I like to draw inspiration from the Googlesearchterms that bring people to this blog. And someone recently arrived by way of the following philosophical doozy, a question close to the heart of every frustrated 9th-grader and every soul-searching tenure-track mathematician:

It’s helpful here to contrast math with science. That’s because they’re completely unalike. Sure, they share a symbiosis, and fans of one are often fans of the other, but their definitions of “truth” couldn’t be further apart.

Science’s measure of truth is the experiment. A theory may be perfectly consistent, wonderfully elegant, and intellectually satisfying—but if it fails to predict what happens in the world, then we toss it on the compost heap and move on. To a scientist, “truth” is whatever we see happening in reality, and all our explanatory frameworks must ultimately answer to the cold, hard facts of nature.

Mathematics’ measure of truth is the proof. Frankly, mathematicians don’t care about physical reality. They ask only one type of question: “Assuming that A is true, can we conclude that B is also true?” A proof is an airtight argument, deploying logic to show exactly why B must follow from A. It shows that the fact could be no other way—logic simply would not permit for alternatives.

Thus, proof doesn’t merely support a fact. Proof immortalizes a fact. Proofs, like diamonds, are forever.

Look back 200 years. Much of the science from 1814 has fallen out of favor, supplanted by new and superior ideas—the periodic table, the theory of evolution, general relativity, the modern atom. Even some cutting-edge science from as little as 20 years ago has been discarded as new evidence came to light. (Remember “junk DNA”?) [Edit: Apparently “junk DNA” is still a thing. See the helpful comments section below. My bad!]

By contrast, math never goes stale—not in 20 years, not in 200, not even in 2000. We’re still teaching kids about Euclid’s geometry, a two-millennia-old system that’s as fresh and true as the day Euclid sighed and said, “All done.”

So why do we prove things in math class? For the same reason that we run experiments in science class: because that’s how we pin down the truth.

22 thoughts on “Why should we use proofs in the first place?”

Great post! Deductive vs. inductive reasoning always makes me think of the song “Circumstances” by Rush: “Now I’ve gained some understanding of the only world that we see [inductive], things that I once dreamed of have become reality [deductive].”

Sorry, but junk DNA is still mostly junk DNA, the hype-driven frenzy about the ENCODE misstatement notwithstanding. The way that biologists tell whether DNA serves a biological function (as opposed to having accidental chemical interactions of no significance) is by looking for indications of selection for or against the DNA sequence. Without evidence of selection, it is extremely unlikely that the DNA has meaningful biological function. The amount of “junk DNA” varies enormously from organism to organism—currently the best estimates of the amount of junk in human DNA is around 90% of the genome. Each new find of a new DNA function changes the estimate slightly (affecting usually around 0.1% or less of the genome).

A bit of an aside: There is good reason to suspect that, ENCODE’s hype notwithstanding, junk DNA is really fairly common. See Dan Graur’s perhaps somewhat biased 🙂 but still quite nice take on this at his “Judge Starling” blog (http://judgestarling.tumblr.com/). I’d point towards a particular post, but this topic is at least half of his posts on his blog, so… 🙂 (A more polite and calm recent paper in Trends in Genetics makes much the same point about the wild overselling of ENCODE: Lawrie D.S. and Petrov D.A. (2014) Comparative population genomics: power and principles for the inference of functionality. Trends Genet. 30, 133–139) One lesson? We aren’t just trying to forge theories to predict and explain the world; we are also trying to find out what the world is like! The actual state of the world is part of what science is about, and sometimes it is harder to figure even that out than we might hope…

One interesting wrinkle re: proofs. What do we take to be a proof in math and logic? That is, what *counts* as “airtight reasoning” and why? The question is not entirely trivial — from those who feel that only constructive proofs should ‘count’ to the more general question of how we can justify our most basic rules of inference, what it means for B to follow, in a deductively certain fashion, from A is, at least in some sense, not given by the system itself, or demanded by the world more generally. Nor is the idea of a contradiction quite as straightforward as we sometimes pretend… This doesn’t make math more like science, per se, but it should influence our understanding of, and ways of thinking about, foundational issues in math and logic.

