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Mathematical Realism (PRO) vs. Mathematical Anti-Realism (CON)

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Voting Style: Open Point System: 7 Point
Started: 3/12/2015 Category: Philosophy
Updated: 3 years ago Status: Post Voting Period
Viewed: 1,377 times Debate No: 71508
Debate Rounds (4)
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I intended this debate to be Realism (more specificly Platonism) vs Nominalism, but later I decided it to be more general, and broad.

1. Rules
(1) - No arguments in first round, but Con should give some defintions, and explanations (also can give some examples).
(2) - Con doesn't have to argue for whole anti-realism, (s)he can choose a specific form of anti-realism, and argue for it.
(3) - Sources must be cited for any quotation.
(4) - No youtube (or other) videos.
(5) - Images can be included (width:height=2:1).

2. Some Definitions, Explanations, and Examples
Mathematical Realism holds that mathematical entities exist independent of (human) mind, it includes numbers, formulaes, geometric shapes, and etc.
For example, when we ask a question, if there exists whole number between 3 and 5, and if answer is yes, that means existence of whole numbers is assumed, and if answer is no, that means 4 is not whole number.
Basically, mathematical realism says, mathematical truths are discovered, not invented.
There are generally, two views of mathematical realism - platonism and empiricism.
Mathematical platonism is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Mathematical objects are independent of intelligent agents and their language, thought, and practices.
Empiricism denies that mathematics can be known a priori. Mathematical facts is discovered by empirical research, just like facts in any of the other sciences. (Rationalists have some good arguments against it)




I accept the debate challenge and look forward to an interesting debate.

Since my opponent did not state which view they will be defending, I will present several versions of both Realism and Anti-Realism, and we can select the ones we desire.


Mathematical Realism - Subdivided:

Mathematical Platonism - there exists abstract mathematical objects, and our theories simply describe them

- our theories are true decriptions of mental objects

Mathematical Physicalism - our theories are true descriptions of non-mental physical objects

Mathematical Anti-Realism - Subdivided:

Conventionalism - mathematical sentences are analytically true; i.e. true by definition of the words in them

Game Formalism - mathematics is the result of a game of symbol manipulation

Mathematical Formalism - mathematics gives us truths about what holds in formal systems

Deductivism - mathematics gives us truths of theorems as resultant from axioms

Fictionalism - mathematics describes the properties of ideas that simply fail to refer to existent concepts

I shall argue for a sort of combined conventionalism and fictionalism, which I will state as:

"mathematical statements are descriptive of the properties of mathematical ideas by definition, but fail to refer to existent concepts"

I await which view of Realism my opponent will choose and their opening argument.


Debate Round No. 1


Mathematical realism is the view that our mathematical theories are true descriptions of some real part of the world. However, Mathematical Anti-Realism is just a view that Mathematical Realism false. [1] Mathematical Realism can be divided into two categories:
Mathematical Platonism
- the view that, there exists abstract mathematical objects (non-spatiatemporal mathematical objects, and mathematical theories provide true descriptions of such objects); [1]

Mathematical Anti-Platonism - the view that mathematical theories true descriptions of concrete objects (Psychologism and Physicalism). [1]

I am Platonist, and when I use term "mathematical realism" I generally refer to Platonism. Mathematical Platonism can be defined as the conjunction of following three theses [2]:

(a) There are mathematical objects (EXISTENCE)
(b) Mathematical objects are abstract. (ABSTRACTNESS)
(c) Mathematical objects are independent of intelligent agents and their language, thought, and practices. (INDEPENDENCE)

