say is the "language" refered to in definition/description provided for formal systems, and would this language be objective?
A language, in this context, is just a set of symbols. It doesn't really make sense to ask about the objectivity of such a thing. As stated in the description, this debate will focus on mathematical truth.

C) The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following (Jech 1997, p. 1),
stands for exists,  means for all,  stands for "is an element of,"  for the empty set,  for implies,  for AND,  for OR, and  for "is equivalent to."
I am not sure what the question is. ZFC is the most commonly studied formal system, and it is a formal system like any other: It has a language, and a deductive system.

A)  Are these symbols and meanings used through out the world? Do you know what existed or was used before these symbols were promoted by Ermelo?
The symbols are beside the point. Mathematicians consider the symbols to simply be a way of representing the underlying mathematics. The term "symbol" is a bit misleading when used in reference to the language of a formal system. If we use a different physical symbol, it will still be considered the same formal system, so long as it has the same meaning.

B) there are and/or have been different formulas & methods used to find the same answer. A common running joke in media is a parent trying to help their kid do math but realizing the process to find the answer is different.
Yes. It is possible for there to be more than one objectively accurate method. For example, in the physical, there may be more than one way to bake a cake. That doesn't mean that only one actually makes a cake, it just means that there is more than one way to make a cake. It may be true that what counts as a cake is not objective, but it is not a perfect analogy, and hopefully you can still understand what I'm getting at.

Ancient societies like methods used by Byzantines had similar mathmatical processes to today's standard. Would a change in formula or method to determine an answer demonstrate subjectivity in math because the  formulas/methods in doing math change?
As I explained above, it would not.

In your given presentation for F  and P. You give only two options. These options are
i p is in f
ii p is not in f

Is this accurate depiction for your presentation?
No, it is not. P is necessarily a proposition in the language of F. The two possibilities I presented were P is a theorem in F, or P is not a theorem in F. That is, either P can be proven using the deductive system of F, or it can't.

2 + 2 = 4 ..  yes? Can only be 4 then. That's objective.
I agree, 2 + 2 = 4. It can only be 4.

Here is a good one.
0 = 0
Lets add some numerical values so zero is zero .

4-4 = 10-10
We can rewrite this as a product of 2

2*2 & 2*5
Lets put it into a solving equation.

(2-2)(2+2)= 5(2-2).
Interesting enough. What if we cancel (2-2) from each side?

We get
(2+2)=5
This is rather simple to debunk. Let's take a look at this step:

(2-2)(2+2)= 5(2-2).
Interesting enough. What if we cancel (2-2) from each side?

We get
(2+2)=5
(2-2)(2+2)=5(2-2) is correct. Both sides are zero. You then attempt to apply the cancelation property: If ab=ac than b=c whenever a is not equal to 0. Unfortunately for you, your use of it is invalid, as you attempted to cancel 2-2 from both sides, which is equal to 0. Now, I could stop right here, but for any who feel that this is just mathematicians putting arbitrary rules in place, here is where the cancelation property comes from:

Suppose ab=ac. Then we subtract ac from both sides: ab-ac=0. Now we apply the distributive property: a(b-c)=ab-ac=0. So we see that a(b-c)=0. When two things multiply to zero, one of them must itself be equal to zero, so either a=0 or b-c=0. At this point we can conclude that if a is not equal to 0, than b-c must equal 0, and so b=c. This, however, requires that a be non-zero, as if a=0, than there is no need for b-c to be equal to zero, and so b is not necessarily equal to c.
Con
#4
I just realized I made error in organizing my post. Will correct what I can.

1. Questions
A. Well, maybe I should say I consider your introduction to mean we are looking at modern math which includes formal systems. Math has many parts to it, we are looking at a part. Am I incorrect with this understanding?

So I was asking if all parts of formal systems pertain objectivity to it.  My follow up is to look at what is meant by "theoretical organization of terms"

Are we supposed to look at formal systems or parts of it (like language which is the symbols, or other parts) as a hypothetical set of facts that can be proven/tested/observed, but in some way still open to future discovery?

In science, theories are open to change when applicable, does math operate in the same way?

B. A language, in this context, is just a set of symbols. It doesn't really make sense to ask about the objectivity of such a thing. As stated in the description, this debate will focus on mathematical truth.

Language evolves, sometimes changing meaning or other aspects that can alter a series of premises and conclusion. Or at least our ability to understand it. Which in this case would change mathematical truth. Worth a question.