Two irritating questions:
1) numbers are arbitrary. So do proofs ever really prove anything?
2) why do we have to do them in school when we have to take a year of rhetoric later anyway? Just tell me 3+3=6 and I’ll believe you and move on.

Most proofs you see in school don’t concern specific numbers. (Bertrand Russell famously spent 250 pages proving 1 + 1 = 2, but that’s very exceptional.) Instead, you prove things like this:

-The square root of two cannot be written as a ratio of integers.
-There is an infinite quantity of prime numbers.
-If you add up the first N whole numbers (1 + 2 + 3 + … + N), the total is (N+1)N/2.

Also, in general, I’m not sure I’d agree that numbers are arbitrary. Their names are arbitrary; the base 10 system is arbitrary; our notation is often arbitrary; but numbers are kind of just numbers.

Not advocating against it; I think math is crucial to our current social functionality, but I think I’d rather leave proofs to those who like them; and get back to teaching kids multiplication tables instead of how to count on their fingers.
And numbers can be manipulated to say whatever you want, especially in economics where everything is make-believe 😉

Proof is fact, the numbers produced by them are not “make-believe”, instead it’s language that is manipulated to spin the results.
It’s a fact that about 50% of students are below-average (by the definition of average). I’ve seen an article that claimed that schools are failing because of this fact.
As a slightly more serious case, p-values in statistical analysis were once very popular and even used in court cases. This lead to wrongful convictions, it wasn’t that there was any fault in the calculations or proofs but instead that the results were misinterpreted.

If math taught me anything, it taught me that my answers, although correct, were incorrect because of the Way I found my answer. I constantly argued with teachers about it. As an adult, math taught me that I could use the same steps, but maybe round them ever so slightly different; or start at a different place from the decimal point to get the facts I wanted.

I also agree that interpretation is a problem, but is it a problem entirely because of perception, or because numbers aren’t really proof until we say the are?

Proof is not fact. Proof is what we accept as fact.

Given that knowledge, I can calculate numbers to get any result I want, especially for political gain. Numbers are a language, and any language can be spun.

As an example, I can take a Charles Murray quote that isn’t about students at all, and translate it into a numerical argument, maybe make a number of it claiming that 50% of students are idiots.

Then someone can use this number I write and article. The other people will see this number in articles that were aggregated from ten other articles all reporting the same unchecked, make-believe number I created from a quote that had no number at all, which in reality is an imaginary number, like infinity; or arbitrary, like “half.”

And Murray said half of all children. But who’s? Did they actually count them? No number of children was given, so the number is factually zero. If half of zero children are below average, then numerically, No children are below average. Numbers don’t lie, right?

But let’s revisit another part of the fact. What is average? Neither Murray nor your article provided a specific number, so let’s use zero again. Ok, now all children are in the negative numbers for brains. So, using the numbers provided, we can factually conclude that all kids are dumber than my end table.

Now, your article changed Murray’s words into a number to represent students specifically. But out of how many students? Let’s say all US school children: did someone count them all? Certainly not, since stats and figures are actually estimates like the census since no one is going to pay the labor for anyone to actually count heads.

So…where were we again? Oh yeah, proof; it’s all based on made up numbers.

l will concede that it’s easy to say that with things like polls and stats, but what about hard measurement? I know my ceiling is 8 feet high. How do I know that? Because my ruler says so. Oh, ok, who made the ruler? How did they determine the length of an inch or how many would equal a foot? What proof or fact was that based on? Oh, right, someone made it up. Before that, my plot of land was just expensive as your plot of land because you measured it’s worth in footsteps, but I measured by how many cows were in it. Then I gave one of my cows to the chieftain, and he agreed that one-cow land was worth more than 43,550 square feet.

We need math, to be sure, because this type of arguing is silly. But we have to have universal agreement and acceptance first.

But to non-math people, proofs are simply busy work for number nerds (respect). The basic proofs are good to include in the “average” math curriculum because they’re thinking exercises we need (like gym class; we all hate it, but it’s necessary for development). But math, numbers, proofs, all of it isn’t some end all be all: we had nine planets, ’til we didn’t; the numbers that told us cigarettes, Risperdal, and petrochemicals in our water were harmless were right, until they weren’t; and even Einstein’s EMC was spot on, until someone redid the numbers.