The Frege's argument for Platonism asserts that it is only tenable view of mathematics [3]. Frege's argument can be written as following:
(i) Most sentences accepted as mathematical theorems are true. (TRUTH)
(ii) Let S be one of such sentence.
(iii) The singular terms of the language of mathematics purport to refer to mathematical objects, and its first-order quantifiers purport to range over such objects. (CLASSICAL SEMANTICS)
(iv) By Classical Semantics, the Truth of S requires that its singular terms succeed in referring to mathematical objects.
(v) Hence there must be mathematical objects, as asserted by Existence.
Classical Semantics enjoys strong prima facie plausibility. For the language of mathematics strongly appears to have the same semantic structure as ordinary non-mathematical language.
For example, the statement "there exist prime numbers between 12 and 35" is the same as "there exist trees in forest".
Another example, if statement "there exist solution for x-2=0" is false, then there is no solution. It is a simple statement and and it can be true only if the object to which term refers exists, the solution can exist only if 2 exists (x=2 is the solution). Therefore, 2 exists. If 2 exists, then it is abstract object (it is non-spatiatemporal and independent, x=2 is objectively true). We can generalize it for all arithmetic objects.
When a random person says, there exists a prime number which is sum of cubes of two natural numbers, it is not a product of his imagination, or matter of his opinion, but rather it is just a simple fact (sum of cubes of 3 and 4 is 91).
"There exists" is one of the most common phrases in mathematical discourse, and whether one is talking about a prime number with a certain property or the solution to a differential equation, a statement about existence is a statement of fact, not a matter arbitrary choice or opinion [4]. So numbers do have some sort of objective and independence existence.
Quine's Indispensability Argument is also good argument for Platonism [5]:
(i) We should believe the theory which best accounts for our sense experience. (science, empirical method)
(ii) If we believe a theory, we must believe in its ontic commitments. (ontological commitment)
(iii) The ontic commitments of any theory are the objects over which that theory first-order quantifies.
(iv) The theory which best accounts for our sense experience first-order quantifies over mathematical objects.
(v) Therefore, we should believe that mathematical objects exist.
Basically, if you believe Newton's theory of gravitation to be true, and formula to calculate gravitational force to be correct, then you should accept the existence of mathematical objects.
Putnam's Success Argument [5]:
(i) Mathematics succeeds as the language of science.
(ii) There must be a reason for the success of mathematics as the language of science.
(iii) No positions other than realism in mathematics provide a reason.
(iv) So, realism in mathematics must be correct.
So, if we assume science is best arbiter of what exists, and we know that science depends on Mathematics. Therefore, if science says numbers exist, then numbers do exist.

[1] - Philosophy of Mathematics ( ) - Mathematical Realism and Ant-Realism by Mark Balaguer (pages 35-101)

[2] - Platonism in the Philosophy of Mathematics, Stanford Encyclopedia of Philosophy -
[3] - The Fregean argument for Platonism, Encyclopaedia Britannica -
[4] - Do Mathematical Entities Really Exist?, Lee Lady -
[5] - The Indispensability Argument in the Philosophy of Mathematics, Independent Encyclopedia of Philosophy -


I would like to begin by making two arguments against Platonism.

The Epistemological Argument

The Epistemological Argument asserts that Platonism advances a view of reality which would entail an inability to understand mathematics at all. The argument can be formulated as follows:

Premise One: Humans exist entirely within spacetime.

Premise Two: If there exist any abstract mathematical objects, they exist outside of spacetime.

Premise Three: Humans can only obtain knowledge of items that exist within spacetime.

Premise Four: If there exist any abstract mathematical objects, humans could not obtain any knowledge of them.


Premise Five: If Mathematical Platonism is correct, then human beings could not have mathematical knowledge.

Premise Six: Human beings have mathematical knowledge.


Conclusion: Mathematical Platonism is incorrect.

I do not believe that there is any objection to Premises One, Two, or Six, so I will not waste time on them. Rather, I will focus on the justification of Premises Three and Four as a whole, as these will likely be the most contested.

Human beings obtain knowledge through two means: emprical and rational. The empirical means require a usage of the senses, which are entirely based on physical phenomena. Thus, the human senses cannot be used to determine the nature of the non-spatiotemporal. When arguing on the ability of humans to perceive ideas such as those in mathematics, the discussion inevitably involves the rational faculties, so we must ask how they are involved.
The answer is that by itself, pure logic cannot deduce anything outside of pure logic. Logical axioms such as DeMorgan's Law only apply when there is something non-abstract to apply it to. Otherwise, much like formulae in physics cannot deduce anything without values being substituted in for the variables, logical axioms cannot make deductions about real-world non-abstract phenomena unless they are being applied to known facts outside of logic. Hence, logic cannot be used to derive knowledge of items beyond known spacetime unless already given knowledge about said items.

