Is mathematics forced upon us or arbitrary?

Question

Math at its core begins with calling something true or false and following logic. WE for example call an odd number 2n+1, but what if we called an odd number 2n and flipped it for it to become an even number?

The same can be said for any definition. Hence why the term "let" is used in nearly all proofs. What if we didn't "let" n be an arbitrary integer?

This idea gets even more confusing for me, because is logic then arbitrary? Can we have another set of rules that might lead to different math? If we never agreed that something is false or true would that change math?

Answer 1

I will distinguish two parts of this question. The first question is, do the examples you presented show that mathematics is arbitrary. Second is, whether there can be alternative mathematics in such a way that choosing between the actual mathematics and the alternative is an arbitrary choice. I’ll start by addressing the first one.

Switching the definitions of odd and even merely changes the words that we use to refer to the properties of numbers. So the property of being odd would still be the same property regardless of what we want to call it. Of course, the choice of words is not entirely arbitrary, since it is at least in some sense conventional. What I mean by that is that the definition “Number is even if and only if it is some equal to two times some other number” corresponds with the English meaning of the word “even”, so it is advisable to use this definition, since it makes reading theorems easier. Of course, not all definable properties of numbers correspond neatly with an expression in English, but even in these cases, there are practical considerations to take into account when choosing a word to refer to that property.

Even if you would want to maintain that the choice of words is arbitrary, this would still not show that mathematics is arbitrary since mathematics is about numbers (and sets, functions, lines, spaces etc.) and not about the words we use to refer to numbers. Like Shakespeare once wrote: “Even number by any other name is still devisable by two.”

What about the case of “let n be an arbitrary integer”? This is, in fact, a valid inferential step, but I feel it is not usually explained very well during introductory courses which can be a source of confusion. The rationale behind this expression can be demonstrated this way. To show that no card has a number on it greater than 10 you can pick a card from a standard deck. Now without turning the card over you know that it is either number card or a face card. If it is a face card there is no number on it. If it is a number card it is the highest value is 10 which is less than 11. By the transitivity of less-than relation, the statement holds for all cards with the value less than 10. Now you probably realize that there is no longer any point of turning the card over since whatever it is it the statement holds for it. Therefore, one can infer that the statement holds for all the cards.

Inferences made with arbitrary integers are analogous to the above example. There are a lot of things that can be known about n just from the fact that n is an integer. Arbitrariness here just means that we don’t assume it to be any specific integer and this allows us to make universal claims about all integers. This does not show that these inferences are arbitrary or that mathematics is arbitrary.

To turn to the second question, is mathematics and/or logic is arbitrary? I take it that you mean that since in mathematics some sentences are called axioms and from those, we derive other sentences with some rules of inference, could we then just change the axioms or rules of inference arbitrarily? This sort of idea is usually associated with formalism in the philosophy of mathematics. While it enjoyed some popularity during the beginning of the 20th century it is nowadays considered to be problematic. I’ll sketch some objection typical raised against it.

The first objection is if any axioms will do, why we are so interested in certain axioms like axioms of arithmetic or geometry? What makes these axioms special and axioms that talk about centaurs and unicorns uninteresting? The second objection is that, at least to many mathematicians, axioms like “Successor of a natural number is a natural number” seem to be true and obvious. (How we can know that they are true is problematic, but axioms obviousness is somewhat hard to explain if the choice of its truth value is arbitrary.) The third objection has to do with the application of mathematics. How can arbitrary set of sentences and rules of inference produce verifiable predictions about concrete objects? If I put 7 marbles to a bag that contains 5 marbles is it really a coincidence that the bag now contains 12 marbles?

Also depending on precise formulation, formalist may need to deal with some technical objections such as those relating to incompleteness theorems.

This is by no means a refutation of the idea, nor a comprehensive presentation on the subject. For further reading, I’d suggest these articles from Stanford Encyclopedia of Philosophy. They are comprehensive, but a bit difficult.

https://plato.stanford.edu/entries/formalism-mathematics/

https://plato.stanford.edu/entries/frege-hilbert/

If you are interested in more introductory style text, then Steward Shapiro’s Thinking About Mathematics has a chapter on formalism and deductivism.

