Can math be subjective?
Often times in math, ever since Kindergarten and before, math has been defined by the fact that there are only one answer for problems. For example: $1+1=2$ and $\frac{d}{dx}x^2=2x$. What I am showing by these two examples are two questions that are from completely different areas of math. However, they both have have only one solution. Problems with multiple answers doesn't necessarily mean they are subjective though, such as $|x|=2,$ which has two solutions. My question is that are any such problems that depend entirely on perspective? If all subjective math problems follow a certain pattern, please tell me what that pattern is. I really have no idea of any examples of this and I would really be interested to see one. Thank you very much.
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$\begingroup$There's plenty of room for subjective opinion in mathematics. It usually doesn't concern questions of the form Is this true? since we have a good consensus how to recognize an acceptable proof and which assumptions for such a proof you need to state explicitly.
As soon as we move onwards to Is this useful? and Is this interesting?, or even Is this likely to work?, subjectivity hits us in full force. Even in pure mathematics, it's easy to choose a set of axioms and derive consequences from them, but if you want anyone to spend time reading your work, you need to tackle the subjective questions and have an explanation why what you're doing is either useful or interesting, or preferably both.
In applied math, these questions are accompanied by Is this the best way to model such-and-such real-world problem? -- where "best" again comes down to usefulness (does the model answer questions we need to have answered?) and interest (does the model give us any insight about the situation we wouldn't have without it?).
The subjective questions are important in research, but can also arise at more elementary level. The high-school teacher who chooses to devote several lessons to presenting Cardano's method for solving the generic third-degree equation will certainly have to answer his students' questions why this is useful or interesting. Perhaps he has an answer. Perhaps he has an answer that the students don't agree with. In that case, he cannot look for a deductive argument concluding that Cardano's formula is interesting -- he'll have to appeal to emotions, curiosity, all of those fluffy touchy-feely considerations that we need to use to tackle subjective questions.
$\endgroup$ 2 $\begingroup$Every single statement, question, and claim in mathematics is subjective because they are always based on a set of axioms, which are arbitrary, and are picked to observe their consequences.
However, once you phrase the claim in the form of an implication, (such as: "if [the axioms of Euclidean geometry], then [the Pythagorean theorem]") then you have an objective truth. This is assumed to be the meaning when any mathematician states a theorem - we understand what axiomatic framework they are working in and understand that their claim is contingent on those axioms.
Given a particular axiom system, three of the possible results for a mathematical claim are:
We prove the claim true. [Ex: The Pythagorean theorem]
We prove the claim false. [Ex: "The integers under multiplication form a group"]
We prove that the claim is independent of our axiomatic system. [Ex: the continuum hypothesis].
(see Mario Carneiro's comment for other possibilities).
There are no claims that can be subjective if we take for granted that we are working in an axiomatic system. Some people might argue that the continuum hypothesis is "subjective" under ZFC, but I prefer to think of it as simply having no truth value.
$\endgroup$ 15 $\begingroup$There are a few more points of subjectivity that are related to, but slightgly different from the sometimes subjective choice of axiom system: I am talking about definitions of some standard objects. For example, people may have different "opinions" whether or not $0\in\mathbb N$. Or whether they accept an answer to a question asking for an explicit solution only if it is elementary (a combinations of polynomials, trigonmetrics, esponential, logarithm) or if they would also accept something involving the Lambert $W$ function or the error function ...
And then there are things that are just personal preferences for different notation (or cultural preerences - I personally have great difficulties reading something as simple as a long division if it is written the "American way" that looks to me rather like a $\sqrt .$)
$\endgroup$ 2 $\begingroup$It depends on what you mean.
First: You give the equation $\lvert x \rvert = 2$ as an example of an equation/problem with two solutions. In a sense you could say that the problem only has one solution in that the solution set is $\{-2, 2\}$.
