Large Values of 2

So, in my last post, I asked you to ponder the following question:

  • How many tons do you have when you put two 14-ton dump trucks on the same scale?

Now, because of my stunning prescience, I already know your answer:

“Wait…a question? What question? THAT question?!? You thought
I’d actually ponder some question? Bahahahahahahaha…[passes out]”

Now that you’re back…

Skyrim’s Ralof doing his best Samson impression.

…let’s actually address this question.

Sooooo…(14 tons) + (14 tons) = 28 tons. Easy enough, right? Right?

Well, first, what is meant by “14-ton dump truck?”* Do you really think that it is exactly (14 tons) × (2000 pounds/ton) = exactly 28,000 pounds? Do you really think that two of them give exactly 56,000 pounds? As in exactly 56,000.000000000000000000 pounds?

*To make sure that we’re not just playing games, let’s assume that these trucks are empty, and we won’t be putting anything into the dump bed.

Sure, that’s the pure calculation of “14 × 2000 = 28,000” and of “2 × 28,000 = 56,000,” but exactly 56,000.000000000000000000 pounds? Oh, c’mon now. That 18th decimal place is so small and so sensitive to change that the impossibly small amount of rust that accumulated (or the impossibly small amount of paint that flaked off in the gentle breeze) while you were reading this sentence changed that value somewhat.

Vehicle weights are typically rounded off to the nearest ton. Consequently, a “14-ton” truck could really be anywhere between 27,000 and 29,000 pounds.*

*As a personal example, my family owns two Nissan Pathfinders. My wife drives the considerably newer one, and it is considered a 2-ton vehicle. “So it weighs 4,000.0000000000 pounds?” Uh, no. It’s a “chubby” two tons, coming in around 4,300 pounds.

What would happen if both of those dump trucks were, say, 28,600 pounds. By using conventional terminology, these trucks would both be classified as “14-ton” trucks.

But their combined weight would be 28,600 + 28,600 = 57,400 pounds. Now, 57,400 pounds is closer to 58,000 pounds (29 tons) than it is to 56,000 pounds (28 tons).

Thus, we have an example here of (14 tons) + (14 tons) = 29 tons.

“14 + 14 = 29? That’s heresy!” Well, yes. Uh…I mean, no… Uh…I mean, yes, “14 + 14 = 29”; but, no, it’s not heresy. I’m afraid you’ll just need to look for another witch (I guess I’d actually be a warlock in this case.) to burn.

But recall all of this in the context of the previous post:

  • In the context of the integers under normal arithmetic, 2 + 2 is certainly 4.
    • Similarly here, in the context of the integers under normal arithmetic, 14 + 14 is certainly 28.
  • In the context of addition modulus 3, 2 + 2 is certainly 1.
    • Similarly here in the context of addition modulus, say, 20, then 14 + 14 is certainly 8. [Without rehashing all of the last post, this calculation is 14 + 14 = 28 (“too big!”). Subtracting 20 gives 28 – 20 = 8.]
  • In the context of adding water and alcohol, 2 + 2 is certainly 3.84 (within rounding)
    • In the context of adding water and (a LOT of) alcohol, 14 + 14 is certainly 26.88.

But now we add a new context entirely:

  • In the context of approximation,
    (approximately 14) + (approximately 14) could possibly be (approximately 29).
    • Consider the 28,600-pound trucks in the original example.
  • Of course, we could also have
    (approximately 14) + (approximately 14) could possibly be (approximately 28).
    • Consider one truck being about 27,800 pounds and the other being about 28,500 pounds, giving 56,300 total pounds.
  • Or finally, we could have
    (approximately 14) + (approximately 14) could possibly be (approximately 27).
    • Consider two trucks being about 27,400 pounds each, giving 54,800 pounds total.

There’s a joke among mathematicians* that “Two plus two equals five (for large values of two).”

*Yes, those exist. That is, I mean “jokes among mathematicians”–not “mathematicians.”**

**Wait, I mean, yes, mathematicians also exist. It’s just that I meant…uh…[head…hurts…SO…BAD……]

Well, considering the above scenarios, it’s quite possible that two “2-pound” objects could technically weigh approximately 2.35 pounds each (“large values of 2” in the joke). Their combined weight would be 4.7 pounds, rounding off to 5 pounds.

Now, I can anticipate an objection: “If we know they’re 2.35 pounds each, why don’t we just say so and have a total of 4.7 pounds? That way, we don’t have to resort to rounding (and your subsequent mathematical witchcraft…)”

I would certainly agree with that. But–leaving aside our unhealthy excitement over decimal places–consider that sometimes we do not know that they’re 2.35 pounds each. We often do not know that the combined weight is 4.7 pounds. We often cannot know such information.

Why not? Well, simply put, you need a scale that can measure to that level of accuracy.

Let me concede that this isn’t necessarily a problem when measuring things around 4-5 pounds. Grocery scales, for example, often measure to the fraction of an ounce. If you were to rummage around my house, you would find a simple home scale* that measures to the gram (approximately 1/28 of an ounce).

*And I would invite you to do so as well. I haven’t been able to find that thing for ages.

But, what a lot of people do not comprehend is that even these “precise” values are rounded off, too.

It’s not that hard to understand that, if a scale only displays the nearest pound, then anything between 4.5 and 5.5 pounds will round off to “5” on the display.* This is more likely to been seen on a bathroom scale that shows a weight of 153.** In this instance, the displayed “153” could be anything between 152.5 and 153.5.

*”But I learned in school that 5.5 rounds up to 6?” You’re not wrong, but I’ll get to why that doesn’t ever really come into play here.

**My personal experience considers a weight of 153 to be purely theoretical.

But a scale that displays an extra decimal is also rounding off as well. For instance, the bathroom scale at my home might display 153.2 (again, purely theoretical). Here, the displayed “153.2” could be anything between 153.15 and 153.25.

Well, if precision matters, why not just add another decimal place?

Or another?

Or…another?

When does it stop?

Unfortunately, it never stops. And when it comes to things that are measured (such as weight or height), we can only ever know things within a certain range of values.* Even a scale that measures something as precisely as “153.18720043”** is just rounding off an actual value that is somewhere between 153.187200425 and 153.187200435.

*And that’s why theoretically we don’t worry about rounding, say, 5.5 up to 6. Nothing measured is ever exactly 5.5 anyway. It’s always a little bit more or a little bit less than 5.5.

**And something that can measure that accurately would be fantastically expensive.

There is a massive difference between things that can be measured and things that can be counted. I can know with absolute certainty that, having counted them, there are exactly 70 mugs in my office right now.*

*That is not an exaggeration, and that only counts the ones that I can actually use to drink my (obscene amounts of) coffee.

What I cannot know with absolute certainty is the exact volume of any of them. Does my “Happy Narwhal” mug* hold 15 fluid ounces? or 15.2? or 15.18? or 15.183? or 15.1831? or… I hope by now that you get the point.

*Thanks, kids! I know it’s actually my “Hypodermic Narwhal” mug, but there’s just not time for an explanation here. Besides, this blog is ultimately for you, and you already know the meaning behind “Hypodermic Narwhal.”

Yes, I drink from this in public.

And this is where the uncertainty creeps in. Not in the actual calculation, per se, but in the numbers that can be put into that calculation in the first place.

Without belaboring the point (for now*), the term for measurable things is continuous, and the term for counted things is discrete.

*After all, I can’t help but belabor points. I’m just going to do that…uh…’belaboring’ in the next post.

The mathematics of counting values is pretty straightforward [i.e., (70 current mugs) + (15 new mugs from my rabid fan base) = (85 total mugs)]. The mathematics of measured values, however, is a lot trickier. For example, pouring about 12 ounces of coffee out of a carafe that holds about 60 ounces leaves about 48 ounces in the carafe.

But what if it’s a “small value of 12 ounces” and a “large value of 60 ounces”? What does that say about the remaining “48 ounces”?

If that sounds like an overly-complicated way to think about this situation, I’d wholeheartedly agree. After all, who really cares about the precision of pouring a cup of coffee? Especially if I’m just doing so at home or the office, why would I care about 11.7 versus 11.8 ounces?

But where it matters is if we know that approximately 48% of voters will vote one way and approximately 47% vote the other. This is especially because “48%” in these contexts does not mean “between 47.5% and 48.5%” but rather “between 45% and 51%.” And, similarly, “47%” means “between “44% and 50%.”

There’s a lot of overlap in those ranges, and a “small value of 48%” combined with a “large value of 47%” can lead to an immensely different political landscape than the original “48% v. 47%” would imply.