The reality is that nothing is really definite, and nothing can actually be proven without presupposing the proof.

Pedantically, “It’s a fact that about 50% of students are below-average (by the definition of average)” is incorrect. If, on a test, 19 students score 50% and 1 scores 100%, the average is 52.5% and 95% of the students scored below average.

is 20 a properly representative sample?

“Science’s measure of truth is the experiment. A theory may be perfectly consistent, wonderfully elegant, and intellectually satisfying—but if it fails to predict what happens in the world, then we toss it on the compost heap and move on.” You may be fully aware of this but I’ll say it anyway: That’s a drastic simplification. In fact many scientific theories and models fail to predict what happens in the world at some level, but still are useful. Newton’s theory of gravity, for example, is demonstrably wrong; general relativity provides a better description of gravity under a wider range of circumstances (but is also presumed wrong, since it is inconsistent with quantum mechanics). Nevertheless Newtonian gravity is used, for example, by astrophysicists studying dark matter, since it is for all practical purposes as good as GR in describing motions of stars and galaxies and is far more tractable. Similarly for many other situations. Models are constructed that certainly do not correspond to reality, but encapsulate some salient features of reality. Some theorists even make a living studying, say, quantum fields in two dimensions, or solutions to GR that do not correspond to the real universe… because they provide insight into quantum fields or relativity beyond the specific, nonphysical example.

Thanks – that’s a good discussion of the issue. “Toss it on the compost heap” is probably overboard.

I’d say the basic distinction still holds: mathematicians care about idealizations and abstractions for their own sake, while scientists employ abstractions and models only for the sake of gleaning insight about reality. But I probably undersold the variety of methods that scientists draw on. (Obviously, astronomers can’t limit themselves to conducting “experiments.”)

how we can justify our most basic rules of inference

See “What The Tortoise Said To Achilles”. In summary, every time Achilles proposes a new rule of inference, the Tortoise promptly flattens it into another theorem of the system.

If you affirm “Socrates is a man” and “Phaedrus is a man” and deny “Socrates and Phaedrus are men”, I will think that you do not understand the term “and”, not that you actually have different reasoning than I do. Similarly, if you affirm both “Socrates is dead” and “Socrates is alive”, I will think you are just confused about the meanings of English words. If you don’t accept the usual rules of inference, what method of inference could I possibly use to convince you of their correctness?

John: Your questions are well taken. (Though the article “What Achilles Should Have Said to the Tortoise” is interesting in this context, too.)

But consider: many of us are happy to affirm that there are the same number of natural numbers as odd natural numbers, while some people are not, and affirm, quite adamantly, that there must be twice as many natural numbers as odd natural numbers. Not all of the people in the latter class are mathematically naive. Which group doesn’t understand what “counting” means, or what it means for two sets to have the same number of elements?

Or consider the question of whether we should accept the axiom of choice in set theory. Does it lead to contradictions? (If it does, we should reject it!) Or are the results merely surprising, revealing perhaps that certain terms didn’t mean what we thought they meant? Could we get the benefits of choice w/o having to accept the oddities that result by picking a different axiom? If we could, should we?

Finally, some very serious people think that proofs should be restricted in ways that respect *relevance.* This is a weaker constraint than demanding that all proofs be constructive, but still more demanding than what we normally think of as standard. Whose right? Why? Is who is right given obviously by the *meanings* of our terms, in any ordinary way?

In both cases, it is a question of how best to extend terms from a finite set (of natural numbers in the first case, of anything in the second case) to an infinite set. We know that there are half as many evens and naturals in every finite set of natural numbers, and that the axiom of choice holds for every finite set.

But we don’t in general have direct intuitions about infinity, so the question is, “What is most useful?” It turns out that there are useful theories in both directions: thinking about “The number of evens is the same as the number of natures” leads to infinite cardinality; thinking about “The number of evens is half the number of naturals” leads to Lebesgue measure. Likewise, there are non-standard (but still interesting) set theories that reject the axiom of choice in favor of some other axiom, since its negation is known not to be a theorem of ZF set theory.

