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The foundations of mathematics are unproven
Philosopher and logician Kurt Gödel upended our understanding of mathematics and truth.
- In 1900, mathematician David Hilbert laid down 23 problems for the mathematics world to solve, the biggest of which was how to prove mathematics itself.
- Far from solving the issue, Kurt Gödel showed just how groundless the axioms of mathematics are.
- Gödel's theorem does not devalue mathematics but reveals that some truths are unprovable.
Everything's a bit crazy at the moment. We're drowning in a sea of lies, half-truths, polarization, debate, argument, and uncertainty. But at least there's math, right? That one sanctuary of truth and certainty. It's the algebraic flotsam we can grip on to, before we're swept away.
Well… look away now if you like it like that, because Kurt Gödel might be about to snatch even that away. His incompleteness theorems shook the foundations of the (math) universe. In fact, he rather did away with those foundations altogether.
In the early 20th century, the famous mathematician David Hilbert laid down 23 problems for the mathematics world to solve. Some of them are particularly esoteric, but the big ones concerned the issues of math's consistency and completeness. Hilbert hated the fact that the whole of mathematics depended on certain "axioms" that were, themselves, not proven. He wanted no loose ends, paradoxes, or unproven items. This was math after all!
Gödel, though, rather quashed all that.
Gödel would have hated what the postmodernists made of his work.
To see how, we have to first know that "axioms" are those statements that we accept as true before we go about doing math. They're like the letters needed to make words. For example, A + B = B + A is an axiom, as are all the functions of arithmetic and so on. Simply put, axioms are the building blocks of mathematics. They're as true for Euclid, drawing squares in ancient Greek dust, as they are for a pained 15-year-old, frowning over some calculus.
The problem is that these axioms are not proven. They're true because they always work, and we observe them as true all of the time. But they're not proven.
Imagine the whole of mathematics as a huge sack, and inside are all the possible things math can do. It's a mighty big sack, indeed. What Gödel proved is that, first, there exists in this sack a set of things which cannot be proven or disproven, such as axioms. Second, there is no possible way to prove these axioms from within that sack. It's impossible for math, on its own, to prove its own axioms.
Essentially, it's a problem of self-reference. It's an issue seen, too, in Russell's paradox about sets. More famously, the liar paradox imagines a sentence like, "This sentence is false." When you examine it closely, it creates a logical circularity. If the sentence is true, then it's false; but then if it's false, it's true. It's enough to make a robot's brain explode.
Gödel applied a similar logic to the whole system of mathematics. He took the sentence, "This statement is unproven," and converted it into a number statement about numbers (with a code system known as "Gödel numbering"). He discovered that this proposition cannot be proven within that system.
Going even further than this, Gödel concluded that in every system that's rich enough to allow for arithmetic, there will be a proposition within it that cannot be proven by it's own tools. We need some kind of "meta language" to prove the rules by which a system operates. It's a bit like how we can't see our own eyes or draw around the hand that's holding the pencil.
How postmodernists weaponized Gödel
Gödel has been misrepresented, even in his lifetime. For instance, certain postmodernist philosophers used him to say, "There is no truth! Even math is groundless!" They wanted to show how everything was meaningless, and truth amounted only to opinion.
But this isn't the point. Gödel only showed that truth does not always need to be proven. This is, of course, no small thing. To pull apart truth and provability, to allow for "unproven truths," seems highly counterintuitive
Gödel, himself, thought there were objective truths. His theory only went to show the limitations of mathematics but not that it was flawed in any way. He would have hated what the postmodernists made of his work.
"Deepfakes" and "cheap fakes" are becoming strikingly convincing — even ones generated on freely available apps.
- A writer named Magdalene Visaggio recently used FaceApp and Airbrush to generate convincing portraits of early U.S. presidents.
- "Deepfake" technology has improved drastically in recent years, and some countries are already experiencing how it can weaponized for political purposes.
- It's currently unknown whether it'll be possible to develop technology that can quickly and accurately determine whether a given video is real or fake.