I will follow up with this consideration:

I get to school at 7 o’clock in the morning. A clock may show hour and minute hands. The smaller hand representing hours is pointing at 7.

I leave at 5pm later that day. Looking at te same clock as before,  this will be the second time the hour hand points at 5 on the clock's face. The first 5 AM.

Although there are 24 hours in a day, we know older clock styles only show hours 1 to 12. These clocks use a system (1 to 12) to represent the 24 hour clock. We can say 1PM is 1300 hours; 2 PM is 14 PM or 5 PM is 17 o’clock.

We’re can identify the watch’s system “Modulo 12”, meaning we only use 1 to 12.

Therefore we need to change how we look at 17. For those who do not know we can subtract 12 from 17 to get 5.
5  in our modulo system is understood to mean 5 or 17. We can now say:
12 + 5 = 5.

The modulo or modular system used is developed and may change. Yes?

2. Which Previously was C.
I am not sure what the question is?

No question, just error. I copy & paste from provided link. The symbols did not transfer. the following question A. refered to the symbols within this

A. Mathematicians consider the symbols to simply be a way of representing the underlying mathematics. The term "symbol" is a bit misleading when used in reference to the language of a formal system. If we use a different physical symbol, it will still be considered the same formal system, so long as it has the same meaning.

Makes sense. How do we know the meaning or ensure we have the same meaning?

B) Yes. It is possible for there to be more than one objectively accurate method ..
Hm. I contemplate on this for now.

C). The two possibilities I presented were P is a theorem in F, or P is not a theorem in F. That is, either P can be proven using the deductive system of F, or it can't.
Why would those be the only possibilities?

3. (2-2)(2+2)=5(2-2) is correct. Both sides are zero. You then attempt to apply the cancelation property: If ab=ac than b=c whenever a is not equal to 0. Unfortunately for you, your use of it is invalid, as you attempted to cancel 2-2 from both sides, which is equal to 0.

Ah. So if (2-2) did not equal 0 (zero) then conclusion would have been correct. Then I could put in any equation with any numbers. If at any moment I attempt to cancel out numbers that could equal zero from each side, this is automatically incorrect - this is what you are saying?

Even if we look at equation 4-4 = 5-5
Use distribution property.
4(1-1)= 5(1-1)
Cancel (1-1)
To get
4=5
You would say this is incorrect?

Why is this the case?

4.  If we use a modulo system containing restricted numbers similar to above clock example, we can develop equation 2+2= 1. 1 represents 4 in this system.

In what way would you say "language" is being used in this mathematic equation?
What would demonstrate as an example wat "language" is or includes?

The modulo or modular system provide is not commonly used where as some may or some may not. Without explanation or context, would the equation be viewed as incorrect?

5.  Can you give an equation as an example for your position?

6. Where do we get information about math, equations, etc. from? If this is something objective, and possibly outside of us, how would we deduce what is true?

Round 3
Pro
#5
I will need to be brief, since I had almost forgotten about this, and I am now short on time. Let's tackle three main things:

Modular arithmetic:

If we use a modulo system containing restricted numbers similar to above clock example, we can develop equation 2+2= 1. 1 represents 4 in this system.
This seeming subjectivity is simply because of a flaw in notation. The "2" in mod 3 arithmetic is different from the "2" in standard arithmetic. We just use the same notation for each.

Cancelation:

Even if we look at equation 4-4 = 5-5
Use distribution property.
4(1-1)= 5(1-1)
Cancel (1-1)
To get
4=5
You would say this is incorrect?

Why is this the case?
1-1 is equal to zero just like 2-2. You cancelation property does not apply to zero. Zero is zero no matter how you write it.

Language:

You seem to be a little confused about what a language is. It is simply the set of symbols used in a formal system. No more, no less. Also, when I said as long as they have the same "meaning" they are the same, I meant that the rules of deduction are the same, just with the symbols swapped out. The logic will always work the same, regardless of our chosen physical symbols.
Con
#6
1-1 is equal to zero just like 2-2. You cancelation property does not apply to zero. Zero is zero no matter how you write it.

Why?

This seems to be the accepted method, but is there information into why cancelation property does not apply to zero?

How do we know if this is not a long line of circular reasoning?
Round 4
Pro
#7
1-1 is equal to zero just like 2-2. You cancelation property does not apply to zero. Zero is zero no matter how you write it.

