An Escher sentence?
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Tom Ace writes:
(Possibly) an Escher sentence:
"The concept of mathematics being purely objective is unequivocally false, and teaching it is even much less so."
It appears on page 65 of this document.
I coined (or re-coined) the phrase "Escher sentence" in a post back in 2004 (links updated):
"More people have written about this than I have." It's interesting, as Geoff Pullum observes, that such sentences go down so easy, since they're completely incoherent.
These sentences remind me of the pictures of stairways that spiral up endlessly within a finite space, and the Shepard tones whose pitch seems to go up and up without ever getting any higher. All these stimuli involve familiar and coherent local cues whose global integration is contradictory or impossible. These stimuli also all seem OK in the absence of scrutiny. Casual, unreflective uptake has no real problem with them; you need to pay attention and think about them a bit before you notice that something is going seriously wrong.
Like Escher stairways and Shepard tones, these sentences are telling us something about the nature of perception. Whether we're seeing a scene, hearing a sound or assimilating a sentence, there are automatic processes that happen effortlessly whenever we come across the right kind of stuff, and then there are kinds of analysis that involve more effort and more explicit scrutiny. This is probably not a qualitative distinction between perception and interpretation, but rather a gradation of processes from those that are faster, more automatic and less accessible to consciousness, towards those that are slower, more effortful, more conscious and more optional.
In that sense of the phrase, I'm not sure that Tom's example qualifies. The writer apparently started (at least conceptually) from something like this:
"Mathematics is not purely objective, and teaching it is even less objective."
In that phrasing, "is even less so" works as anaphoric for "is even less objective".
But then to add emphasis, they revised the first clause from "Mathematics is not purely objective" to "The concept of mathematics being purely objective is unequivocally false".
And now readers will interpret "even less so" as "even less false", which of course is not at all what was intended.
This is different from sentences like "More people have written about this than I have", which are naturally interpreted as the writer intended, even though there's no overall grammatical route to that interpretation.
See also:
"Plausible Angloid gibberish", 5/6/2004
"An Escher sentence in the wild", 5/8/2004
"Escher sentences: Prior art", 5/15/2004
"More people than you think will understand", 12/27/2009
"Sharks and New Yorkers", 4/30/2012
"More people have thought about this than I have", 8/6/2018
Tim Leonard said,
February 15, 2021 @ 9:04 am
"Still, few (including this author) have ever tested their theories by eating nightshade berries." [John Eastman, The Book of Forest and Thicket, p. 139]
Roscoe said,
February 15, 2021 @ 10:16 am
From "Alice's Adventures in Wonderland":
Alice had been looking over his shoulder with some curiosity. "What a funny watch!" she remarked. "It tells the day of the month, and doesn't tell what o'clock it is!"
"Why should it?' muttered the Hatter. "Does your watch tell you what year it is?"
"Of course not," Alice replied very readily: "but that's because it stays the same year for such a long time together."
"Which is just the case with _mine_," said the Hatter.
Alice felt dreadfully puzzled. The Hatter's remark seemed to have no sort of meaning in it, and yet it was certainly English.
Chuck Pergiel said,
February 15, 2021 @ 11:08 am
"More people have written about this than I have." Color me puzzled. This sentence makes perfect sense to me. Are we living in parallel universes?
[(myl) So fill in the blank: "More people have written about this than I have ___"?
Tom Ace said,
February 15, 2021 @ 11:25 am
The sentence appears on page 65 of the document at
https://equitablemath.org/wp-content/uploads/sites/2/2020/11/1_STRIDE1.pdf
Greg said,
February 15, 2021 @ 11:42 am
@Chuck Pergiel: what are the two things being compared in that sentence?
For me it should be probably be "More people have written *more* about this than I have," but I suspect there's more than one way to fix it :)
Aaron said,
February 15, 2021 @ 12:55 pm
On first read, for me it produces the meaning "More people than just me have written about this", i.e. "I'm not the only one who's written about this."
Chas Belov said,
February 15, 2021 @ 4:50 pm
For someone who is fluent in other languages, unlike me, it might be interesting to find out how Google Translate manages with Escher sentences. Conversely, there must be Escher sentences in other languages. Are there any languages where you could not create an Escher sentence without violating the grammar rules of that language?
Chuck Pergiel said,
February 15, 2021 @ 4:52 pm
[(myl) So fill in the blank: "More people have written about this than I have ___"?
'written about this' could go in the blank, but it's unnecessary because we already know that we are talking about 'written about this' because it was stated earlier in the sentence. It's probably a predicate clause or something.
Terry Hunt said,
February 15, 2021 @ 7:23 pm
@ Chuck Pergiel
So, if it's actually a meaningful sentence, you should be able to construct an exact paraphrase of it, with no less and no more information in it, yes? Care to try?