The Non-Uniqueness Argument

The Non-Uniqueness argument demonstrates that while Mathematical Platonism

Premise One: If there are any sequences of abstract objects that satisfy the axioms of Peano Arithmetic, then there are infinitely many such sequences.

Premise Two: There is nothing "metaphysically special" about any of these sequences that makes it stand out as the sequence of natural numbers.

Premise Three: There is no unique sequence of abstract objects that is the natural numbers.

Premise Four: Mathematical Platonism entails that there is a unique sequence of abstract objects that is the natural numbers.


Conclusion: Mathematical Platonism is incorrect.

The structure for the first and second premises is best summed up by Benacerraf, who argues for these ideas in terms of set theory. If we imagine infinite sequences of sets that obey a consistent change in structure concordant with the changes advanced by Peano Arithemtic's description of the natural numbers, then we can concieve of a number of sets with that sort of structure, e.g.

Ø, {Ø}, {{Ø}}, {{{Ø}}},...


Ø, {Ø {Ø}}, {Ø {Ø {Ø}}}, {Ø {Ø {Ø {Ø}}}},...

All of these sets and many more like them have similar structural features that obey the structure of the set of the natural numbers as outlined in Peano Arithmetic. Ultimately, when seeking to represent the numbers of a sequences, the contents of our sequence of sets is not important, but the structure is. As long as the structure stays true, then the sequence can be thought of as the sequence of natural numbers. Knowing this, we can see that none of these infinite sequences have any redeeming characteristic to make them be seen as the sequence of natural numbers.

I will now move on to rebutting my opponent's arguments.

Rebuttal of Frege's Argument:

This argument relies largely on a conflation between the logical sentences describing mathematics and the sentences used outside of it. While it is true that Classical Semantics does imply that all objects in statements must refer to real objects, there is no principle that excludes the referral to a conception of something. If we did not accept this, then Classical Semantics would result in absurd scenarios and deductions.
For example, consider the statement: "Santa Claus lives at the North Pole." This statement is technically true due to how we have defined Santa Claus, in that he is a fat bearded man with elves and reindeer who lives at the North Pole. Within this definition, the statement is true. Clearly, if we did not allow ourselves to refer to conceptions, then Classical Semantics would entail that this statement refers to an actual Santa Claus. Clearly it isn't, and the reason why is that we are not referring to a real Santa Claus, but merely to our conception of Santa Claus.
How does this apply to mathematics? Simply put, my stance of Fictionalism merely advocates that the objects of mathematics themselves are not refered to, however this says nothing about our conceptions of mathematical ideas. Mathematics is a system of rules that we have invented, and when we describe the properties of mathematical objects within those rules, we are describing the behavior of our conceptions. To give another example, consider Chess:
Chess, like mathematics, has logical rules and consistency to it. We can describe the behavior of objects within Chess just as in mathematics and use the same language. We can say that "There exists a piece that can move both diagonally and horizontally." However, no one is willing to imply that there exist actual abstract Pawns and Kings, and so similarly we have no reason to believe in abstract number objects either.

Rebuttal of Quine's Indispensability Argument:

The main problem with this argument is that it is a misrepresentation of what mathematics really is. As I said before, mathematics is a system of logical rules that intertwine with each other. This system happens to be the best system that humans have invented to describe the way that reality behaves, but even so, it is not really a "theory" of reality. It is more of a tool that allows us to provide an input (the observations that we have about reality) and allows us to describe other properties of reality. In the same way, computer programs that sort lists have real-word counterparts, but we have to understand how to translate them. For example:
Suppose a programmer writes a program designed to sort rubber balls by their color. The input for his code might be a list of the colors of the balls, or it might be a picture of them, or list of HEX color-codes. However, he cannot input the balls themselves. He must translate the balls into some form of information, feed that into his code, and then converts the information he recieves into a batch of sorted balls. But he can never simply give the balls to the program. He must translate them into information first.
Similarly, our mathemtical formulae that describe reality do not do so directly. We cannot simply give two stars into Newton's formula for gravitation and recieve the amount of force between them. We must first translate the stars into two numbers representing masses and a number representing distance, and then we can determine a number representing force, which we can translate back into our conclusions. But mathematics by itself cannot do this.