And this in turn leads to the perception that mathematics and science (and even art) aren’t so different after all. For one thing, mathematics isn’t a timeless monolith. There was a definite point in time and a definite person who discovered group theory: all the theorems of group theory were just as true on the day before Évariste Galois’s death as they are now, but nobody had thought of them. For another, proof is the mechanism mathematicians use to convince other mathematicians, and in practice what matters is that proofs are convincing, not that they are in some incomprehensible sense correct.

Great post! Deductive vs. inductive reasoning always makes me think of the song “Circumstances” by Rush: “Now I’ve gained some understanding of the only world that we see [inductive], things that I once dreamed of have become reality [deductive].”

Thanks – gotta go listen to that Rush song now!

Sorry, but junk DNA is still mostly junk DNA, the hype-driven frenzy about the ENCODE misstatement notwithstanding. The way that biologists tell whether DNA serves a biological function (as opposed to having accidental chemical interactions of no significance) is by looking for indications of selection for or against the DNA sequence. Without evidence of selection, it is extremely unlikely that the DNA has meaningful biological function. The amount of “junk DNA” varies enormously from organism to organism—currently the best estimates of the amount of junk in human DNA is around 90% of the genome. Each new find of a new DNA function changes the estimate slightly (affecting usually around 0.1% or less of the genome).

Dan Graur is one researcher who is particularly irritated by the constantly stated “death of junk DNA”. For example, see http://judgestarling.tumblr.com/post/67599627086/a-pre-refuted-hypothesis-on-the-subject-of-junk-dna

You can find several of his other posts on the subject by googling

judge starling junk DNA

Ah, thanks! I’ll edit the post.

A bit of an aside: There is good reason to suspect that, ENCODE’s hype notwithstanding, junk DNA is really fairly common. See Dan Graur’s perhaps somewhat biased 🙂 but still quite nice take on this at his “Judge Starling” blog (http://judgestarling.tumblr.com/). I’d point towards a particular post, but this topic is at least half of his posts on his blog, so… 🙂 (A more polite and calm recent paper in Trends in Genetics makes much the same point about the wild overselling of ENCODE: Lawrie D.S. and Petrov D.A. (2014) Comparative population genomics: power and principles for the inference of functionality. Trends Genet. 30, 133–139) One lesson? We aren’t just trying to forge theories to predict and explain the world; we are also trying to find out what the world is like! The actual state of the world is part of what science is about, and sometimes it is harder to figure even that out than we might hope…

One interesting wrinkle re: proofs. What do we take to be a proof in math and logic? That is, what *counts* as “airtight reasoning” and why? The question is not entirely trivial — from those who feel that only constructive proofs should ‘count’ to the more general question of how we can justify our most basic rules of inference, what it means for B to follow, in a deductively certain fashion, from A is, at least in some sense, not given by the system itself, or demanded by the world more generally. Nor is the idea of a contradiction quite as straightforward as we sometimes pretend… This doesn’t make math more like science, per se, but it should influence our understanding of, and ways of thinking about, foundational issues in math and logic.

D’oh! Gasstation beat me to it! W/ much the same cites, no less!

Well, my error probably deserved two thwacks upside the head, so it’s just as well you both chimed in.

Two irritating questions:

1) numbers are arbitrary. So do proofs ever really prove anything?

2) why do we have to do them in school when we have to take a year of rhetoric later anyway? Just tell me 3+3=6 and I’ll believe you and move on.

Most proofs you see in school don’t concern specific numbers. (Bertrand Russell famously spent 250 pages proving 1 + 1 = 2, but that’s very exceptional.) Instead, you prove things like this:

-The square root of two cannot be written as a ratio of integers.

-There is an infinite quantity of prime numbers.

-If you add up the first N whole numbers (1 + 2 + 3 + … + N), the total is (N+1)N/2.

Also, in general, I’m not sure I’d agree that numbers are arbitrary. Their names are arbitrary; the base 10 system is arbitrary; our notation is often arbitrary; but numbers are kind of just numbers.