What fun…

Russian Potato Soup

So last week’s blog began a series of posts dealing with certainty and ended on a cliffhanger*: Is mathematics certain?

*OK, considering most of you fell asleep before the end of that post, cliffhanger might be a tad strong.

Frankly, that’s a pretty stupid question.

Now, some would remark, “Of course, it’s a stupid question. Mathematics is certain.”

Congratulations…you’re as stupid as that question.

“What?!? You’re saying mathematics is not certain?!?”

Additional congratulations on plumbing even further depths of stupid because I’m not saying that either.

I’m not saying it’s a stupid question because the answer is obvious. I’m saying that it’s a stupid question because it creates a false dilemma. It forces a “yes” or “no” answer to a question that cannot be answered either way.

“OK, heretic! I’m done with your blog!”

Photo: the late Powers Boothe OWNING it on screen

OK, now that we’ve gotten rid of the morons who proved the Dunning-Kruger Effect far better than I ever could, we can get down to business.

As I’ve made quite clear so far, “Is mathematics certain?” is a pretty stupid question. Yes, I’ll admit that it seems harmless enough. After all, isn’t mathematics certain?

Well…MOAR MEMES!!!!

Pirate Captain commenting on his magnificent beard

What I mean by all of this is that some things in mathematics are certain but some are not.

Let’s start with the most basic of all: “Is 2 + 2 = 4?”

Yes, assuming the integers under normal arithmetic.

“Well, how could it possibly be anything else?”*

*”Paging Drs. Dunning and Kruger. Paging Drs. Dunning and Kruger.”

Let’s consider a couple of examples where 2 + 2 is not 4.

The first comes from…uh…chemistry? physics? chemysics? My hesitation stems from the fact that the chemistry involved in this example is explained by the physical properties of the chemicals. Let’s just say it comes from…SCIENCE! (insert dramatic theme music here)

Anyway, let’s grab two gallons of water and (going full-on Russian) two gallons of alcohol*–say, ethyl alcohol specifically.

*Anytime you need to explain a joke, that joke fails. So let me acknowledge that joke in my title probably failed because it actually references this right here. Russians love their alcohol–especially vodka, which is brewed from potatoes. Get it? GET IT?!? “potato soup” “vodka” HAHAHAHAHA…huh?…what’s that? Yes, I’ll shut up now and get back to the post.

What happens when you mix together the two gallons of water and the two gallons of alcohol? You get approximately 3.84 gallons of solution. As tempting as it is to blame Vladimir for sipping that extra 0.16 gallons of alcohol,* the fact is that the only correct way to do the arithmetic here is to accept that 2 + 2 ≈ 3.84.

*That translates to about 2 1/2 cups of pure alcohol. Good night, Vladimir! Do svidaniya, Vladimir’s liver!

That’s why the stipulation above (“assuming the integers under normal arithmetic”) is so critical. And that’s why units are so important in the above example. The statements “2 + 2 = 4” and “(2 gal. water) + (2 gal. alcohol) = (3.84 gal. solution)” are very different things.

So my typical answer to “Is 2 + 2 = 4?” is usually, “Under normal arithmetic? Well, then, obviously ‘Yes.'”

I don’t do this to play semantic games or to try to appear smarter than someone else. I just find it good practice to make sure that we’re all speaking on the same terms.

Jesus would occasionally employ the same tactic. In Matthew 19:16, someone asks Jesus, “Good Master, what good thing shall I do, that I may have eternal life?” This is every witnessing Christian’s dream. Instead of knocking on door after door time and time again,* someone just walks up and asks how to be saved!**

*And having those doors forcefully knock back on noses time and time again.

**Of course, the answer is not in a “good thing to do,” but at least the conversation is off and running.

In a very interesting twist, Jesus throws the question right back at the questioner, “Why callest thou me good? there is none good but one, that is, God” (v. 17).

In other words, before we can talk about the “good things” you must do, what do you even mean by good in the first place? If there is none good but God, then how can the things you do–beneficial though they may be–really measure up to this Ultimate Standard of good?

Jesus is clarifying (for the benefit of the questioner and the surrounding audience) that good in this context means “gaining God’s complete approval” instead of the typical use of good to mean “a preferential outcome.”

Similarly, I like to establish the boundaries of some conversation just as that conversation gets underway–not for some “holier-than-thou Jesus-did-it-now-watch-me-do-the-same” reason, but usually because I’m just pedantic.

Another way in which 2 + 2 could differ from 4 would be using modular arithmetic. I certainly do not want to get into a deep discussion here, but let me mention that we all use modular arithmetic on a daily basis. And I mean literally a “daily basis.”

Why? Because of the clocks on which we base our days. Ignoring military time,* what is 8 o’clock plus 6 hours?**

*For those of you who’d like to out-pedant me. I know your little game…

**Again…no cheating! I already told you that “14 hundred” is off limits. And, yes, I also know that every sergeant every where and at every time pronounces it as “14 hunert.”

The answer is clearly 2:00. In the context of “clock arithmetic,” 8 + 6 = 2.*

*And for you stubborn military time adherents, I ask you, “What is 8 ‘hunert’ plus 20 hours?”

“Clock arithmetic” uses what is called “modular 12 arithmetic” or “mod-12.” Basically, 12 counts as 0. We write this as 12 ≡ 0 (mod 12).*

*[Technical nerd info. follows. Continue at your own peril.] The ≡ symbol is not a mistake. It indicates “congruent to” or “equivalent to.” Addition in modular arithmetic is technically the combination of classes (or collections) of numbers. Every multiple of 12 (…, -24, -12, 0, 12, 24, 36, …) is part of the same “class” as the number 0. But keep in mind that this is still addition; though the “=” sign might be replaced with the “≡” sign, the addition sign “+” remains the same.

In “clock time” (or “mod 12”), the numbers go as follows: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, etc.* (Remember that 12 and 0 are the same thing in this context.)

*Please note that once I successfully take over the world, I will replace all the 12’s at the top of each clock with a 0. You might not like it initially, but I’m doing this for your own good. You’re welcome.

In terms of arithmetic, every integer can be converted to a number between 0 and 11 (remember that 12 means the same thing as 0) by dividing that number by 12 and looking only at the remainder.

For example, consider 38. Well, 38 ÷ 12 = 3 remainder 2. Thus, 38 ≡ 2 (mod 12). What does this mean? Suppose it is 7:00 right now. Then you can calculate the time 38 hours from now by just adding 2, giving 9:00.

Obviously, 7:00 a.m. today would convert to 9:00 p.m. tomorrow night. Similarly, 7:00 p.m. tonight would convert to 9:00 a.m. the day after tomorrow. Such details might matter immensely to you, but the clock doesn’t care. It doesn’t care about your days or nights; it doesn’t care about the day of the week, month, or year. It’s a clock, for cryin’ out loud.

If you think about it (and I hope this blog makes you do that…), that’s more-or-less how we do this calculation anyway. If adding 38 hours, we basically “throw away” all the 12-hour increments as irrelevant. We “throw out” the first 12 hours, giving 38 – 12 = 26. Theoretically, we’d continue throwing hours away as follows:

  • 38 – 12 = 26 (26 is “too big” for the clock)
  • 26 – 12 = 14 (still too big)
  • 14 – 12 = 2…Aha! 38 hours from now is essentially just two hours on the clock!
  • 7:00 + 2 hours = 9:00…Eureka!*

*For the record, I strongly suggest that your “Eureka!” reaction be less…uh…”memorable” than Archimedes’ (in)famous “Eureka!” moment.

What about 38 hours after 11:00? Well, “38 hours” still converts to “2 hours” on the clock. 11:00 + 2 hours = 13:00 (“Ahhh!…’too big’…”), but “13:00” you know to just be one hour past 12, ergo, 1:00.

This just scratches the surface of modular arithmetic, and–who knows?–maybe sometime in the future, I’ll cover it more in depth.*

*Note that virtually all internet security is based upon using modular arithmetic of REALLY big numbers; look us “RSA encryption” if you are so inclined. The point here is that the topic of modular arithmetic is a LOT more than just the trivial situation of figuring out times.

But the current discussion is focused more on “2 + 2 = 4,” so let’s return to that.

Obviously, in “clock time” (“mod 12”) the numbers 2 and 4 never get near the number 12 that causes things to reset to 0.

But what about “mod 3”? Here, the number 3 is the same thing as 0. Thus, the numbers cycle back even more quickly than on a clock: 0, 1, 2, 0, 1, 2, 0, 1, etc.