After former U.S. President William Henry Harrison delivered his inaugural speech on March 4, 1841, he posed for a daguerreotype, the first widely available photographic technology. It became the first photo taken of a sitting American president.
As for the eight presidents before Harrison, history can see them only through artistic renderings. (The exception is a handful of surviving daguerreotypes of John Quincy Adams, taken after he left office. In his diary, Adams described them as "hideous" and "too true to the original.")
But a recent project offers a glimpse of what early presidents might've looked like if photographed through modern cameras. Using FaceApp and Airbrush, Magdalene Visaggio, author of books such as "Eternity Girl" and "Kim & Kim," generated a collection of convincing portraits of the nation's first presidents, from George Washington to Ulysses S. Grant.
Modern Presidents George Washington https://t.co/CURJQB0kap— Magdalene Visaggio (@Magdalene Visaggio)1611952243.0
What might be surprising is that Visaggio was able to generate the images without a background in graphic design, using freely available tools. She wrote on Twitter:
"A lot of people think I'm a digital artist or whatever, so let me clarify how I work. Everything you see here is done in Faceapp+Airbrush on my phone. On the outside, each takes between 15-30 mins. Washington was a pretty simple one-and-done replacement."
Ulysses S Grant https://t.co/L1IGXLI3Vl— Magdalene Visaggio (@Magdalene Visaggio)1611959480.0
"Other than that? I am not a visual artist in any sense, just a hobbyist using AI tools see what she can make. I'm actually a professional comics writer."
Did another pass at Lincoln. https://t.co/PdT4QVpMbn— Magdalene Visaggio (@Magdalene Visaggio)1611973947.0
Of course, Visaggio isn't the first person to create deepfakes (or "cheap fakes") of politicians.
In 2017, many people got their first glimpse of the technology through a video depicting former President Barack Obama warning: "We're entering an era in which our enemies can make it look like anyone is saying anything at any point in time." The video quickly reveals itself to be fake, with comedian Jordan Peele speaking for the computer-generated Obama.
While deepfakes haven't yet caused significant chaos in the U.S., incidents in other nations may offer clues of what's to come.
The future of deepfakes
In 2018, Gabon's president Ali Bongo had been out of the country for months receiving medical treatment. After Bongo hadn't been seen in public for months, rumors began swirling about his condition. Some suggested Bongo might even be dead. In response, Bongo's administration released a video that seemed to show the president addressing the nation.
But the video is strange, appearing choppy and blurry in parts. After political opponents declared the video to be a deepfake, Gabon's military attempted an unsuccessful coup. What's striking about the story is that, to this day, experts in the field of deepfakes can't conclusively verify whether the video was real.
The uncertainty and confusion generated by deepfakes poses a "global problem," according to a 2020 report from The Brookings Institution. In 2018, the U.S. Department of Defense released some of the first tools able to successfully detect deepfake videos. The problem, however, is that deepfake technology keeps improving, meaning forensic approaches may forever be one step behind the most sophisticated forms of deepfakes.
As the 2020 report noted, even if the private sector or governments create technology to identify deepfakes, they will:
"...operate more slowly than the generation of these fakes, allowing false representations to dominate the media landscape for days or even weeks. "A lie can go halfway around the world before the truth can get its shoes on," warns David Doermann, the director of the Artificial Intelligence Institute at the University of Buffalo. And if defensive methods yield results short of certainty, as many will, technology companies will be hesitant to label the likely misrepresentations as fakes."
The author of 'How We Read' Now explains.
During the pandemic, many college professors abandoned assignments from printed textbooks and turned instead to digital texts or multimedia coursework.
As a professor of linguistics, I have been studying how electronic communication compares to traditional print when it comes to learning. Is comprehension the same whether a person reads a text onscreen or on paper? And are listening and viewing content as effective as reading the written word when covering the same material?
The answers to both questions are often “no," as I discuss in my book “How We Read Now," released in March 2021. The reasons relate to a variety of factors, including diminished concentration, an entertainment mindset and a tendency to multitask while consuming digital content.