Not advocating against it; I think math is crucial to our current social functionality, but I think I’d rather leave proofs to those who like them; and get back to teaching kids multiplication tables instead of how to count on their fingers.

And numbers can be manipulated to say whatever you want, especially in economics where everything is make-believe 😉

Proof is fact, the numbers produced by them are not “make-believe”, instead it’s language that is manipulated to spin the results.

It’s a fact that about 50% of students are below-average (by the definition of average). I’ve seen an article that claimed that schools are failing because of this fact.

As a slightly more serious case, p-values in statistical analysis were once very popular and even used in court cases. This lead to wrongful convictions, it wasn’t that there was any fault in the calculations or proofs but instead that the results were misinterpreted.

If math taught me anything, it taught me that my answers, although correct, were incorrect because of the Way I found my answer. I constantly argued with teachers about it. As an adult, math taught me that I could use the same steps, but maybe round them ever so slightly different; or start at a different place from the decimal point to get the facts I wanted.

I also agree that interpretation is a problem, but is it a problem entirely because of perception, or because numbers aren’t really proof until we say the are?

Proof is not fact. Proof is what we accept as fact.

Given that knowledge, I can calculate numbers to get any result I want, especially for political gain. Numbers are a language, and any language can be spun.

As an example, I can take a Charles Murray quote that isn’t about students at all, and translate it into a numerical argument, maybe make a number of it claiming that 50% of students are idiots.

Then someone can use this number I write and article. The other people will see this number in articles that were aggregated from ten other articles all reporting the same unchecked, make-believe number I created from a quote that had no number at all, which in reality is an imaginary number, like infinity; or arbitrary, like “half.”

And Murray said half of all children. But who’s? Did they actually count them? No number of children was given, so the number is factually zero. If half of zero children are below average, then numerically, No children are below average. Numbers don’t lie, right?

But let’s revisit another part of the fact. What is average? Neither Murray nor your article provided a specific number, so let’s use zero again. Ok, now all children are in the negative numbers for brains. So, using the numbers provided, we can factually conclude that all kids are dumber than my end table.

Now, your article changed Murray’s words into a number to represent students specifically. But out of how many students? Let’s say all US school children: did someone count them all? Certainly not, since stats and figures are actually estimates like the census since no one is going to pay the labor for anyone to actually count heads.

So…where were we again? Oh yeah, proof; it’s all based on made up numbers.

l will concede that it’s easy to say that with things like polls and stats, but what about hard measurement? I know my ceiling is 8 feet high. How do I know that? Because my ruler says so. Oh, ok, who made the ruler? How did they determine the length of an inch or how many would equal a foot? What proof or fact was that based on? Oh, right, someone made it up. Before that, my plot of land was just expensive as your plot of land because you measured it’s worth in footsteps, but I measured by how many cows were in it. Then I gave one of my cows to the chieftain, and he agreed that one-cow land was worth more than 43,550 square feet.

We need math, to be sure, because this type of arguing is silly. But we have to have universal agreement and acceptance first.

But to non-math people, proofs are simply busy work for number nerds (respect). The basic proofs are good to include in the “average” math curriculum because they’re thinking exercises we need (like gym class; we all hate it, but it’s necessary for development). But math, numbers, proofs, all of it isn’t some end all be all: we had nine planets, ’til we didn’t; the numbers that told us cigarettes, Risperdal, and petrochemicals in our water were harmless were right, until they weren’t; and even Einstein’s EMC was spot on, until someone redid the numbers.

The reality is that nothing is really definite, and nothing can actually be proven without presupposing the proof.

Did somebody say Economics?

https://mathwithbaddrawings.com/2014/05/07/the-mathematical-dialect-quiz/

https://mathwithbaddrawings.com/2013/08/21/five-math-experts-split-the-check/

Haha excellent

Pedantically, “It’s a fact that about 50% of students are below-average (by the definition of average)” is incorrect. If, on a test, 19 students score 50% and 1 scores 100%, the average is 52.5% and 95% of the students scored below average.

is 20 a properly representative sample?

