Seeing X as Y
Is How Ideas Are Creatively Connected
The assertion "All Seeing is Seeing As" is commonly credited to G. N. A. Vesey, who authored the paper “Seeing and Seeing As” in the Proceedings of the Aristotelian Society, Volume 56, Issue 1, 1 June 1956, Pages 109–124.
However, the concept that our perception of the world is shaped by our perspectives and contexts predates Vesey. It is already implied, for instance, in Kant's notion of "Ding an Sich" and Nietzsche's "Will to Power."
Philosophically, the notion that "All Seeing is Seeing As" neither denies nor affirms an objective reality.
Wittgenstein examined the duck-rabbit illusion that was featured in the 23 October 1892 edition of Fliegende Blätter, a German humor magazine.
The capacity to shift from one viewpoint, such as a duck's, to another, like a rabbit's, exemplifies the broader skill of alternating between perspectives, contexts, frameworks, mindsets, or even paradigms, which is believed to be linked to creativity.
But in the case of the duck-rabbit illusion, what we have is a subproblem of “All Seeing is Seeing As”, namely “Seeing X as Y”
My Interest in “Seeing X as Y” was sparked when reading Douglas Hofstadter’s “Le Ton Beau de Marot: in Praise of the Music of Language” where he shared an anecdote:
‘…reports that Ulam and his mathematician friend Gian-Carlo Rota were once having a lively debate about artificial intelligence, a discipline that fascinated Ulam but whose approach he thought was far too closely tied to logic and strict deduction. Ulam, a very visual thinker and a great believer in the centrality of imagery to thought, said to Rota,
“The time has come to enrich formal logic by adding to it some other fundamental notions. What is it that you see when you see? You see an object as a key, a man in a car as a passenger, some sheets of paper as a book. It is the word ‘as’ that must be mathematically formalized... Until you do that, you will not get very far with your AI problem .”’
Perceiving a man in a car as a passenger, sheets of paper as a book, life as a journey or a game, and a threat as an opportunity are all instances of "Seeing X as Y."
The concept of "Seeing X as Y" can have numerous meanings; it might be metaphorical, offer a different perspective, signify an interpretation or re-interpretation, represent a shift in mindset or reframing, or even indicate a paradigm change (as in the case of Thomas Kuhn).
Additional examples of "Seeing X as Y" include:
Plato perceived the world as the shadows of ideal forms.
Daniel Dennett viewed meaning as a product of evolutionary processes.
"Seeing the glass as half empty" versus "seeing the glass as half full" reflects an our attitude
Light is perceived as a wave, or simultaneously as both a particle and a wave.
"Seeing nature as a teacher, as a source of wisdom, is a fundamental aspect of transcendentalism." - Ralph Waldo Emerson
“Seeing X as Y” in Mathematics.
The notion of "Seeing X as Y" is a core element of human cognition and perception. This concept is also pertinent to Mathematics, where it can be defined with greater precision due to the abstract structures in Mathematics that facilitate analogies across different domains and the generalization of concepts.
In Mathematics, "Seeing X as Y" implies that if X is part of structure A and Y is part of structure B, then some operations that apply to Y in domain B have equivalent operations for X in domain A.
Functions can be envisioned as points within a different space that is endowed with the concepts of distance and continuity, leading to the creation of functions of functions, or functionals.
Numerical series can be interpreted as generating functions.
A recent piece in Quanta Magazine, titled “A Rosetta Stone for Mathematics” recounts the story of the French Mathematician André Weil, who envisioned connecting Number Theory, Riemann Surfaces and Finite Fields. Such connections are very profound and not at all obvious in the beginning. These connections became natural once established
“Seeing X as Y” is the way we connect ideas.
In summary, we can say that “Seeing X as Y” is much more than making hyperlinks, it is connecting ideas in a not always obvious way.
The analogy of connecting ideas to connecting dots with simple points and lines is quite deceptive. This is because ideas (represented as points) often possess various meanings that impose limitations on their potential connections with other ideas. A more accurate representation would be to imagine these points as spiked viruses, where the spikes can either facilitate or inhibit connections.
If "Connecting the Dots" symbolizes creativity, then truly creative thinking lies in "Seeing X as Y."