The point of saying that a problem only has one solution/answer is that you can't have two answers that contradict each other. If you, for example, ask if a given equation has a solution, then the answer is only yes or no. It can't be both yes and no.
There are, of course, questions that mathematicians are interested in that will have several answers. You could ask how you best mathematically describe something using an equation. Here the key work is "best". It is, to some extent, subjective what is best.
So in general you will not have answers that depend on something subjective in (pure) mathematics.
Now, you can drive the question about subjectivity and different answers to a philosophical one. Mathematics is (can be) built on a set of axioms (say of set theory) and the rules of logic. Some will argue that mathematics is just a game of how to use the rules of logic given the set of axioms. But then questions arise about how to best pick the axioms and questions about exactly what rules of logic we should allow. These discussions are interesting to some mathematicians and some will say that this is inherent to the nature of mathematics. And in these discussions answers will have a level of subjectivity to them. Much more can be said about this, but when we fix a system of axioms and a set of rules, then we avoid these discussions (I might be oversimplifying things).
Another way that subjectivity can arise in mathematics is when the discussions turn to what areas deserve the most attention. If an institution is hiring a new faculty, then the existing faculty will have discussions about the directions of the department. Here arguments can be made for the different areas and the discussions can turn political.
Note, some might say that $1+1 = 2$ is subjective because we have $1+1 =0$ when we do modular arithmetic (mod $2$). But understand that the elements $1$ are completely different in the two situations. We are taking about two different sets of "numbers" and so there isn't any subjectivity here.
$\endgroup$ 3 $\begingroup$There are some notions which have no common accepted definition or actually many competitive definitions which do not agree. Some examples are:
Perhaps, one day, mathematicians will agree in each of these cases which is the correct definition, but of course this is kind of subjective (what makes a definition correct?). In any case, the mathematics done with each of these notions is objectively true.
$\endgroup$ 6 $\begingroup$Some people does not aggree with usage of Axiom of choice in proofs*
There is Constructivism (* for example this ones ) which does not accept proofs by contradiction as a proof of existence (mathematics)
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.
Consequences, within the meaning of subjectivity is left to the reader ;)
$\endgroup$ $\begingroup$Are there problems whose solutions are subjective?
It depends on how far you are willing to stretch the word "subjective." An idea may have two essentially different meanings to two different mathematicians. To one mathematician, the fundamental theorem of algebra might mean the field of complex numbers is algebraically closed. To another, it might mean that every complex $n\times n$ matrix has a complex eigenvalue. Both meanings are equivalent, but they are not "the same," in a very restrictive sense, so one might convey this as subjective.
Otherwise? Not really. I think, however, that there is a larger question to be answered here.
Is mathematics subjective?
No...but yes.
As far as "the facts," as you call them, are concerned, once we agree on definitions and axioms, things are pretty much set in stone. The theorems are either right or wrong (or unprovable), and there isn't any wiggle room.
Here's something that most laypeople, and even a good handful of mathematicians, don't get: mathematics is not just "the facts." The myth that mathematics is solely about the theorems is as wrong as it is dangerous; it scares away many people that only know mathematics by the bland material fed to them in high school and discourages mathematicians who decide that their value is based solely on what Thurston called "theorem-credits."
While on the topic of theorem-credits, I think many, if not most, questions of the form "Is mathematics [blah-dee-dah-dee-dah]?" can be resolved by reading On proof and progress in mathematics. In fact, if you haven't read it, then you probably should.
Mathematics, or at least the mathematics that I do, is about thinking and sharing ideas with other mathematicians. This is where subjectivity creeps in. Typical questions of a mathematician, amidst the deep mathematical thoughts, are
- "Does this look presentable?"
- "Is this notation clear?"
- "Is this worth further study?"
- "Does mathematics honestly need another use for the word 'normal'?"
These are about as subjective as questions can get.
$\endgroup$ $\begingroup$Mathematics is not subjective because it is perceived differently by different mathematicians. That's a misuse of the word subjective.