“Science’s measure of truth is the experiment. A theory may be perfectly consistent, wonderfully elegant, and intellectually satisfying—but if it fails to predict what happens in the world, then we toss it on the compost heap and move on.” You may be fully aware of this but I’ll say it anyway: That’s a drastic simplification. In fact many scientific theories and models fail to predict what happens in the world at some level, but still are useful. Newton’s theory of gravity, for example, is demonstrably wrong; general relativity provides a better description of gravity under a wider range of circumstances (but is also presumed wrong, since it is inconsistent with quantum mechanics). Nevertheless Newtonian gravity is used, for example, by astrophysicists studying dark matter, since it is for all practical purposes as good as GR in describing motions of stars and galaxies and is far more tractable. Similarly for many other situations. Models are constructed that certainly do not correspond to reality, but encapsulate some salient features of reality. Some theorists even make a living studying, say, quantum fields in two dimensions, or solutions to GR that do not correspond to the real universe… because they provide insight into quantum fields or relativity beyond the specific, nonphysical example.

Thanks – that’s a good discussion of the issue. “Toss it on the compost heap” is probably overboard.

I’d say the basic distinction still holds: mathematicians care about idealizations and abstractions for their own sake, while scientists employ abstractions and models only for the sake of gleaning insight about reality. But I probably undersold the variety of methods that scientists draw on. (Obviously, astronomers can’t limit themselves to conducting “experiments.”)

how we can justify our most basic rules of inferenceSee “What The Tortoise Said To Achilles”. In summary, every time Achilles proposes a new rule of inference, the Tortoise promptly flattens it into another theorem of the system.

If you affirm “Socrates is a man” and “Phaedrus is a man” and deny “Socrates and Phaedrus are men”, I will think that you do not understand the term “and”, not that you actually have different reasoning than I do. Similarly, if you affirm both “Socrates is dead” and “Socrates is alive”, I will think you are just confused about the meanings of English words. If you don’t accept the usual rules of inference, what method of inference could I possibly use to convince you of their correctness?

John: Your questions are well taken. (Though the article “What Achilles Should Have Said to the Tortoise” is interesting in this context, too.)

But consider: many of us are happy to affirm that there are the same number of natural numbers as odd natural numbers, while some people are not, and affirm, quite adamantly, that there must be twice as many natural numbers as odd natural numbers. Not all of the people in the latter class are mathematically naive. Which group doesn’t understand what “counting” means, or what it means for two sets to have the same number of elements?

Or consider the question of whether we should accept the axiom of choice in set theory. Does it lead to contradictions? (If it does, we should reject it!) Or are the results merely surprising, revealing perhaps that certain terms didn’t mean what we thought they meant? Could we get the benefits of choice w/o having to accept the oddities that result by picking a different axiom? If we could, should we?

Finally, some very serious people think that proofs should be restricted in ways that respect *relevance.* This is a weaker constraint than demanding that all proofs be constructive, but still more demanding than what we normally think of as standard. Whose right? Why? Is who is right given obviously by the *meanings* of our terms, in any ordinary way?

In both cases, it is a question of how best to extend terms from a finite set (of natural numbers in the first case, of anything in the second case) to an infinite set. We know that there are half as many evens and naturals in every finite set of natural numbers, and that the axiom of choice holds for every finite set.

But we don’t in general have direct intuitions about infinity, so the question is, “What is most useful?” It turns out that there are useful theories in both directions: thinking about “The number of evens is the same as the number of natures” leads to infinite cardinality; thinking about “The number of evens is half the number of naturals” leads to Lebesgue measure. Likewise, there are non-standard (but still interesting) set theories that reject the axiom of choice in favor of some other axiom, since its negation is known not to be a theorem of ZF set theory.

And this in turn leads to the perception that mathematics and science (and even art) aren’t so different after all. For one thing, mathematics isn’t a timeless monolith. There was a definite point in time and a definite person who discovered group theory: all the theorems of group theory were just as true on the day before Évariste Galois’s death as they are now, but nobody had thought of them. For another, proof is the mechanism mathematicians use to convince other mathematicians, and in practice what matters is that proofs are convincing, not that they are in some incomprehensible sense correct.