Rebuttal of Putnam's Succes Argument:

This argument mostly falls under the same rebuttals as the previous one, but I would also like to point out that this is also an Argumentum ad Ignoratum. Simply because we haven't proven an explanation false does not automatically entail that said explanation is the correct one. This argument thusly fails.



Debate Round No. 2


tahirimanov forfeited this round.


Extend all arguments.
Debate Round No. 3


Sorry I have to cut my argument short, still having headache.

1. The Epistemological Argument

I may agree with the first and second premise, but there is no way premise three can be proved. Firstly, human mind is not something physical, therefore for human mind it is possible somehow to acquire information about abstract objects. Secondly our knowledge about space-time is empirical, and we don't have external objective way of verifying the truthness (or correctness) of our senses about space-time (it is what we see, and what we see is not always true). Thirdly, our knowledge of space-time comes from physics, and physics uses mathematics as its basis, and we cannot judge the primary based on tertiary. And lastly, what we call space-time is just three spatial and one temporal dimensions. And according to physics we can acquire knowledge of things which are beyond that.

2. Non-Uniqueness and Peano
Firstly, Peano axioms doesn't tell everything about natural numbers. Secondly, what makes a sequence to stand out is its structure. And arithmetic is not about some particular sequence of objects.


Okay, so my opponent completely dropped my rebuttals to their arguments. I understand that they have a headache, but I'm writing this with a headache too.

The Epistemological Argument

My opponent makes four objections to this argument, which I will address individually.

1. Well, the human mind is derived from the electric interactions of neurons, so arguably it is physical. But even if it weren't, that wouldn't imply that it can acquire information about any non-physical entities. There could easily be multiple types of non-physical entities, and numbers could be a completely different type than the human mind. In order for this objection to hold, you would have to prove that numbers and the mind are the same type of entity, and even then you would have to prove that being the same type of entity grants us the ability to learn about other such entities. This would only be possible by providing an example, and no such examples exist.

2. This isn't really relevant to the argument. Our ability to know about the spatio-temporal with certainty is completely unrelated to our ability to know about the abstract, which is what this argument is focused on.

3. This is irrelevant for the same reasons I just mentioned.

4. Physics does not necessarily say that we can gain knowledge of all things that are beyond spacetime. And even then, most theories in physics describing higher dimensions classify them as alternate timelines and universes, so anything within them would still be physical. [1]

The Non-Uniqueness Argument

My opponent has two objections to this argument.

1. That's not really feasible for any description of number theory. The more that we work with our understanding of math, the more we find ways that our descriptions of it are woefully inaccurate. That does not mean, however, that they fail utterly at describing anything. It just means that their applications have limits. This occurs outside of math as well, e.g. Newton's laws don't describe fast-moving particles but engineers still use them for everyday math because they work well enough at that level. Likewise, Peano works well enough for the purposes of this debate, and is the best working model of number theory for natural numbers at the moment. [2]

2. Yes, a sequence does stand out because of its structure. That's why there are an infinite number of potential sequences of natural numbers. Arithmetic is about a particular sequence of objects if you accept Mathematical Platonism. Because the natural numbers are a sequence of numbers, and according to MP, those are all objects.


Vote for me because:

1. My opponent dropped my rebuttals to his arguments.

2. My opponent's rebuttals missed the point and/or were insufficient.



Debate Round No. 4
4 comments have been posted on this debate. Showing 1 through 4 records.
Posted by Surrealism 3 years ago
I agree, but I meant in the vote section. Conduct is one of the categories people vote on.
Posted by tahirimanov 3 years ago
:) Conduct is irrelevant, but exchange of knowledge and ideas are relevant.....
Posted by Surrealism 3 years ago
That's okay, it just means you won't win conduct.
Posted by tahirimanov 3 years ago
Sorry about forfeit. It is holiday here, and plus I got sick. And totally forgot.... I will post my argument within day....
No votes have been placed for this debate.