In the case of “mod 3” arithmetic, 2 + 2 = 1. “HUH?!?” Well, 4 = 3 + 1, but 3 and 0 are the same. Thus, 4 = 1.*

*[More technical info.] Technically (that’s what “technical info.” means, I guess…), 4 ≡ 1 (mod 3), and technically 2 + 2 ≡ 1 (mod 3).

Now, to appease the Salem jurors accusing me of witchcraft, I’m gonna try to have my cake and eat it, too.* Notice that in these two examples–(a) mixing water and alcohol and (b) addition modulus 3–there is still complete certainty as to the final answer.

*I’m still trying to fit three completely unrelated metaphors into one sentence, but–so far, even in my best attempts–I’ve only been able to fit two.

The “uncertainty in mathematics” I’ve been alluding to arises in these cases only from failing to establish the context of the arithmetic.

  • In the context of the integers under normal arithmetic, 2 + 2 is certainly 4.
  • In the context of addition modulus 3, 2 + 2 is certainly 1.
  • In the context of adding water and alcohol, 2 + 2 is certainly 3.84 (within rounding)

Wait…WHAT? “Within rounding?”

Yup, “within rounding.” And this is where the genuine certainty that we “know” and love from mathematics begins to be distinguished from the genuine uncertainty when using mathematics. But that’s for next week.

For now, let me leave you with this question as an appetizer:

  • How many tons do you have when you put two 14-ton dump trucks on the same scale?

Let’s Talk About a Woman’s Age…[slap]

As established in my very first post, the point of this blog is to teach my kids how they can know things with certainty. I’ve defined proof, argued about arguments, failed to convince about the meaning of convince, and wasted four months avoiding discussions of qualified and judges.

The problem with speaking about certainty is that very few things are actually “certain” if you really dig down and think about those things.

Consider your weight* as you step on a digital scale. That scale probably returns this rock-solid number, perhaps even to the tenth of a pound. Now that’s certainty!

*This is only for the guys here. Talk about a woman’s weight at your own (eternal) peril.

OK, Chubbs, step off of it and then back on. There’s a pretty good chance that the number has changed somewhat–especially if it is a more “accurate” scale that measures into the fractions of pounds.

Did your weight change? I suppose that if the number changed you could–and probably did–chalk it up to having shifted your feet slightly. And you’d probably be right.

Of course, you could force the scale to change by chugging a quart of water, adding two pounds right there. Or you could have–while your were in the bathroom–uh…”availed yourself of the facilities” and lost a pound or two.

In a sense your weight just did change. After all, the scale doesn’t know the difference between what has been absorbed into your body versus that which is sloshing around in your digestive system.*

*I suppose you could take off your shirt and shoes and change that number as well, but I think we can all agree that we should restrict the phrase “my weight” to only be the body itself.

In fact, boxers and wrestlers take advantage of this very fact by starving and sweating themselves–sometimes to the point of genuine physical harm–in order to “make weight” and qualify for a particular weight class. There is often a bit of a party atmosphere after the official weight-in as the participants then pack back on all of that temporarily dropped weight in preparation for the next days’ matches.

But I dare suggest that your weight actually did change. During the few brief moments of stepping on, stepping off, and then stepping on again, your body kept maintaining its various functions: heart beating, lungs breathing,* etc.

*And if the numbers on your scale looked anything like mine, that “lungs breathing” was an audible gasp.

At the very least, your body took in some atmospheric air and exchanged a portion of the oxygen in that air for the carbon dioxide in your body. Of course, the resulting change in weight is usually so slight that only the most sensitive of equipment (measuring to the thousandths of a milligram*) can detect that change.

*Or “millionths of a horse-drawn hamburger” for my fellow American readers.

But even if it requires that sort of equipment to detect, the point remains that your weight DID change.

Now you might say, “But that teeny tiny amount doesn’t really matter.” Of course, it doesn’t matter. But that wasn’t the original question. The original question was, “Did your weight change?”

But enough about weight. How about a different example? What about your age?*

*Between questions on weight and age, it’s remarkable that I ever got married. Between questions on weight and age, my wife also finds it remarkable that I ever got married.

Well, that number certainly seems to stay constant for a while–a year, in fact.

Suppose you answer, “I’m 75.”*

*I’m under no illusions that people under retirement age bother to read my posts.**

**Come to think of it, I’m under no illusions that people at or beyond retirement age bother to read my posts, either.***

***Except for my mom. Thanks, mom!

But are your really “75”? Or are you “75 years, 108 days” and tomorrow you’ll be “75 years, 109 days”? Or are you “75 years, 108 days, 5 hours, 44 minutes, and 19 seconds” and very soon will be “75 years, 108 days, 5 hours, 44 minutes, and 20 seconds”?

There’s a good chance that even for fast readers, that that last sentence took a few seconds to read and process and that the “20 seconds” part was already outdated by the time you got to it.

Even better, consider this: You are older now than when you first began this sentence.*

*Your face right now.

Of course, there are some things that do NOT change. But I’m afraid you’ll find those to be a lot more rare than you might suppose.

You might offer up definitions as things that do not change. Well, for one, a lot of definitions have changed. For example, when the Fairy Godmother wished Disney’s Cinderella to “Be happy! Be gay!” I suppose that such a remark might be taken a bit differently today than it was back in 1950.

But let me give you the benefit of the doubt and offer up an example of a definition that isn’t nearly so malleable. How about dozen?

We all know that dozen means “12 items.” Now that’s a constant! That doesn’t change!

Agreed! But it’s also completely pointless. Definitions–or more precisely, the term associated with some definition–is nothing more than a mental shortcut.

In the above example, using the phrase “12 items” is clearly cumbersome in everyday language, especially because so many objects are sold in packages of “12 items.”*

*Of course, someone could make the argument that “so many objects are sold in packages of ’12 items'” precisely because the word dozen is so familiar.

And because saying “12 items” all the time is so cumbersome, we’ve created the word dozen merely as a shortcut to the concept of “12 items.”

In the Cinderella example, the Fairy Godmother’s use of the word gay offers up a flavor of happiness that was not conveyed through using the word happy a moment before.

We think of happy as being…well…”happy.” But gay meant “to make an obvious, external show of that happiness.” And I think we can all agree that “Be happy! Be gay!” is much more rhythmical than “Be happy! Make an obvious, external show of that happiness!”

In fact, the brief “Be happy! Be gay!” is actually clearer to interpret than the longer statement–at least, it is clearer to those who have already internalized the term and its meaning.

That illustrates part of the struggle with learning a new language. One of my colleagues grew up in France and–in addition to his regular work teaching biology–also teaches French.

He remarked to me the difficulty he had learning English. He initially “knew” the words, but needed to convert each of them from English into French in order to make sense of those words. For example, he needed to convert the English word book into the French word livre before he could decipher that book means “a bound collection of papers.” Of course, he’d already interpreted livre to mean “a bound collection of papers,”* so he just needed to get from book to livre and the process was done.

*Or, more precisely, “une collection d’articles reliés.”

But the thing that really struck me was when he said, “I remember when I first started to think in English.”

All of my inner dialogue is in English because I automatically convert English words into their broader meanings. Similarly, my colleague used to think in French for a similar reason. But after months and years of living in the United States–being saturated with English–he eventually began to make the immediate jump from the English words to their fundamental meanings.*

*As an interesting aside, he also noted that, when he goes back to France, the reverse happens. After about a week or two, he begins to think in French again. Returning to the States then requires a week or two to transition back to English.

To sum up this overly long discussion of definitions, they don’t (typically) change because they are by definition* merely shortcuts for longer, more cumbersome concepts.**

*Yes, I know: the definition of a definition. And down the rabbit hole we go!

**Which is why we say “Pull up a chair!” instead of “Pull up an object designed solely to be sat upon!”

Now, the perceptive reader* might have noticed that the definition I chose as absolutely unchanging–a dozen–is related to mathematics. And if there’s one thing that’s unchanging, it’s the truths of mathematics.

*Again…Hi, mom!

Right? Right?!?

Silver…Where?

“When you come to a fork in the road…take it.”

Yogi Berra

It is unfortunate that the great Yogi Berra is better remembered for his humorous quotes than for his Hall-of-Fame playing (and coaching) career: 18 All-Star Game appearances, 10 (!) World Series championships, caught Don Larson’s perfect game (dominating the video of the final out as the jubilant catcher leaping from home plate and bounding into Larson’s arms), and so on.*

*As a Pittsburgh Pirates fan, he even shows up in the Pirates’ greatest moment as the left fielder (!) looking up dejectedly as Bill Mazeroski’s home run won the 1960 World Series.

