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"Through Other Eyes", a novelette by R. A. Lafferty (ISFDB, Wikipedia, Great Science-Fiction & Fantasy Works,The Ants of God Are Queer Fish), available at ralafferty.com. It was first published in Future Science Fiction, February 1960, available at the Internet Archive. Here is a review by Andrew Ferguson.
A scientist invents a machine which lets you see the world through someone else's eyes by putting your consciousness into their brain somehow.
The Cerebral Scanner, newly completed by Charles Cogsworth, was not an intricate machine. It was a small but ingenious amplifying device, or battery of amplifiers, designed for the synchronous—perhaps "sympathetic" would be a better word—coupling of two very intricate machines: two human brains.
Here the only concern was with the convoluted cortex of the brain itself, that house of consciousness and terminal of the senses, and with the quasi-electrical impulses which are the indicators of its activity. It had been a long-held opinion of Cogsworth that, by the proper amplification of a near score of these impulses in one brain, a transmission could be effected to another so completely that one man might for an instant see with the eyes of another—also see inwardly with that man's eyes, have the same imaginings and daydreams, perceive the same universe as the other person. And it would not be the same universe as the seeking man knew.
He then invites a party of 7 people to try it out.
The Scanner had been completed, as had a compilation of the dossiers of seven different brains: a collection of intricate brain-wave data as to frequency, impulse, flux and field, and Lyall-wave patterns of the seven cerebrums which Cogsworth would try to couple with his own.
The seven were those of Gregory Smirnov, his colleague and counselor in so many things; of Gaetan Balbo, the cosmopolitan and supra-national head of the Institute; of Theodore Grammont, the theoretical mathematician; of E. E. Euler, the many-tentacled executive; of Karl Kleber, the extraordinary psychologist; of Edmond Guillames, the skeptic and bloodless critic; and of Valery Mok, a lady of beauty and charm whom Cogsworth had despaired of ever understanding by ordinary means.
The main character describes the visions of reality through everyone else's eyes - one person sees everything as living, moving, and vibrating.
"[. . .] Every tree has a strong smell in her world. This was an ordinary elm tree, and it had a violent musky obscene smell that delighted her. It was so strong that it staggered. And to her the grass itself is like clumps of snakes, and the world itself is flesh. Every bush is to her a leering satyr, and she cannot help but brush into them. The rocks are spidery monsters and she loves them. She sees every cloud as a mass of twisting bodies and she is crazy to be in the middle of them. She hugged a lamp post and her heart beat like it would fight its way out of her body.
Another sees everything as cold, dead molecules, etc.
Later he saw through the eyes of Theodore Grammont, and felt a surge of pity.
"If I am blind compared to Gregory, then this man is blind compared to me. I at least know that the hills are alive; he believes them to be imperfect polyhedrons. He is in the middle of a desert and is not even able to talk to the devils who live there. He has abstracted the world and numbered it, and doesn't even know that the world is a live animal. He has built his own world of great complexity, but he cannot see the color of its flanks. The man has achieved so much only because he was denied so much at the beginning. I understand now that only the finest theory is no more than a fact gnawed on vicariously by one who has no teeth. But I will return to this world too, even though it has no body to it. I have been seeing through the eyes of a blind hermit."
There is one bit where one character sneaks an early use of the machine, or is found staring back at the main character somehow.
The attempt to see into the world of Karl Kleber was almost a total failure. The story is told of the behaviorist who would study the chimpanzee. He put the curious animal in a room alone and locked the door on it; then went to the keyhole to spy; the keyhole was completely occupied by the brown eyeball of the animal spying back at him.
Something of the sort happeded here. Though Karl Kleber was unaware of the experiment, yet the seeing was in both directions. Kleber was studying Cogsworth in those moments by some quirk of circumstance. And even when Cogsworth was able to see with the eyes of Kleber, yet it was himself he was seeing.
"I am looking through the eyes of a peeper," he said. "And yet, what am I myself?"