Circular reasoning
Georg Cantor and his set theory shook the foundations of mathematics by highlighting circular reasoning, that is to say, self-referential. At the turn of the 20th century, his paradoxes emerged under the pen of Burali-Forti, Russell and Richard, throwing Frege into dismay. Kurt Gödel showed that certain assertions cannot be proved or disproved within a given axiomatic framework, bringing a negative answer to a famous problem posed by Hilbert. Self-reference is also found in literature, painting, sculpture and… in the press where, intentional or not, it makes us smile.
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The paradoxes that shook mathematics
What a devilish theory Georg Cantor created with set theory! So thought some mathematicians at the turn of the 20th century, when a series of paradoxes seemed to threaten the foundations of mathematics. This prompted a reassessment of the role of logic.
Disputes and paradoxes | Tangente
Even the greatest mathematicians have fallen into the traps laid by logic and its paradoxes.
When works speak for themselves | Tangente
Advertising regularly makes use of it, as do graphic designers and artists. Self-reference has become a genre in its own right, appealing not only to our eyes but to all our senses! Let's take a wild tour of this seemingly limitless creativity.
Gödel's incompleteness theorems
In mathematics, although there are many conjectures and hypotheses, it is generally assumed that any well-formed statement must have either a proof or a refutation. Kurt Gödel showed that this is not so. His proof uses self-reference.
Pure and impure self-reference | Tangente
Self-reference is not "pure". It must be tied to an object, a situation or a concept—here, for instance, artistic creation, and photography in particular.
Self-reference at every level | Tangente
We do not always pay attention to them, but self-referential sentences and turns of phrase surround us every day!