MadMath

It's common nowadays in conversations about the method of proof-by-contradiction for someone to pop in and say, "People think Euclid's proof of there being infinite prime numbers uses proof-by-contradiction, but it doesn't, it's a direct proof". For example, the current Wikipedia article on Euclid's theorem says this:

Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the finite set initially considered contains all prime numbers, though it is actually a proof by cases, a direct proof method. The philosopher Torkel Franzén, in a book on logic, states, "Euclid's proof that there are infinitely many primes is not an indirect proof..."

Okay, admittedly Euclid's theorem is not in its entirety structured as a proof by contradiction. Yes, there's a proof by cases, in which a number of the form \(lcm(ABC) + 1\) is assessed as being either prime or not prime. But the core of that second case is clearly a proof by contradiction!

If we look at Euclid's original text for the second case, we see the following. Given that \(lcm(ABC) + 1\) is not prime, Euclid takes \(G\) to be some prime number that divides it. He then reasons like this (looking at the Fitzpatrick translation of Heiberg's presentation of the Greek):

I say that \(G\) is not the same as any of \(A, B, C\). For, if possible, let it be... [some logic here]... The very thing is absurd. Thus, \(G\) is not the same as one of \(A, B, C\).

This form is patently a proof by contradiction, and use of the phrase with "absurd" (in the original Greek, "ὄπερ ἄτοπον") highlights that fact.

While the overall superstructure of Euclid's theorem is not a proof by contradiction... Yes Virginia, Euclid's theorem uses a proof by contradiction, and it's an essential part of his proof that there are infinite primes.

Let's say you're an instructor in a low-level math course; maybe something like college algebra, a liberal-arts math course, or something similar. It's possible that you consider this to be beneath you and you're bored in class. Here's a little game you can play with yourself that will spice things up a bit:

When you ask a question to the class, and someone answers incorrectly, see if you can conversationally edit the question in a way that the student would have been right. That is, respond by starting with, "Well, that would be right if the problem said ____", and fill in the end of that sentence in appropriate way. 

I'm very fond of this technique. It actually accomplishes several things:

• Makes things a bit more challenging for the instructor, keeping them on their toes
• Cushions the "no you're wrong" response to the student (a bit like the "shit sandwich" feedback protocol)
• Force you to diagnose & clarify the misdirected mental pathway for you and the student (and in fact usually the student has misperceived some pattern that's just adjacent to the given problem).

Try it and see how it feels. To be clear: I don't do this because I'm bored in class, but nevertheless I've found it to be a compelling and clarifying technique.

Here's a fact that I've never seen expressed clearly, or in this way, in any of the several college-level algebra books from which I've taught. Say we're working in the domain of real numbers, and have a radical of some index over a variable to a power. In broad strokes, we can divide the power by the radical index -- however, in some cases, distressingly, you'd need an absolute value to express the result. The question is, exactly when do you need that absolute value?

$$\sqrt[n]{a^m} = a^{m \over n} \text{ or } |a^{m \over n}|? $$

Say we're in this situation, with m and n whole numbers, and m is evenly divisible by n. The primary issue is that when the initial power m is even, it makes any product nonnegative, wiping out any negatives that the base a might represent. So when the reduced power m/n is odd, then it would fraudulently claim to possibly produce negatives, which our initial expression cannot do -- and so require the absolute value as a correction. 

Let's give more detail by inspecting all the permutations of even/odd possibilities between the starting and ending powers in the expression:

1. Odd m, odd m/n: The odd starting power m can produce values of any sign, and so can the odd reduced power m/n. So all is fine here, and we don't need the absolute value.
2. Even m, even m/n: The even power m wipes out any negatives, and the reduced even power m/n does the same thing. So again they're aligned, and no absolute value is needed.
3. Even m, odd m/n: This is the case alluded to above -- the even start power m wipes out negatives, but the odd ending power m/n would deceive us into thinking negatives could be a possible product. This is the situation in which we need the absolute value as a correction.
4. Odd m, even m/n: This case is impossible. If m/n is even, then any multiple (e.g., by n to produce m) is also even.
   

So it's only that third case in which the power switches from even to odd where we need the absolute value bars for full fidelity. Interestingly, since the fourth case can't happen, we could express the protocol briefly as follows:

When the powers switch parity, then we need absolute value bars.

Comedian Nate Bargatze talks about his experience taking remedial classes at a community college in Tennessee:

The Iron Law of Stack Exchange (Stack Overflow):

They hate hard questions from new users.

Fundamentally, voters and respondents on Stack Exchange like scoring points with answers that are obvious, easy to write, and don't take excessive amounts of thought. So there is some amount of irritation to questions that are fundamentally hard, and resist such easy answers. This perceived annoyance is exacerbated by a new (relatively low-ranked) user asking a question on any site, and the first instinct by members is often to look for some way in which the question can be rejected as being poorly-formed.