Source: AP

But the ability of his quotes to outshine such a career only illustrates just how brilliantly charming–yet surprisingly introspective–they really were:

  • “It ain’t over ’til it’s over.”
  • “Nobody goes there anymore. It’s too crowded.”
  • “A nickel ain’t worth a dime anymore.”

The point is this: I came to a fork and failed to take it. Way back when I was actually writing this blog, I was working through a definition of proof: an argument that convinces qualified judges.

I worked through the meaning of argument.

I worked through the meaning of convinces.

Then the “fork”: continue through the meaning of qualified (as started in my previous post) or work through the meaning of judge? I successfully chose neither.

I sat on it for a while, but “a while” soon became “a month.” “A month” soon became “a few months.” I’m now trying to keep “a few months” from growing into “many months” or “a year” or “several years” or–most likely–“six weeks past Rapture.”

Sooooo…let’s just go with the meaning of judge.

Having made this choice, I now realize just how easy this part is. The much tougher part is going to be returning to the meaning of qualified.

My problem, though, is that I suffer from the same issue that I expect you suffer from: I have a hard time distinguishing between judge and qualified judge.

It turns out that anyone can be a judge. In fact, everyone is a judge.

“But the Bible says, ‘Judge not’!” O, ye of little brains. Have ye not read? The rest of the verse (Matthew 7:1 and the subsequent verse 2) continues “that ye be not judged. For with what judgment ye judge, ye shall be judged.”

Simply put, a judge is “one who makes a decision.” As a verb, judge means “to reach a conclusion.” And Scripture’s warning in Matthew 7 is basically a rewording of the Golden Rule: “Judge others as you would have them judge you.”

The more I think about it, the better this warning makes sense in light of the proof definition. Think about it: this whole blog (four-month absence and all) has basically been me making claims about how we can know things. This started with some posts on the Dunning-Kruger Effect (the idea that the more we learn about some topic, the less confident we are that we have mastered it).

Every claim made about the Dunning-Kruger Effect was essentially me passing some judgment about it. Consider, for instance, that there are only two possibilities about the Dunning-Kruger Effect: (a) the effect is real or (b) the effect is NOT real.

I made a decision here. I–hold on to your biblical hats–made a judgment here: I deemed the Dunning-Kruger Effect to be real.*

*Keep in mind that determining “how accurately” it describes the relationship between knowledge and confidence is an entirely different question from determining IF there is a relationship at all.

I don’t claim to speak for God, but I really doubt that He’s about to split the sky open and vaporize me with a bolt of lightning for having made this judgment.*

*My kids and especially my wife can testify that I’ve done a LOT more to warrant the Holy Bolt of Disciplinary Lightning (level 4).

But what I have done is put myself into a position to be judged in turn by others–in particular, by those with an understanding of the Dunning-Kruger Effect. And I’m certainly in a position to be judged by those who understand the Effect more deeply than do I.

There’s even a small part of me that somewhat wants Dunning and Kruger themselves to read those first few posts of mine so that they could correct my own misunderstandings of the concept.

There are several ways to describe how well someone knows material, and one of the most common is Bloom’s Taxonomy of Educational Objectives (where taxonomy simply means “classification”).

Bloom’s Taxonomy has undergone several revisions over the years, but one of the highest levels of learning is Evaluation. Evaluation literally means “to evaluate” or…wait for it…”to judge.”

*Evaluation was actually the highest level in the original formulation of Bloom’s Taxonomy.

Thus, in a weird sort of way, I made a judgment way back when about the Dunning-Kruger Effect partly to introduce others to the concept. But there was a small, private reason as well: once I put my ideas out into the vast wasteland of the internet, I have put myself in a position for others to, in turn, critique (or judge) my own observations.

If my observations–if my judgments–can bear the critique of others, I am in that much better of a position to promote those observations as genuinely correct.

The real issue with judging, however, is in being qualified. Any idiot can criticize, but most of those criticisms are just the stupid mutterings of the ignorant.

The judgments I welcome as a retort to my own judgments are those from qualified judges–not the “expert opinions” of “YouTube researchers” or graduates of the “University of Facebook Moms” who’ve “uncovered the vast Illuminati-based conspiracy to use Amazon Echoes to ultrasonically inject us with Martian vaccines.”*

*No, I’ve not seen this specific claim anywhere And, yes, I’m making it up now, hoping that some idiot will read it and think “Maybe he’s onto something…”

But I’ve struggled a bit in my writings about qualified, allowing that struggle to grind this blog to a halt for these last several months.

Thus, I’m going to move on for the time being. Once I finally figure out what I want to say about qualified, I’ll return to it, and formally close the discussion of proof.

For now, just keep in mind that whenever I use the word proof, I use the following definition: an argument that convinces qualified judges.

And I’ll be sure to be back in just a few short months…

Eating Words is a Weird Diet

So, after reading my last post, an astute reader* remarked to me that it seemed like a big set-up for a discussion about transcendence. Well, the reader wasn’t exactly wrong, because I do plan to talk about transcendence–eventually.

*I think that an honest conversation among ourselves would reveal that not a single person on the planet really knows what “astute” means. We just use this word because it seems like a way to both praise someone else while making sure that others know about our extensive vocabulary. After all, because others don’t know what “astute” means, they assume (ever so wrongly) that we do simply because of our using that word. And we keep using that word, but I do not think it means what we think it means.

But the real purpose of my last post was to get people to think about the need to go beyond ourselves (to transcend ourselves) in order to better observe ourselves.

As mentioned last time, mirrors and cameras help us do this. But in each case, we observe ourselves indirectly. We see pictures of ourselves. We see reflections of ourselves. We hear descriptions of ourselves from others. But at no point do we actually see ourselves.

Oh, sure, we see parts of ourselves. I can see my hands right now as I type this. I see a bit of my nose every time I open my eyes.* I see my feet when I’m putting on my socks.

*Just like you can see part of yours right now. Your nose is always there** and it is always in your field of vision. We’ve just become conditioned to ignore it…unless some crazed blogger mentions it…and then you can’t unsee it for a while.

**Voldemort excepted, of course.

Therefore, I can use direct observation to make a direct judgment about my hands: my fingernails are too short; my skin is too dry; that blister is healing nicely; etc.

But the point here is that my eyes are beyond my hands, giving them a perspective for judgment that my hands just don’t have. My eyes are better judges of my hands than my hands are of themselves.

And the whole point of a mirror is that when I have, say, an eyelash in my eye, to give my eyes a perspective that they just don’t naturally possess.

In other words, my eyes too busy seeing that they are unable to be seen themselves. Thus we use mirrors as a work-around for that.

And, frankly, that is the ultimate purpose of this entire blog: to act as a mental “mirror” to my own ideas. The ideas milling around in my head get taken out of my head (no surgery required!) and put into print.

And the analogy to a mirror holds up. Just as a mirror lets my eye see a reflection of its real self, so also are these words a “reflection” of the real ideas tucked away in my head.

For example, I’m looking right now at a water bottle. I can (and I hope already have) communicate that idea to you by saying those words: “I’m looking right now at a water bottle.”

But do really think that I am saying to myself, “I’m looking right now at a water bottle”?

Of course not. My eyes see a water bottle, and my brain interprets that as a…water bottle. But it is only through the words that come out of my mouth or through the typewriter that allow my brain to convey my idea of a water bottle to others.

And it is only through the words that come out of my mouth or through the typewriter that allow my brain to convey my ideas about proofs and arguments and research and unicorns and well-manicured beards and paper-cut pain and rusted anchors and expensive cellphone plans and well-manicured beards and dirty forks and scented candles and tufted pillows and spicy tacos and well-manicured beards and Elvis songs and political opinions and chicken feed and marbled counters and marbled floors and marbled ceilings and well-manicured beards to others.

And usually once I see those words, I realize just how wrong my ideas really are.

Now, what you are reading here is something like the 4th or 5th draft of this post. It is not uncommon for the counter on my private WordPress editing page to show “28 revisions” by the time I publish a post. Yes, many of those “revisions” are just from me saving my work along the way.

But many of those revisions are the result of me reading what I just wrote and thinking, “The person who just wrote that is a real waste of carbon.”

And that’s where I (finally) get to the title of this post. All of those revisions…all of the changes…even all of the “I really wish I’d written that differently” re-evaluations that I do after I’ve finally publicly published a post…all of these are me “eating my words,” so to speak.

Basically, every post is this:

I write. I read what I wrote. I look at it in disgust. I edit what I wrote. I look at it again in disgust. I edit it a second time. I look at it again in disgust. I realize that what I’ve written can’t be salvaged by edits. I scrap entirely what has been written. I write something brand new. I read what I wrote this second time. I look at it in disgust. I edit what I wrote. I look at it again in disgust. I publish because I’m sick and tired of the whole process.

