Math is Crazy Hard to Define



52 thoughts on “Math is Crazy Hard to Define

  1. Very nice!

    Logic is the study of facts that must be true in each mathematical model.

    I’m going in loops.

  2. Mathematics is a language.

    A language used to describe concepts and arrangements in logic, science, in fact more or less anything.

    1. No. Math is not a language. Math does include its own language development, but it is not merely a language.

      1. Well, if “language” can be defined as:

        “[…] method of human communication, either spoken or written, consisting of the use of words in a structured and conventional way”, (

        … then that would seem to be a fair description of what maths is. It’s an extremely well-defined set of “words” (or symbols), with an extremely well-defined syntax (or rules used to combine those symbols).

        After all, maths is something people use to *describe* something (e.g. logic), just like a language.

        If math “is not merely a language”, what else is it?

        1. You confuse the symbols with the math. For example, numerals and numbers are different. Numerals symbolize numbers. But numbers are not merely numerals. The concept of number pre-exists the symbol. We even have psychology experiments that show that babies understand numbers even though they do not use mathematical language.

  3. How about this:
    Mathematics is the study of abstraction and generalisation, with occasional, unintentional and accidental applications to the real world.

    1. Follow up: for example, couldn’t the “structure” of your political science example be modeled by Venn Diagrams or partial orders or social network graphs? Math is the study of structure. Subjects that don’t use math to study structure just haven’t caught up yet.

      1. Hmm – I quite like your working definition of structure as “the abstraction of reality.” I could see that leading to a fruitful definition of mathematics.

        If you use the Webster’s definition of “structure,” though (“the arrangement of and relations between the parts or elements of something complex”), then I think you run into trouble. E.g., a literary critic is interested in the structure of a novel, and the purpose/impact of this structure. George R.R. Martin’s flexible use of multiple point-of-view characters lends a sense of breadth and fullness to his fantasy world, while his choice to begin the first novel with a magical event (followed by hundreds of pages in which nothing particularly magical happens) helps to set the reader’s expectations about the elusive but potent nature of magic in this world.

        This account feels “structural” to me, but not particularly mathematical, and I don’t think it would much benefit from translation into more quantitative terms. The complexity of a novel just seems too idiosyncratic to benefit from a mathematical account.

        (This is why your “abstraction of reality” definition may work better here – under the dictionary definition, I’m certainly analyzing the relations between elements of something complex, but I’m not really stepping up the abstraction.)

        1. See, I think that those structural qualities ARE mathematical: order, incidence, proximity, concatenation, sequencing. Certainly, there is more to those arts than structure, but the structure itself can be thought of as mathematical. For example, perhaps we could model that structure using graphs as in social networking and reveal some of what the literary critic is trying to convey. With graph databases, we definitely have some exciting mathematical modeling in the near future.

        2. Addendum: from the dictionary definition, we have the mathematical terms of “arrangement”, “relation”, “elements”.

  4. Math is the foundation upon which everything else is built – that allows for anything else to be built. It is the study of everything and the myriad methods we use to do so.

  5. Thought-provoking. I differ in a couple of definitions. I would say that science and structure are not synonymous, but you imply they are. To me, science is founded on observation, not logic. Scientific experimentation has created a need for math to quantify results.

    Also, only in a left-brained world would anyone claim that logic is the key to every discipline. Logic is the final analysis of right-brained associative patterns, dreams, and imaginings. Logic is the result of ideas fed across the corpus callosum one byte at a time. You can describe nothing in the natural world by mathematics.

    Another subject: I’ve been reflecting on your earlier post about fractions. I decided fractions make sense if you’re working with angles of an arc, as in geometry. But even then you have to resort to metrics to calculate the area of a circle. For straight lines, mass, and measures like temperature scales, metrics uses standards like water and mass to provide an easily reproduced standard.

    1. Thanks for your thoughts! Interestingly, the ideas you attribute to me are almost the exact opposite of what I’d hoped to express – obviously I could have communicated this better, so I’ll try again:

      I think we’re in agreement on logic’s role. That’s why I tried to emphasize its insufficiency in history, science, law, etc. I intended the word “key” as an adjective meaning “critically useful,” not as a noun meaning “the essence.” To be more explicit: I think logic is a valuable tool just about everywhere, but it can never stand alone (except in mathematics).

      And I think we’re mostly in agreement on fractions vs. decimals, too – my feeling is that measurements should almost always be reported with decimals (because measurements are inherently imprecise, and decimals are good for capturing our level of certainty), whereas fractions are typically better in a theoretical setting (where we want utter precision).

      1. I understood that part. What struck me most was the lack of clarity about what “science” is. I’ve been wondering who the “scientists” are, as in news articles that claim “Scientists say . . . .” The institutional scientists don’t impress me with their lab-rat approach to life. That “science” is observation puts it within the reach of everyone with any curiosity about how things work. The mathematics part comes afterwards.

        1. I think that “science is observation” is a pretty good start to understand what science is. Most of the things you have to add to science to get it into our modern form is mostly a good deal of caution—being aware of the most common “gotcha’s” which have caused people to deceive themselves in the past. There are a lot, and it takes time and practice to do it right.