I think mathematics is the most objective thing we have. If that's true, then if the word objective means anything, we can simply say that mathematics is objective, without qualification.
I want to bring up two related questions in the mathematics realm. The first is why mathematics so often relates directly to fundamental properties of the natural world. This happens so often it's spooky - someone goes off and researches an obscure mathematical problem, and the answer turns out to predict the curve of a NDA molecule's spirals or something like that.
The second is whether all reality is socially constructed. I say no, that mathematics and much of science are entirely or mostly objective. Usefulness, implementation, interest, our readiness to take a given piece of work further - those are much more socially constructed. (But influenced by objective qualities of the work.) The fact that the results are perceived differently for reasons that are significantly socially constructed does not make them inherently subjective or socially constructed.
Sorry for not reading/taking into account all the other comments, I have to get back to work...
$\endgroup$ $\begingroup$I would argue that the number of solutions does not influence the subjectivity of mathematics. Rather that mathematics is an inherently subjective art in the sense that it is perceived differently by different mathematicians. However, what I think is the true beauty of it, is that we all (most of us) operate under the same vast framework, where even with differing perspectives, all our results are consistent with one another.
In comparison with other fields such as psychology or economics, the results obtained in those fields can be disputed, many economists hold different viewpoints and the results obtained are not consistent for the most part. This is something that does not occur (or if it does, occurs infrequently) in the field of Mathematics.
For example, you said that $1 + 1 = 2$ has only one solution, yet in certain field of mathematics you will often see $1 + 1 \equiv 0 \pmod{2}$. Yet this does not mean that it is not consistent with $1+1 =2$, Mathematics just works under incredibly precise foundations.
My point is, I think you need to reconsider your definition of subjectivity in mathematics.
$\endgroup$ 3 $\begingroup$One important subjective question is when exactly an argument is sufficiently rigorous. I'm very fond of reminding people that rigor is a continuum, not a binary property.
Take the Pythagorean theorem. Is this proof by rearrangement a rigorous proof? I would say yes, in the sense that I would consider it very pedantic of someone not to find it convincing. For all practical purposes, once you've seen that image, you know the theorem is true.
But maybe you can't quite squish the nagging doubt that if you picked a really oddly shaped right triangle, it would reveal an implicit assumption in that diagram, and so you couldn't do the rearrangement and it wouldn't work. Or maybe you just find precision aesthetically appealing. Then maybe the Euclidean proof would suit you, or some more modern formalization of it.
Or, perhaps you find the physical universe so complicated and ambiguous that you just prefer not to think of it at all, and you won't be satisfied unless you've defined triangles as subsets of the set of all pairs of real numbers, and proven the theorem formally in that setting, thus escaping from empiricism and reality altogether and reducing the theorem to a statement about real numbers (which are defined in some other way, etc). Of course, a disadvantage of this approach is that it becomes hard to justify that your argument gives you any knowledge about real world triangles, since you deliberately cut away any connection to them. But whether or not this is really bothersome is subjective.
It all depends on why you're studying math and what you want to do with your theorems, as well as your sense of aesthetics.
$\endgroup$ $\begingroup$Many of the answers posted here cover much what I would say about subjectivity in mathematics. For the most part, the subjectivity of mathematics arises from the acceptance of certain collections of axioms. One possibly controversial one is the Axiom of Choice. There are several "paradoxes" associated with it, such as the Banach-Tarski paradox. Whether or not you accept this as a paradox depends on how weird you think the statement is, but it is not a paradox in the mathematical sense. Once axioms are accepted, the deductions that follow are not subjective.
There is another vein of skepticism in mathematics comes from the point of computability. For instance, in mathematics, we often talk about real numbers. We can produce logical formalisms to purport the existence of the set of real numbers, but a valid question is whether such a system exists in the real world. A computer can only compute in rational numbers (an then only a small subset of rational numbers can be represented by a computer), and measurements made in the physical world can only be expressed as rational numbers. Thus even though we talk about representing real numbers as limits of rational numbers, no such limit can be measured. This leads to other number systems such as the $p$-adic completion of the rational numbers.