I think this idea was put well by the filmmaker David Fincher describing his own films: “Movies aren’t finished, they’re abandoned.”*

*https://www.filmindependent.org/blog/movies-arent-finished-theyre-abandoned-david-fincher-on-why-he-hates-watching-his-films/
In all fairness, I’ve heard this quote from others. In fact, Fincher himself introduces the quote with “They say…” I’ve even seen seen it attributed all the way back to Leonardo da Vinci stating that “Art is never finished, only abandoned.” However, I wanted to have a formal citation somewhere so that you could verify my sources.

This post? What you see now is something that was ultimately just abandoned.

But I guess the more important part of the previous sentence is the “what you see now” part. Abandoned or not, my ideas are still out there.*

*And, yes, I know you can read “out there” in two completely different ways.

Consequently, both you and I can now read what I wrote and both you and I can determine if my ideas are correct or not.

And that is the set-up for this entire post and the connection to the last few posts dealing with proof. Recall that I’m working with this definition of proof: an argument that convinces qualified judges.

The connection is that I am actually the last person who is qualified to determine if the ideas milling around in my head are valid or not. It is only by getting them out of my head by putting them on paper (or electrons, in this case) that I can now try to judge…to evaluate…to qualify those ideas.

Yes, by publishing this blog, I’m welcoming you into my thought processes. As a result, you are in a position to evaluate my ideas as well.

But your evaluation isn’t my primary concern. Don’t take that personally, though, because my primary concern is just getting these ideas out in the open. And these posts accomplish that goal.

The plan? It’s basically this: ideas –> words –> evaluation

More detailed: ideas (in my head) –> words (out of my head) –> evaluation

How many times have you said to yourself, “That sounded better in my head”? I’ve said it thousands of times to myself.

And that’s the ultimate point. Qualification is always an external process.

You would think that, if there was ever something you would be an expert on, it would be your own ideas. But even that turns out not to be the case.

Don’t believe me? Try explaining the simplest of tasks to someone else. An old adage is that “You never understand something until you’ve had to teach it.”

Think you know arithmetic? Offer to tutor a 5th grader in arithmetic.

Think you know American history? Offer to tutor someone in the subject.

Think you know “kids'” Bible stories? Offer to teach a Sunday School class.

Only in those situations do we find out just how UN-qualified we are in even the basics. Only in those situations do we finally expose our hidden–yet tainted–ideas.

“Sunlight is the best disinfectant,” the adage states.

But the great thing about it is that doing so forces us to confront those failures and ultimately forces us to improve.

And it’s then that we are finally in a position to begin understanding what qualified really means.

The Camera Adds a Few Pounds

Do you have any favorite words? I don’t mean something like pizza. I mean, I love pizza, but I don’t necessarily love pizza.

You likely responded to that with “Huh?” You certainly responded to that with “Did you take your meds today?”*

*It’s even more possible that you asked, “Where on earth have your posts been?” Good question, and I’ll admit that it’s been three weeks since my last post. In my defense, the election has been really entertaining–and I’ll also admit that this is a terrible defense. To give context back to this post, I am [s–l–o–w–l–y] working through a definition of proof: “an argument that convinces qualified judges.” The last few posts have dealt with “argument” and “convince.” This post begins (in a very oblique manner) a look at “qualified.”

What I mean is that while I love the object of the word pizza (i.e., I love the round, cooked dough that has a lot of tomato sauce, a lot of pepperoni, and a LOT of mozerrella mozzerrela mozz… cheese) the word itself…pizza…actually sounds a bit stupid: PEET-zuh

But I love the word transcend. There’s a certain rhythm to it, starting with the somewhat harsh “chir” sound (from the “tr”) and cascading down from there.

Of course, we think some words are beautiful because of the feelings they evoke. One example I came across was sumptuous. I mean, who doesn’t like a sumptuous meal? But the word itself sounds a bit odd: SUMP-chew-us*

*Put it all together and say “sumptuous pizza.” We Americans would melt at the very thought of that, but the sounds themselves are impossibly dumb: SUMP-chew-us PEET-zuh

Now, did I overthink that? Of course, I did…because I overthink everything.

But the point is, transcend is a beautiful word to say or hear. And, while I readily admit that there are many other beautifully sounding words, the word transcend seems to go beyond most others.

That seems appropriate: transcend effectively means “to go beyond.”*

*It literally means to “climb across”: trans- (“across”) and scend (‘climb”). The current use of the word carries over from the idea of being above and looking down (while climbing over).

In mathematics, there are transcendental numbers that “go beyond” the normal algebraic numbers.* There are transcendental functions that “go beyond” the normal algebraic functions.**

*An algebraic number is any number that is the solution to some polynomial
anxn + … + a2x2 + a1x + a0 = 0 where each ai is an integer.
And let that be a warning: if you ever complain about this blog again, I’ll throw another equation at you.

**You complained, didn’t you? I’m a gracious blog host and will let you off this time with just a definition: an algebraic function is any combination of addition, subtraction, multiplication, division, or taking roots.

We often speak of “transcendent beauty” or hear of a “transcendent athlete” because the description of such beauty or of such an athlete “goes beyond” normal words.

We sometimes try to describe the majesty of this blog, and the only acceptable answer is “Words fail me.”

But it is only when we transcend (or “go beyond”) something that we are able to truly describe or understand it.

Admittedly, that seems like a strong claim.

But I want you to consider how you would describe…you.

I don’t mean describing your habits. I don’t mean describing your likes or dislikes. I certainly don’t mean describing your political views.

I mean describe the physical you…starting with your facial features.

Did you ever stop to realize that you have never EVER seen your own face? I mean, yes, you’ve seen pictures of your face…you’ve seen a reflection of your face. But you’ve never EVER directly observed your own face.

Every concept you have about your own face (and every nightmare I have about my own) comes from those pictures or reflections that come from somewhere away from your own face. Pictures or reflections that…go beyond your own face. Pictures or reflections that…transcend your own face.

It is here that I can park for a moment and address the title of this post.

Did you ever notice that “the camera adds a few pounds”? And did you ever wonder why?*

*Of course you didn’t. Every single activity mankind has ever attempted–no matter how insignificant–is of more importance than asking why “the camera adds a few pounds.” But I’m here to prove that–no matter how insignificant you think you are–it could be worse: you could be me.

Consider that almost every time you “see” your own face, it is from straight ahead: either a reflection in a mirror or a photo where you are looking at the camera.*

*By the way, photos of ourselves often look a little bit “off.” Why? Usually it’s because photos seem to have some things “backwards.” For instance, “My hair is parted the wrong way!” No it isn’t. You’re just used to seeing it in the mirror going one direction. Everyone else–and the camera, too–sees it going the other direction.

Those “straight ahead” images always fail to capture the items that accurately show our actual weights–usually that sagging chin is only visible from the side.

In other words (or pictures in this case), I usually think I’m this gangster…

…when I’m actually this one:

Thus, the only way to know what I really look like is to “get outside of myself,” so to speak.

And this concept only gets deeper.

Usually, when I want to feel the texture of some fabric, I (being right-handed) touch it with my right index finger. But, if you really think about it–and I mean REALLY think about it–my right index finger has no idea of its own “texture.”

Of course, I know what my right index finger feels like, but only because I’ve touched my right index finger with other fingers (or any other part of my body for that matter).

Even though I use my right index finger to gauge the texture of all sorts of things, my right index finger can never tell–on its own–if its skin is too dry or oily.

Oh, sure, my right index finger can determine if something is different–i.e., I feel something on that finger. But it is only through the context of external observation–looking at my finger, touching it with other fingers, etc.–that I can determine if that different something is water or oil or grease or salsa or glue or…(you get the idea).

In other words, the one item I use the most for determining the texture of other items…cannot determine its own texture.

But that is always the case for describing…anything.

I can describe how soft a sweater feels–but the sweater itself can’t (even if it could talk).

I can talk about how spicy a taco is–but the taco itself can’t (even if it could talk…and even then, I don’t speak Spanish).

I know I’m about to greatly abuse your imagination when it comes to these objects being able to speak or think, but please hold on for a bit.*

*After all, reading this blog has been a great amount of abuse already. What’s a little bit more?

Even if the sweater could talk and think, it could only describe its own texture by having two different parts of the sweater come in contact. But Part A could only describe the texture of Part B, while Part B could only describe the texture of Part A.

In other words, Part A is able to make a judgment about Part B, but cannot make any judgment about its own self.