 Symptoms of this reaction include:

• Closing or down-voting a question on poorly-justified grounds.
• Editing the question to change it to an easier one.
  
• Accusing the question of being an "X-Y problem", that is, the inquirer is confused and really should have asked a different, easier question.
• Complaining about interactions that are common across the Stack Exchange network, but which a new user might not know is commonplace, and so be cowed in that way.

As one personal example: the Stack Overflow coding site is not my top network destination, but over the years I have asked a number of questions there. As a CS faculty member and past professional developer, by the time I need to reach out externally for help, I've exhausted a rather deep search for answers, and my questions are likely to be fairly hard to crack. This almost always results in exasperation and negative votes from users of the site.

In my last question, I asked about a feature of a certain piece of software, which seemed like it should have a rather obvious behavior (based on how relative pathnames should work in the OS), but I couldn't get it to work right. Several comments suggested the obvious behavior, which I was pointing out was failing. The only actual answer came from the actual developer of the software, who again asserted how it should work -- and whom, after some back-and-forth, I ultimately convinced about there being a bug in their work, that they agreed to fix it in the next version.

Despite this rewarding result, no one else could successfully answer this question, and it was (as usual) downvoted into negative territory. Immediately thereafter my Stack Overflow account was actually locked out from asking further questions because of the history of negative votes it garnered.

Given the downward trend of traffic to Stack Exchange in the last year or so (even predating the earthquake of generative AI in that time), it seems like potentially a difficult problem for the site. Over time, it would seem that most of the low-hanging fruit will already be answered, really only leaving hard problems yet to be dealt with -- and these are specifically the ones that are met with more hostility and likely to be ejected by the most dedicated users of the site.

Have we arguably stepped into the singularity? As of last November, OpenAI's release of the ChatGPT language-model system has upended most everything in sight, and in particular, sent educators everywhere scrambling to deal with the ramifications. This chatbot can seemingly craft custom essays, reports, scientific papers, newspaper articles, programming code, and solutions to many (although not all) mathematical problems. Immediately, for free, and in ways almost no human can detect.

Here's a roundup of news stories that I may update in the future:

• Many students at Stanford report using ChatGPT for end-of-semester assignments and exams
  
• ChatGPT passes final exam in an MBA course at Wharton
  
• Professor at U. Toronto says, "you can no longer give take-home exams/homework"
  
• ChatGPT added as author to many academic papers 
• Top AI conference forced to ban ChatGPT from writing science papers
• Scientists unable to tell when ChatGPT has written abstracts for academic papers
• More science journal publishers forced to ban use of ChatGPT on submitted papers 
• Stack Overflow forced to ban ChatGPT answers from the site.
• ChatGPT considered an existential threat to Google.
• Microsoft plans to quickly bring ChatGPT features to Bing, Office, and Azure
  
• Kindle novelists start widely using ChatGPT
• NYC Department of Education bans use of ChatGPT in all schools
• CNET caught quietly using AI to write articles
• BuzzFeed (which broke story above) says it will use ChatGPT to write articles, stock jumps 150%.
• Member of Congress reads a speech written by ChatGPT into official record
• Judge in Columbia use ChatGPT to write a legal ruling
• ChatGPT has the fastest-growing user base of any application in history
• ChatGPT passes Google employment coding interview
  

Image courtesy Craiyon. :-)

Ed Knight is a teacher in New York state. Writing at Medium, he points out the disturbing fact that the vaunted "New York Regents" exams required to graduate from high schools in the state have become completely trivial to pass. For example: In the Algebra Regents, you can ignore all of the (already simple) open-response questions, and just mindlessly mark "C" for all the multiple-choice questions, and you'll be given a passing grade.

Answer all Cs, you get 9 right. Each one is worth two points. That’s 18 points, enough to pass.

More than enough to pass, in fact.

I really didn’t know what to say when I first saw this. I suppose I was thinking like Chuck Heston. “They finally, really did it.” This is jaw-dropping malfeasance. I wouldn’t believe that anyone who claims to care about kids could let this happen — not until I saw it happen.

Shame on NYSED and the Regents.

Really, the root of this problem is the insane scaling procedure that the NY Regents has been doing for years to fake up the test scores. Below is the most recent test's table for converting a "Raw Score to a reported "Scale Score". The scale score is 0-100, making recipients thinks it's a percentage, but it's not. For example: if you score a raw 27 out of the possible 86 points (that's 31% correct), this then gets converted to a reported Scale Score 65 -- i.e., a Performance Level of 3 out of 5, which is considered passing.

Think about that: for years, the NY Regents has considered a score of about 30% as passing for a basic (very simple!) algebra test. And yes, this was exacerbated because for the pandemic years (still ongoing), the policy was adjusted to accept even lower scores than that -- now as low as 20% (i.e., Raw 17, reported as a Scale 50). 

Read more at Medium: Guessing C For Every Answer Is Now Enough To Pass The New York State Algebra Exam