A less technical example is prime factorizations. As far as we are aware, there are only a finite number of particles in the universe, this means that there must be numbers that we can never express given any computer we could ever construct. If there is no way to express these numbers, then a valid question is if it is meaningful to work with these numbers. Certainly number theory tells us that formally every integer has a prime factorization, but consider the number $$N=2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2^{2}}}}}}}}}}}}+1.$$ Theoretically, this has a prime factorization, but this number is so large, that no computer can calculate this prime factorization. (Even if this one can be factored, there must be some upper limit on the numbers that can be by the finiteness of the universe.) You could say that the claim that this number has a prime factorization is subjective, since none could be produced (formal proofs aside).
As for myself, I happily work with the real numbers everyday I do calculus, and I thoroughly enjoy Hardy's number theory work. My paycheck depends on my ability to do analysis. However, while I work I must accept the Axiom of Choice, the real numbers, and prime factorizations.
$\endgroup$ $\begingroup$Erdos (well, actually Alfréd Rényi) said that a mathematician is a machine for turning coffee into theorems. If that's literally true, then mathematics is objective.
Erdos also referred to The Book, in which God kept all of the most elegant proofs of each theorem. I believe that the ability to perceive elegance is a subjective thing. The fact that a problem has more than one solution does not make mathematics subjective. But the fact that the two proofs can be compared to each other and arguments can be made over which is the better definitely does make mathematics subjective.
Finally, someone said that mathematics is what mathematicians do. I don't think that is literally true either. It blurs the distinction between The Book and the act of creating an entry for The Book.
I really want to believe that no machine can create elegant theorems as well as a mathematician and a pot of coffee, or a really hot cup of tea. The act of creating or discovering a proof may or may not be a subjective act. The act of deciding whether that proof belongs in The Book is a subjective act.
$\endgroup$ 4 $\begingroup$Yeah, it can:
Consider that polynomials of degree 4 or lower are said to be solvable (or solvable "in radicals" or whatever) whereas the higher-order ones are said to be unsolvable. But this is quite subjective and arbitrary. The definition of $\sqrt{2}$ is literally "the positive real number $x$ that satisfies $x^2 = 2$". What makes solving $x^2 = 2$ any different from solving $x^5 + x + 10 = 0$? I really don't see a meaningful distinction -- both are real numbers and both are irrational. To write either of them out as decimals, you have to use the same kinds of root-finding algorithms for both. To me, the distinction that one of them can be written in terms of radicals and the other one can't seems pretty arbitrary and subjective.
Just look at the definition of e.g. "closed-form solution". It's completely subjective.
etc.
$\endgroup$ 4 $\begingroup$This question is on the edge of being too soft to learn from. Some people are Platonists and will read this question differently. I'd say yes, math should be considered entirely subjective.
You can imagine you encounter a homeless man who loudly recites Alice in Wonderland on the market place all day long. He may be personally convinced that $50^2$ equals $5225$ and not $2500$ or another number. Now if you ask most math students, they'd say $50^2=5225$ is wrong. They all share this opinion, but the beauty is we only need one person with another opinion to label something as subjective. And the mere opinion of the majority that this man is may be "crazy" (i.e. not acting according to our rules and expectations), or that he doesn't change his mind upon being shown a "proof" or "demonstration", doesn't make him any less of a agent with a subjective opinion. His deviant view makes the issue subjective, by definition. We can now ask anybody on StackExchange and any fields medalist what $50^2$ reduces to, or $\frac{d}{dx}x^2$ for that matter, but we'll only find opinions that (probably) agree with each other. Opinions of people that reassure themselves, and nothing more than exactly that.
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