Even if my hand could talk and think, it could only describe its own texture by having two different parts of my hand come in contact. But my right index finger could only describe the texture of my right thumb, while my right thumb could only describe the texture of my right index finger.

In other words, my right index finger is able to make a judgment about my right thumb, but cannot make any judgment about its own self.

I color-coded the above paragraphs to make a point. The red paragraphs are a copy-and-paste of the blue paragraphs, just with the words “sweater”, “Part A”, and “Part B” replaced with “my hand”, “my right index finger”, and “my right thumb”.

You can even make your own!

  1. Choose any noun, labelled A.
  2. Choose one part of that noun, labelled B.
  3. Choose a different part of that noun, labelled C.
  4. Insert into the appropriate spots below:

Even if [A] could talk and think, it could only describe its own texture by having two different parts of [A] come in contact. But [B] could only describe the texture of [C], while [C] could only describe the texture of [B].

In other words, [B] is able to make a judgment about [C], but cannot make any judgment about its own self.

Let’s do this!!!

  1. Choose any noun, labelled A. [for example, A = firetruck]
  2. Choose one part of that noun, labelled B. [for example, B = siren]
  3. Choose a different part of that noun, labelled C. [for example, C = ladder]
  4. Insert into the appropriate spots below:

Even if [firetruck] could talk and think, it could only describe its own texture by having two different parts of [firetruck] come in contact. But [siren] could only describe the texture of [ladder], while [ladder] could only describe the texture of [siren].

In other words, [siren] is able to make a judgment about [ladder], but cannot make any judgment about its own self.

WHAT FUN!!!

Of course, the grammar is perhaps a bit choppy, and you probably read “WHAT FUN!!!” as “…uh, what fun?” but the principle remains the same:

It is only by transcending itself that something can actually make a judgment about itself.

And, if you think about it, “transcend itself” is a really tough thing to do.

But I’ll try to talk about that next time.

Drinking the Kool-Aid

“Oh, yeah?!? Prove it!” So the theme throughout these recent posts has been to explore what is meant by “Prove it!”

*I take it as a given that we all interpret “Oh, yeah” as a big, round, red man crashing through a wall. I am speaking, of course, about Andy Reid (coach of the NFL’s Kansas City Chiefs).

I settled on one of Reuben Hersh’s definitions of proof: “an argument that convinces qualified judges.”*

*I had another math teacher tell me that he preferred one of the other definitions. I agreed that this definition is one of the weaker ones when defining proof in the strict context of mathematics. However, I prefer the above definition for the much broader, common use of the term proof in everyday conversation. The teacher saw what I meant and granted the point. Unfortunately for me, he only understood after the fact–meaning that I guess I didn’t do as good a job of clarifying what I meant the first time around. Mea culpa.

The last post gave an argument for a return to the original meaning of the word…argument.

Here we will take a look a the word convince. As per our usual arrangement, I will break down the word into its original components, thus sucking the life out of this post.

Surprisingly, the prefix con- does not mean “member of the opposite political party.” It means, in fact, simply…”with.”

Let me* park here for a bit…

*Yes, this is purely rhetorical. It’s my blog, and I’ve already received your $0 subscription fee. So Imma do what I want…[blows raspberry]…

Think of how much that prefix illuminates our understanding of the word conscience–especially if you consider what the word science actually means. Once we get to the “pure reason v. scientific method” spectrum first mentioned several posts ago, we will really get into the details of the scientific method.

For now, though, I’m just going to ask you to trust me concerning the following somewhat surprising concept: the word science literally just means “knowledge.”

Thus, the word conscience literally means “with knowledge.” Have you ever noticed how often you “just know” something is wrong, but you have no idea why?* But, is it just me, or have you also noticed that just about every time that you listen to the voice that “just knows,” you later find out why that thing was wrong?

*And, if you’re like me, you’ll just foolishly plow ahead and do the wrong thing anyway. Paul, neatly summarizing my entire life: “The evil which I would not, that I do….O wretched man that I am!” (Rom. 7:19, 24)

Basically, as we age, we are to rely less and less upon that “little voice” that helps us the times that we lack “factual” knowledge of right and wrong and to rely more and more upon what we learn as we mature.

In other words, the conscience is a wonderful thing, but woe to the one who keeps “flying blind” only by his conscience. The conscience is only at its best when it is accompanied “with knowledge.” In fact, my most important job as a parent is to teach my children right from wrong–everything else is just a bonus. Let me again cite the Shema: “And thou shalt teach them [i.e., God’s commandments] diligently unto thy children” (Deut. 6:7).

Of course, knowing what is right (wisdom) and actually doing what is right (discipline) are two completely different things. The following, widely attributed to Teddy Roosevelt (though I cannot find a solid citation), summarizes this very well:

“Knowing what’s right doesn’t mean much unless you do what’s right.”

So…please forgive that digression, but let me remind you that the whole point of this blog is basically to write to my kids. And, frankly, if my kids ignore everything in this post except for the last few paragraphs, I’ll be thrilled.

Now let’s return to the definition of convince. As I spent way too long discussing, the prefix con– means “with.” But what about vince? This comes from the Latin word vincere, meaning “conquer” (i.e., victory).

Perhaps you now respond, “Aha! I knew it! To ‘convince’ someone means to win!”

Well…congratulations! You’ve learned nothing!

Victory does mean “winning,” but convince effectively means “winning with.” In other words, you do not convince someone by winning against that person, you convince someone by winning with that person.

Of course, the whole notion of “winning” (whether “with” or “against”) still implies a contest of sorts. And here–in the context of proof–the contest is in the realm of ideas.

For example, Person A has an idea that differs from that held by Person B. Even if Person A goes forward with the best and kindest of intentions, there will necessarily be some clash of ideas.

But, if genuine convincing is the goal, Person A is not trying to win against Person B–he is trying to win with Person B. The ultimate goal is to have Person B go forward with Person A as part of the same team rather than going with one as the “victor” and the other as the “vanquished.”

And if Person A ever has an “I was right, and he was wrong” mentality, then true convincing was never the goal: sheer, brutal “winning” was the only goal.*

*And, frankly, that has become the sad state of the Church today. We are far more interested in showing that someone has the wrong moral position than we are convincing them** to accept Christ.*** And we have the audacity to call that “winning.” A lost “moral” person is still lost.

**Yes, I know that the Spirit does the convincing, but “How shall they hear without a preacher?”

***And, yes, I know the worn-out adage, “You have to get them lost before they can be saved.” We’re much more interested in letting them know they’re lost than we are in having them be saved.

For a concrete example, consider that right now I am trying to convince you of some idea. In this case, the very “meta” idea of convincing you about the meaning of convince. And now I am in the very awkward position of acting as if I have a better understanding of the word convince than you do.

Obviously I have to tread lightly now because we have this ridiculous notion that “knowledge = worth.” How stupid is that? Am I a “better person” than my students because I know more math than they do?* Am I a “better person” than my kids because I know more than they do? Is “Joe” a better person than “Frank” because Joe knows more about carburetors?

*Can you think of how awful the class would be if I did NOT know more math than they do?

But, unfortunately, that’s the world we live in today: one that thinks every discussion of ideas is a battle of self-worth, and we sadly think that the statement “I know something that you don’t” equals “I’m better than you.” That notion is sheer nonsense, but sheer nonsense seems to be our world’s motto.

Regardless, I need to tread lightly throughout this whole process because I want to convince you of something rather than just “browbeat” you into acceptance. As in the Persons A/B example above, I want both of us to go forward as part of the same team rather than as “victor” and “vanquished.”

And, to be honest, I hate the “team” illustration that I’ve used throughout this post because we just cannot seem to separate it from the tribal “Us v. Them” notion that infects every aspect of our lives.

But throughout this post, I hope I’ve avoided setting up a “Team I-Know-What-Convince-Means” v. “Team You-Don’t-Know-What-Convince-Means” battle. Because it’s NOT a “battle”: the definition of convince is just an idea and most of us just haven’t had the opportunity to really dig down and find out what it means.

So, returning to the beginning, if I’m trying prove something to someone, I can only do it if I convince him of that idea. And I hope I was able to genuinely convince you of this idea…

…because if I didn’t, I’ve got NO shot of succeeding in the next post. Why? Well, in that post we’re going to look at the last part of the definition of prove: “an argument that convinces qualified judges.”

If I can’t convince you of the meaning of convince, it’s going to be a bloodbath trying to convince you of what it means to be qualified.

But (literary) bloodbaths can be fun to watch. Who doesn’t like watching someone not just fail, but fail in epic fashion? Sounds fun, right?