Print versus digital reading
The benefits of print particularly shine through when experimenters move from posing simple tasks – like identifying the main idea in a reading passage – to ones that require mental abstraction – such as drawing inferences from a text. Print reading also improves the likelihood of recalling details – like “What was the color of the actor's hair?" – and remembering where in a story events occurred – “Did the accident happen before or after the political coup?"
Studies show that both grade school students and college students assume they'll get higher scores on a comprehension test if they have done the reading digitally. And yet, they actually score higher when they have read the material in print before being tested.
Educators need to be aware that the method used for standardized testing can affect results. Studies of Norwegian tenth graders and U.S. third through eighth graders report higher scores when standardized tests were administered using paper. In the U.S. study, the negative effects of digital testing were strongest among students with low reading achievement scores, English language learners and special education students.
My own research and that of colleagues approached the question differently. Rather than having students read and take a test, we asked how they perceived their overall learning when they used print or digital reading materials. Both high school and college students overwhelmingly judged reading on paper as better for concentration, learning and remembering than reading digitally.
The discrepancies between print and digital results are partly related to paper's physical properties. With paper, there is a literal laying on of hands, along with the visual geography of distinct pages. People often link their memory of what they've read to how far into the book it was or where it was on the page.
But equally important is mental perspective, and what reading researchers call a “shallowing hypothesis." According to this theory, people approach digital texts with a mindset suited to casual social media, and devote less mental effort than when they are reading print.
Podcasts and online video
Given increased use of flipped classrooms – where students listen to or view lecture content before coming to class – along with more publicly available podcasts and online video content, many school assignments that previously entailed reading have been replaced with listening or viewing. These substitutions have accelerated during the pandemic and move to virtual learning.
Surveying U.S. and Norwegian university faculty in 2019, University of Stavanger Professor Anne Mangen and I found that 32% of U.S. faculty were now replacing texts with video materials, and 15% reported doing so with audio. The numbers were somewhat lower in Norway. But in both countries, 40% of respondents who had changed their course requirements over the past five to 10 years reported assigning less reading today.
A primary reason for the shift to audio and video is students refusing to do assigned reading. While the problem is hardly new, a 2015 study of more than 18,000 college seniors found only 21% usually completed all their assigned course reading.
Maximizing mental focus
Researchers found similar results with university students reading an article versus listening to a podcast of the text. A related study confirms that students do more mind-wandering when listening to audio than when reading.
Results with younger students are similar, but with a twist. A study in Cyprus concluded that the relationship between listening and reading skills flips as children become more fluent readers. While second graders had better comprehension with listening, eighth graders showed better comprehension when reading.
Research on learning from video versus text echoes what we see with audio. For example, researchers in Spain found that fourth through sixth graders who read texts showed far more mental integration of the material than those watching videos. The authors suspect that students “read" the videos more superficially because they associate video with entertainment, not learning.
The collective research shows that digital media have common features and user practices that can constrain learning. These include diminished concentration, an entertainment mindset, a propensity to multitask, lack of a fixed physical reference point, reduced use of annotation and less frequent reviewing of what has been read, heard or viewed.
Digital texts, audio and video all have educational roles, especially when providing resources not available in print. However, for maximizing learning where mental focus and reflection are called for, educators – and parents – shouldn't assume all media are the same, even when they contain identical words.
Humans may have evolved to be tribalistic. Is that a bad thing?
- From politics to every day life, humans have a tendency to form social groups that are defined in part by how they differ from other groups.
- Neuroendocrinologist Robert Sapolsky, author Dan Shapiro, and others explore the ways that tribalism functions in society, and discuss how—as social creatures—humans have evolved for bias.
- But bias is not inherently bad. The key to seeing things differently, according to Beau Lotto, is to "embody the fact" that everything is grounded in assumptions, to identify those assumptions, and then to question them.
Ancient corridors below the French capital have served as its ossuary, playground, brewery, and perhaps soon, air conditioning.