Rudolf Carnap's views on conventionality of mathematics are summed up nicely in “Empiricism Semantics and Ontology”. (Quine’s critique of Carnap's position can be found in “Truth by Convention” and “The Two Dogmas of Empiricism”.) These may give interesting pointers about arbitrariness even though I would distinguish conventionality from arbitrariness.

Answer 2

The statement "let n be an arbitrary something" does not influence the validity of a mathematically logical construct.

The logical part in your example is represented by 2* and +1, while n is just something to confirm that logic no matter the value chosen for it. Since the logic is applied to integers, one cannot argue that it is invalid if we choose a non-integer 'n'.

Also, grammatical naming is just a naming convention, it does not influence the logical or math result of something.

We may call odd numbers Klingons and even numbers Romulans. It would not change anything in the way the logic works.

Logic is something based on validity. And mathematical validity can be verified in multiple ways. Even the 10-base digits themselves have a build-in CRC system. Therefore, logic is not arbitrary.

Answer 3

The word mathematics comes from Ancient Greek μάθημα (máthēma), meaning "that which is learnt", "what one gets to know", hence also "study" and "science". ... Its adjective is μαθηματικός (mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". ... Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "teachers" rather than "mathematicians" in the modern sense.

In my opinion, an important part of the mystery of mathematics is that it is a language which allows to accurately communicate ideas and knowledge. The quote from wikipedia above illustrates that the initial understanding of mathematics already included that idea.

The mystery of how a "language which is able to communicate anything at all" is possible also exists to a lesser extent for normal human languages. A normal human language is often misunderstood and the meaning of words shifts over time. So the mystery can be explained in this case, by having some ideas how the current meaning of a human language got established, and the realization that a normal human language is simply not very accurate and universal. So the mystery for mathematics is how it is possible that it provides an accurate universal language. The modern answer is probably that it (mostly) describes structural properties, and that structural properties are not subject to arbitrariness to the same extent as many other human concepts.

Can we have another set of rules that might lead to different math. If we never agreed that something is false or truth would that change math?

Philosophers and mathematicians have proposed different "concrete examples which suggest a positive answer" to those questions over time. Maybe most annoying among those is Kripkenstein (a portmanteau of Kripke and Wittgenstein) which doubts that we can even define what it means for information or meaning to stay constant over time. Nelson Goodman's new riddle of induction with its concepts of grue and bleen nicely nails the core of that doubt.

Answer 4

There are multiple forms of logic. Kevin C. Klement, after discussing classical truth-functional logic offers modal logics as non-truth-functional propositional logics. His description of classical truth-function logic may be useful:

So far we have focused only on classical, truth-functional propositional logic. Its distinguishing features are (1) that all connectives it uses are truth-functional, that is, the truth-values of complex statements formed with those connectives depend entirely on the truth-values of the parts, and (2) that it assumes bivalence: all statements are taken to have exactly one of two truth-values—truth or falsity—with no statement assigned both truth-values or neither.

Does that mean that logic is arbitrary? What it means is that one can choose different logics to use. What is important is that metatheoretic results, such as completeness and soundness, should hold for that logic.

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There are also different mathematical objects. The integers and the natural numbers are different. In the integers one has the operation of subtraction while in the natural numbers one does not. A finite set such as {0,1} could be considered a mathematical object. It would not be like the integers either because the cardinality is different.

Some mathematical objects seem to be very similar to each other such as the integers and another set where odd integers are called even integers. These are two different objects, but it is likely, unless something else differentiates them, that a function exists from one of these objects to the other that is one-to-one and onto and which preserves the structure, that is, an isomorphism exists between those objects. If that is the case one can say those two mathematical objects, though different, are the same up to isomorphism. This abstracts away some of the differences between those two objects. Wikipedia describes this feature of isomorphism as follows:

The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

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When one says, "Let n be an integer," what one is doing is naming a particular integer so that it can stand for any integer. The authors of forallx describe the first-order logic rule of universal elimination using "F" as the domain rather than the integers in this way:

From the claim that everything is F, you can infer that any particular thing is F. You name it; it’s F.

Whatever results apply for this n, just a name, they would also apply for any other integer. It is a way to talk about all integers.

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Klement, K. C., "Propositional Logic" Internet Encyclopedia of Philosophy https://www.iep.utm.edu/prop-log/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Wikipedia contributors. "Isomorphism." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 14 Mar. 2019. Web. 22 Mar. 2019.