Oh, yeah!

Let’s Get Ready to…Violate Trademarks!

I really wanted the title of this post to be boxing’s iconic opening: “Let’s get ready to ruuuummmmblllllle!”

But somewhere in the deep recesses of my memory—-past the Konami code–past the knowledge that my Dad promised that we’d stop at McDonald’s after the Wednesday evening church service on April 17, 1985…but we never did–even past the date of my anniversary*—-long past all of that was the nagging idea that the “Ruuuuummmble!” phrase had actually been trademarked.

*It’s somewhere in summer, right?

And, guess what? I was right! It’s not “boxing’s” iconic opening. It’s actually the announcer Michael Buffer’s iconic opening. And that’s fine, because when you hear him say it, you realize that he not only owns it…he owns it:

But, regardless, “rumbling” is the American way. At least, it’s the American way when it comes to arguing.

Why? Because we’re total morons. All of us. That includes me, and you’d better come to grips with the fact that it includes you.*

*You might say, “But I’m not American. I’m Canadian!” Well, you’re a moron, too. You just happen to be an exceedingly polite moron who’s “very sorry” for being one.

And how do I know this? Because I already know your reaction to that claim: “Oh, yeah?!? I got your argument right here, jerk!” [shakes fist]*

*Or, for the Canadian reaction: “That’s an interesting notion, you flag-waving, bald-eagle-worshipping redneck, eh?” [sips Tim Horton’s]

In the purest sense, the word argument comes from the Latin arguere which means “to make clear; prove.” The full use of arguere indicates that there is also a negative aspect to the word, for the full meaning is “to make clear; prove; accuse.” Thus, in its original form, the root of the word argument means effectively “to clearly prove an accusation.” But the main emphasis of the definition is upon the “clearly prove” aspect.

Unforturnately–and we’re still sticking with this root word arguere for the moment–we have focused on the negative aspect of that word (the “accusation” part) instead of the positive–and much more important–part of the word: “to make clear; prove.”

This issue has carried over to our (mis)understanding of the word argument. The purest expression of argument would be “a set of reasons with the intention of persuading.” This is usually in the context of determining moral right or wrong, but it would still be proper to use it in the context of determining whether something is factually correct or incorrect.

Unfortunately, the emphasis over time has been upon the “persuading” part of the definition rather than the actual core of the definition: “a set of reasons.” Consequently, the meaning of argument has devolved over time as our notion of “persuading” has devolved over time, so that argument now basically means “bickering” (at best) or outright “fighting” (at worst). This is so much the case that the first definition of argument that is usually found in a dictionary is now some variation of “bickering” or “quarrelling” or “fighting” instead of the original notion of presenting “a set of reasons.”

But that’s us today: “I’d like to present an argument” = “Let’s get ready to rumble!”

All of this is to go back to the last post where several definitions of prove were postulated by Reuben Hersh, and I* settled upon this one as the best, overall representative definition: “an argument that convinces qualified judges.”

*Well, it is my blog…

Well, if we don’t know what argument really means, then we have no hope of knowing what prove actually means.

If you take the true meaning of argument, then you get the following definition of prove: “a set of reasons that convinces qualified judges.”

However, if take the eroded definition of argument that means “bickering,” then the definition of prove takes on a whole new meaning: “a quarrel that convinces qualified judges.”

But that is actually a contradiction. You can not “browbeat” somebody into being “convinced.” You might be able to browbeat him into acceptance, but not conviction.

I suppose that the best-known example of this is from George Orwell’s 1984. At one point, the main character Winston Smith is told repeatedly and emphatically that “2+2=5”. Over time, Smith comes to accept this “fact,” but it is quite clear that he is never convinced.*

*Two side notes here.
(1) Forget Nostradamus; no one predicted future society as well as Orwell (except maybe for The Simpson’s). There have actually been “debates” on Twitter (that should tell you all you need to know right there) about the “white nationalist implications” of insisting that “2+2=4.”
(2) Though you can certainly force someone to take certain actions, you can never bully someone into a genuine belief. I find it ironically “good” that society is slowly breaking down morally in that it becomes more obvious our society’s need for Christ. During the “good ol’ days,” society pressured people into behaving according to Judeo-Christian mores without addressing the root need for Christ Himself. A “good” lost person is just as lost as a “bad” lost person; but–to our shame–we feel a lot better about ourselves when “good” lost people die without Christ.

At that unfortunately is where our contemporary definition of argument and consequently our contemporary definition of prove have left us. Any time we here someone say, “Let me prove it to you,” it has almost become a code phrase* for “I’m gonna shove this down your proverbial throat until you have no choice but to accept it.”**

*”Newspeak” for you 1984 fans!

**To use my sister’s “favorite” line from our dad: “You don’t have to like it. You just have to eat it.”

And in that case, there is no genuine proving that ever occurs.

I love Hersh’s definition of proof, but I can only do so if I keep in mind what argument truly means.

And I hope that this post has been an example of that. Were there some personal jabs in there? Yeah, I guess. But note that those jabs were not to demonstrate what the words prove or argument mean. Those jabs were mainly to prod you to think about your existing uses of those words.

From that point on, though, I tried (and only hope that I succeeded) to give a demonstration of the meaning of argument that was in keeping with that original meaning: a gradual revealing of reasons (beginning with the root meanings of argument way back with the Latin arguere) that built towards a final goal–namely, that a true argument actively avoids bickering or quarrelling and instead is a reasoned set of facts that lead to a conclusion.

In other words, there is a world of difference between “making an argument” and “being in an argument.” As we continue to look at what it means to genuinely prove something (with proof being “an argument that convinces qualified judges”), I hope that you have a better understanding of what arguments should be rather than unfortunately what they currently are.

Next time, I want to look at more of the definition of proof: What is meant by “convince”? What is meant by “qualified”? (That last one is going to be a doozy…)

Proof Is In the Puddin’

My background is in mathematics* and now I teach mathematics for a living. As any other math teacher can attest, this means that I teach students how to do proofs.

*Two thoughts here:
1. I cannot fathom a more disinteresting five words to open a blog.
2. My background is actually in video games, sarcasm, and an obscene amount of coffee. (I highly recommend Ethiopian Yirgecheffe if you get the chance.) I once read somewhere someone’s definition of mathematician as “a machine for turning coffee into theorems.” Seems a bit too on the nose…

Doing so inevitably leads to the following conversation:

Student: “Sir? What actually is a proof?”

Me: “Y’know something? That’s a really good question. Why don’t you wait until we’ve done a couple first. That way, the definition will make a bit more sense. OK?”

Every parent out there knows that this is the official answer for “I don’t have the foggiest idea.” So all I need to do is stall them long enough for them to forget their original question. And, fortunately, goldfish have longer attention spans than students.

Thus, I don’t really ever need to know what a proof actually is.

…

…

…

At least, I didn’t need to know…

A few years ago, I needed to do some research about geometry teachers.*

*And there it is, friends: the single most depressing sentence in history.

In particular, I needed to research their opinions about using proofs in the classroom.*

*This sentence was also in the running for “most depressing sentence,” but only took the bronze. By the way, second place was awarded to the group of sentences that begin with “As the parent of a junior high boy…”

Have you ever heard the quip, “Opinions are like noses; everyone has one.” Well, definitions of proof among mathematicians are the exact opposite: no one has one.

It’s almost like we math teachers have formed this educational cabal in which the First Rule of Defining Proof is “never talk about defining proof.”

Thus, I had to turn to mathematical philosophers for some ideas about the definition of proof.*

*Good grief… Just look at these sentences. This is turning into the 2020 of blog posts.

Well, believe it or not, someone actually did try to give a definition.

The title of Reuben Hersh’s paper kind of gives away the answer: “Proving Is Convincing and Explaining.” I say kind of because “convincing and explaining” are more what (good) proofs do, rather than what they are.

*Reuben Hersh. “Proving Is Convincing and Explaining,” Educational Studies in Mathematics 24 (1993): 389-399.
Fellow nerds can read the whole thing at https://moodle.tau.ac.il/2018/pluginfile.php/316018/mod_resource/content/1/Hersh%201993%20ESM.pdf

But by page three of the article, it becomes a bit depressing (common theme, eh?) to find out that Hersh has three definitions of proof. Well, I guess having three definitions to sort through is a lot better than the “none” that I had before.

His first definition is rooted in the etymology (big Greek words!) of the word prove. The Latin probus means “good”; the subsequent Latin probare means “to test” or “to approve” or “to demonstrate.” The resulting English word prove is a combination of these two ideas: essentially, “to demonstrate the goodness” (or “soundness” in this context).