          The major thing that the above description is missing, though, is the theoretical side of science. That’s the part that tries to explain the observations using as little extra information as possible. This has also been part of science since as long as we have written records, and it is also the biggest problem in trying to define science. It’s also probably not a coincidence that this is where science meets math, and all the definitions become fuzzy.

        2. I agree, and this is why we need a better cultural definition of what science is. I’m shocked at how naïve Americans are about simple things, like that methane and natural gas are the same molecule. Or that ethanol is the gasoline additive as well as whiskey. What, after all, is a “climate scientist?”

  6. Forget “science of” for a moment. Mathematics is an art, the art of pure pattern. Poetry is pattern in words, drawing is pattern in lines, painting is pattern in colors, music is pattern in sound and time, sculpture is pattern in space. Mathematics is the art of pattern, abstracted from all of these.

  7. Jordan Ellenberg in “How not to be wrong” defines mathematics as “the study of things that come out a certain way because there is no other way they could possibly be.” My son, who is studying mathematics at Cambridge, says it’s “the study of interesting tautologies”, which is more concise but less evocative.

  8. I’ve always been partial to the inductive definition of mathematics due to Thurston in his essay “On proof and progress in mathematics”.

    Mathematics is the smallest subject satisfying the following:
    1. Mathematics includes the natural numbers and plane and solid geometry.
    2. Mathematics is that which mathematicians study.
    3. Mathematicians are those humans who advance human understanding of mathematics.

  9. So a quick googling for the etymology comes back with “that which is learnt” and “the study.” Not “the study of”, just “the study”. I like that.

    “ology” which we commonly say is the suffix for “the study of” really means “the discussion of.” Because logos are words or speech.

    Now if I had to pin a definition of mathematics more precisely than “the study.” How about “the study of truth.”

    More googling… what do you think of these definitions?

    “Mathematics is the art of giving the same name to different things.” — Henri Poincare

    “The subject in which we never know what we are talking about, nor whether what we are saying is true.” — Bertrand Russel

    “Mathematics is the science that draws necessary conclusions.” — Benjamin Pierce

    “Mathematics is the manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules.”

  10. My five year old is at the stage where she questions everything. She asked me what Maths was and I replied with, ‘You know, numbers and counting and stuff.’ …..I’m so sorry.

  11. Most people are not aware of it, but their several types of sense apparatus send them signals of different types which are then moderated by intervening structures to discard material which genetics and experience regard as irrelevant for survival and utility. Obviously individuals differ as to what may be or not be relevant. The resultant signals are then passed on to nervous centers which construct a mosaic in consciousness which we term reality. The environmental signals differ widely from one species to another as each individual has different requirements to survive so what they finally assemble is probably quite different between different creatures. These final assemblies are abstracts of the actual goings on in the universe and mathematics is a rather precise way of relating the various final abstractions themselves and the various elements which compose the mosaics. But science deals with the mosaic formulations out of observed phenomena. Mathematics goes beyond that. It formulates all sorts of relationships out of both observed realities and imagined abstractions that have not been observed. On occasion, as with Einstein’s formulations of the nature of the universe, it has been discovered that abstractions that were purely speculative with no observed phenomena have been discovered to be a match with puzzling observed phenomena and therefore become science. But mathematics, with its own language, is an exploration into many merely imaginary abstract relationships such as in speculative other unobserved universes with constants different from our own. Those exploratory adventurous thoughts is what makes mathematics so fascinating and very valuable.

  12. Hi, love your post about “Math is crazy hard to define”. Humorous and arresting! Going to show to students who can’t stand Math. Keep the humor coming.

  13. So, in an academic setting, I would conclude “Math is the politics of making much ado about nothing.”

    More seriously: “Math is the science of numerical deduction.”

    That gives some short shrift to topology, perhaps, but that can be cast into geometry.

  14. Math is the underlying truth in all that follows a logical pattern. Math cannot be tweaked, it is discipline through and through.

  15. The definition of math is an interesting question because when put along with the other fields of study, we assume that it is something we humans have discovered. We can look at other things in the world and say “Alright look, this is this and now let’s think about it” but the truth is we invented math. All of math is developed and based off of axioms that we create on our own and it quickly becomes clear that math is a figment of our imagination, a tool of science, and some sort of indescribable beauty of life. I think at this point, the best definition is indeed Vihart’s: Math is creating rules and seeing what happens.

  16. As von Neumann said, “In mathematics you don’t understand things. You just get used to them.”

    But also like Biology is the science of life, but what *is* life? When it comes down to it, is there any way we can define terms in this world? Or does it all come down to patterns going on the brain that have no objective meaning?

    Is it even a way to understand things at all? Or is all thought just pattern recognition and “getting used to things” on a deep level?

  17. My definition: “Mathematics is the science of formal models.”

    In this definition, a “model” is a simplified description of reality which makes it possible to deduct facts about it. And a formal model is a model that can be (but almost always is not) written down completely in a few pages of text, and where conclusions about the model can be reached by formal reasoning alone.

    Applied mathematicians build models by using the existing machinery of mathematics, pure mathematicians develop the machinery of mathematics to make it more powerful and easier to use.

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