Tim Leonard said,
February 15, 2021 @ 10:15 pm
@ Chuck Pergiel
How many have you written about this?
GS said,
February 15, 2021 @ 11:04 pm
"More people have written about this than I have fingers," obviously.
Keith said,
February 16, 2021 @ 1:42 am
My parents and grandparents in Yorkshire would have completed the sentence with "had hot dinners". That was the standard way of comparing one very limited set of occurrences with a much larger set.
"I've seen more crash blossoms than you've had hot dinners" is an (invented) example that comes to mind.
Michael Watts said,
February 16, 2021 @ 3:30 am
"More people have written about this than I have [written about this]" is actually a lot more coherent than the original example of "More people have been to Russia than I have [been to Russia]", because the two things in question are both conventionally subject to measurement. You could conceive of two quantities:
1. The number of people who have written about "this".
2. The amount [of material] that I have written about "this".
And then compare them. It's not a comparison that makes much sense, but that's not an obstacle; compare Douglas Adams' "The big yellow ships hung in the air in much the same way that bricks don't.", or a hypothetical unlike-with-unlike comparison like "Warren Buffett is richer than Shaquille O'Neal is tall."
Chuck Pergiel said,
February 16, 2021 @ 9:49 am
Sorry. I finally realized what the problem is. Why I could not see it before is a mystery.
Roscoe said,
February 16, 2021 @ 11:07 am
Warren Buffett makes more money than Shaquille O'Neal…put together!
Daniel Barkalow said,
February 16, 2021 @ 2:20 pm
I don't think "teaching it is even less objective" is what is intended. I think the intended meaning is more like "teaching that it is objective is even more problematic". There's a different argument that teaching mathematics is not objective: you should explain concepts in multiple different ways, give students explanations, examples, and work to do (because different students will learn better from each of these), relate problems to concrete goals that are meaningful to different students, and so forth; however, that's covered elsewhere.
Haamu said,
February 16, 2021 @ 3:04 pm
I guess I'm with Daniel Barkalow on the likely intended meaning of the sentence.
On my first read, the first parsing issue I encountered was determining the referent of it. I did not take "Teaching it" to mean "Teaching [mathematics]," but rather "Teaching [that the concept of mathematics is purely objective]."
That leads to "much less so" resolving to "much less [unequivocally false]." Which in turn leads to other problems: (1) Is this an inversion — i.e., was "much more" intended? and (2) Are there gradations of being "unequivocally false"? For (1) I say, probably, and for (2) I say no, but it's the sort of casual phrasing many people use.
Haamu said,
February 16, 2021 @ 3:23 pm
@MYL: I found this part of your 2004 comments interesting:
(Emphasis mine, obviously.)
I'm reminded of Daniel Kahneman's 2013 book, Thinking, Fast and Slow. I took his model to be a fairly sharp, qualitative dichotomy between the fast/automatic "System 1" of human cognition and the slower/logical/analytical "System 2." I'm wondering if you buy his model (assuming I'm not mischaracterizing it) or if you still see no qualitative distinction among these processes.
My sense is that the qualitative distinction, if one exists, is not between perception and interpretation but between different types of interpretation. I.e., System 1 handles perception and automatic interpretation, and System 2 fires up and attempts rational analysis only when System 1 throws an error. But it does feel like a qualitative distinction, not a mere gradation.
Brett said,
February 16, 2021 @ 4:57 pm
@Kieth: As Monty Python famously inverted it: "I've had more gala luncheons than you've had hot dinners!"
tom said,
February 17, 2021 @ 12:45 pm
"even much less so." seem to refer to the amount of obejectivity.
The "purely objective is unequivocally false" can mean there is little objectivity to it or none , so the amount after "even much less so." is small , zero , or negative (just subjective).
The first reading on this , I thought the Escher sentence , is a paradoxical self contradictory one.
so this was my reaction
One can say that math is objective in the sense that the objective pass thrue the subjectve , and even that sentence such example.
George Amis said,
February 17, 2021 @ 2:58 pm
I'm not sure I quite understand the "Escher Sentence" concept, but how about the question "Do you feel more like you do now than you did before you came?'
Andrew Myers said,
February 17, 2021 @ 5:50 pm
Isn't it just "More people have written about this than I" with an extra "have" tacked onto the end?
Michael Watts said,
February 18, 2021 @ 12:59 pm
You could think of it that way, but it wouldn't do much to explain why people find it acceptable have.
Joyce Melton said,
February 18, 2021 @ 11:59 pm
Is it a count problem, not a mass problem? *More people (countable) have written about this than I*, makes sense. Adding the second have makes less sense because it seems to change it to be about the mass of the writing rather than the count of who has written.
"People have written more about this than I have" makes some sense but really needs an "Other" tacked on the beginning not to also sound weird. To also not sound weird?
Now it all sounds weird.