Hersh summarizes this with his first definition of proof:

(1) Test, try out, determine the true state of affairs.

This use of the words usually applies more to things than it does to people or ideas–and almost in a passive sense. For example, the maxim “The proof is in the pudding” is actually a misquote of a longer statement: “The proof of the pudding is in the tasting.” In other words, the quality of the pudding (a thing) is demonstrated (is proved) by being tasted.

Well, that first definition is fine, I guess. But I usually like to browbeat people with something like “Prove it to me!” And the definition above just doesn’t work in this case. (Though I will certainly accept gifts of pudding at any and all times.)

Hersh’s THIRD definition is the type of monstrosity that could only be cobbled together by something equally monstrous…mathematicians. And a cobbled-together Frankenstein it is. For though I said earlier that no mathematician has a genuine definition of proof, Hersh was able to take the fleshy, sub-human concepts that mathematicians have about proof and put together this hideous eyesore:

(3) A sequence of transformations of formal sentences, carried out according
to the rules of the predicate calculus.

The less said of this, the better.*

*Perhaps we should just call this third definition “The Definition That Shall Not Be Named.”

But the SECOND definition really cuts to the core of what is usually meant by the word proof:

(2) An argument that convinces qualified judges.

The more I look at this definition, the more I just love it. Of course, I’m looking at this as a definition that I can give in the math classroom, and this definition perfectly describes what we are trying to do in the mathematical community. Yes, the Definition That Shall Not Be Named is perhaps more precise (at least, to the two people on the planet who know what the words in that definition mean) in a mathematical sense, but every mathematician still thinks that certain proofs are more…well…”beautiful” than others.

For example, there are dozens of proofs of the famed Pythagorean Theorem. The website linked here has–and I kid you not–118 proofs of that theorem. Some of them are agonizingly stale,* but some of them are just more aesthetically pleasing (at least to mathematicians). I, for one, am a little bit partial to version #3. Version #5 is also pretty interesting–especially once you realize that it was first proposed by President James Garfield in the late 1800s.

*OK…most people would say that all 118 are agonizingly stale.

But what makes a proof more…uh…”beautiful” than another? Well, beauty certainly is subjective (“…in the eye of the beholder”), but certain proofs do more than just convince: they explain. Proof #3, for example, not only demonstrates that the Pythagorean Theorem is true, it helps the reader visualize why the pieces fit together to make it true.

This is comparable to one mechanic who tells you that a certain sound indicates a certain problem versus another mechanic who actually shows you the two parts grinding together. In each case, you might know that the car has a particular problem, but only in the second case would you truly understand why.

Thus, an argument that explains while convincing just resonates more than an argument that only convinces.

Likewise, I prefer the “an argument that convinces qualified judges” definition of proof FAR more than either of the other two. Why? It accomplishes the two-fold goal of first stating what a proof is, but also stating what a proof does.

Now, you may disagree with the definition, but I think this gives us a good test case: I will try to prove to you that this definition of proof is sound.*

*Those of you with some experience in logic know that “proving a definition” is a logical impossibility. But what I’m doing here is demonstrating (i.e., “proving”) why one definition is preferable to another.

Unfortunately, I’m already over 1000 words and my “proof of prove” has at least three parts. Thus, I will pick up with this next time.

But let me finish by trying to put all of this in context. In my last post, I mentioned about the Scarecrow from The Wizard of Oz receiving a mere piece of paper…a diploma…a degree (the delightfully ridiculous “Doctor of Thinkology”).

The last part of the definition of proof is “qualified judges.” What makes one a “qualified judge”? A piece of paper?

If I Only Had a Brain

The Wizard of Oz is universally considered one of the greatest films ever made:

  • It is in the Top 10 on the American Film Institute’s “100 Years 100 Movies” list. Say whatever you will about the state of cinema and argue about its placement on the list (right at #10), but with the 1000s of movies made over the years, you don’t get into the Top 10 without universal acclaim.
  • Yes, I know that it is based upon L. Frank Baum’s book, and that the general ideas and themes are his. But the book is much, MUCH darker than the film–the stuff of nightmares, in fact.
  • Finally, I would go out of my way this week to force a movie whose central element is a windstorm: a tornado, to be precise. Why? I missed last week because Hurricane Sally decided to take a slow, leisurely stroll along the Alabama-Florida border, taking my electricity along with it. Fortunately, Sally was a fickle lady and returned my beloved electrons shortly thereafter.

But why would I reference The Wizard of Oz now when last week two weeks ago, I concluded by asking “How do you know that you can trust me for calculating a [mortgage] payment?” I think we’d all agree that the connection between the Yellow-Brick Road and the Bank of America is a bit tenuous.

Well, the connection comes from my single favorite part of The Wizard of Oz. I deliberately wrote part instead of scene in that sentence because what I love is a theme that stretches ever so subtly through the entire movie: the Scarecrow’s lack of a brain.

Throughout the film, we are clearly told time and time again that the Scarecrow has no brain. And, to be frank, Ray Bolger’s rendition of “If I Only Had a Brain” (which starts at the 2:44 mark in the video below)* is my favorite song in the film, though I’ll clearly acknowledge that “Somewhere Over the Rainbow” is an objectively better and more iconic song.

*For you “techies,” the video above clearly shows that I know how to embed a YouTube video into WordPress. I also know how to create a YouTube link that starts at a particular timestamp (in this case, it would be https://www.youtube.com/watch?v=zyhUHJKfR5Y&feature=youtu.be&t=164). What I don’t know how to do is get the embedded video to start at that timestamp. Any advice on pulling that off would be appreciated. In fact, I’ll cut your subscription rate to this blog in half!

Well, what’s so special about lacking a brain? After all, my entire career (teaching) assumes that somebody’s head is (for lack of a better term) “empty.” And frankly I can only consider myself to be successful in that career when that somebody’s head is no longer empty. To borrow Paul’s phrasing, “Though I teach with the tongues of angels, and my students have not understanding, I am as a crashing cymbal.”

But the subtle, ironic part of The Wizard of Oz‘s treatment of the Scarecrow’s lack of a brain is that the Scarecrow is always the one who comes up with the good ideas.

*And if you pay attention, the Tin Man, despite lacking a heart, is always the one who shows the most emotion. The Cowardly Lion as well always shows the most courage. After all, courage is not doing something that scares others; it is doing something that scares you. Yes, he ran away when first meeting the Wizard…but he went back later. That is courage.

And my absolute favorite moment of the movie is when the Wizard finally gives the Scarecrow his diploma (his “Doctor of Thinkology”). Why then? Because that’s the only time in the movie that the Scarecrow makes an error.*

*According to histories of the making of the film, the error was not actually intended. The Scarecrow was supposed to state the Pythagorean Theorem correctly to show off his new intelligence (or, at least, his newly recognized intelligence). In the film, he should have said, “The sum of the squares of two legs of a right triangle is equal to the square of the remaining side.” Instead, he said (with the errors in bold), “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” But I’d argue that this error on the part of the filmmakers drives home the point much more strongly than it would have been otherwise.

So…what’s the point? Credentials aren’t everything.

Now, let me first put out a “Stupid Alert.” Do not interpret the statement above as me saying, “Credentials mean nothing.” And certainly do not tell others that I think that “credentials mean nothing.” Because, if you do, then you, my friend, are stupid. No, let me tweak that: you, my friend, are STOO-PID.

I would love to get into the meaning of “Credentials aren’t everything” this week, but I feel that I just need to first address the abundance of stupid that is everywhere.

And for that, I need to talk about negation. To negate a statement is to simply change what we call its truth value. The statement, “The Eiffel Tower is in Paris” is either true or it is false. And, clearly, that statement is true.

To change the statement to false, we simply introduce the word not as needed: “The Eiffel Tower is not in Paris.” Simple enough, no?

It gets a little trickier when there are a bunch of things involved, such as the entire population of France.

Consider this: “Every Frenchman is in Paris.” That statement is clearly false–just ask a citizen of Marseille or Toulouse.

Every Frenchman is in Paris!

But creating the negation is a little more difficult this time. After all, the statement, “Every Frenchman is not in Paris” is still false (now ask a Parisian).

Every Frenchman is NOT in Paris!

The negation of “Every Frenchman is in Paris” is actually “Not every Frenchman is in Paris.”

In other words, the negation of all is not none. There is some middle ground.

Well, my statement “Credentials aren’t everything” is a negation: “Credentials are not everything.” What is it negating? The simple statement, “Credentials are everything.”

Thus, “Credentials aren’t everything” leaves room for “Credentials still mean something.”

But what do they mean?