Why?

This seems to be the accepted method, but is there information into why cancelation property does not apply to zero?

How do we know if this is not a long line of circular reasoning?
I answered this in round 2:

Unfortunately for you, your use of it is invalid, as you attempted to cancel 2-2 from both sides, which is equal to 0. Now, I could stop right here, but for any who feel that this is just mathematicians putting arbitrary rules in place, here is where the cancelation property comes from:

Suppose ab=ac. Then we subtract ac from both sides: ab-ac=0. Now we apply the distributive property: a(b-c)=ab-ac=0. So we see that a(b-c)=0. When two things multiply to zero, one of them must itself be equal to zero, so either a=0 or b-c=0. At this point we can conclude that if a is not equal to 0, than b-c must equal 0, and so b=c. This, however, requires that a be non-zero, as if a=0, than there is no need for b-c to be equal to zero, and so b is not necessarily equal to c.

Con
#8
Unfortunately for you, your use of it is invalid, as you attempted to cancel 2-2 from both sides, which is equal to 0. Now, I could stop right here, but for any who feel that this is just mathematicians putting arbitrary rules in place, here is where the cancelation property comes from:

Suppose ab=ac. Then we subtract ac from both sides: ab-ac=0. Now we apply the distributive property: a(b-c)=ab-ac=0. So we see that a(b-c)=0. When two things multiply to zero, one of them must itself be equal to zero, so either a=0 or b-c=0. At this point we can conclude that if a is not equal to 0, than b-c must equal 0, and so b=c. This, however, requires that a be non-zero, as if a=0, than there is no need for b-c to be equal to zero, and so b is not necessarily equal to c.
See this is what I mean. My question is why not how. Providing text book explaination goves us how the cancelation process occurs, but does not give us why it occurs in that manner.

Some years ago, Mr. X developed the cancelation process. For reasons unexplained, we have continued to teach the same process since. I've taken math. We all have taken math. We know how math class works.

An individual stands infront of the room to tell us how things are done. Try anything outside the methods provided and we get negative scored on homework and tests. Who else remembers having to show our work on a seperate sheet of paper for tests/quizes?

Requested to ensure our work is done the exact same way. This unity is a blessing in different ways, ensuring universal practices by neglecting and punshing out of the box thinking.

That's what really makes this debate easy for the pro. We are all taught the same way to do the same thing. How can we think and accept a different method or truth if we have been conditioned to be bias?

We can see this bias in this debate. Why is answered as how. If we were talking about morals or universal laws that can be demonstrated by science, then we can examine the "why's."

Consider debates on morality for example. The person arguing for objective morality can produce examples, reasonings, history, and more. A debator providing any thing less, like how a moral is explained, would be considered to have a weak position.

Here now, pro is providing excallent explaination on how we do cancelation process, but does not provide evidence as to why. The method suggested can easily be interchanged with method I suggest. Only our bias is telling us no.

Round 5
Pro
#9
Your complaint seems to be that we are simply told "cancelation works!" and the why is never explained. The proof that I provided explains exactly why it works. It is a completely rigorous derivation of this incredibly valuable principle. On that note, if you want to know why it is used, that is about all there is to it. It is known to work as per the proof I provided, and it is useful. I see no need to continue this argument, as your main counterargument has failed. I have provided a step by step proof of the cancelation property, including an explanation of why it does not work for a = 0.
Con
#10
Pro does not see the need to continue this argument/debate during the very last post of the debate...

Yeah, that makes sense. We are at the end of the debate.

Pro fails to notice that my " main counterargument" is an unanswered question.

Simply put, the step by step proof is a copy & paste definition that does not explain why. Instead it just repeats the notion "do it this way, not that way"

That is the pro's argument for "math is objective." Math is X, everyone does math by X. Why? Because math is X and everyone does math by X. Paraphrasing, I hope you don't mind.

For example,
I performed cancelation process to a subjective perspective to receive a reasonable and subjective solution.  Pro's response? "Do it this way, not that way "
Again this is not step by step proof. Not proof.

Since pro has repeated themselves, I can reference round 4 as viable response.  I will save the audience time rereading something in a different way.

Thank you everyone for taking time to read and for pro to provide this debate.

I am not a crook. Vote for N- uh no I mean vote con. For me.
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Yes...