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Does size limit intelligence?
I may make a small, but very intelligent species for a journal thing that I am making.
So could a microorganism achieve human-like intelligence? Or a small fly? Or does the brain have to be a specific size for things to be considered intelligent?
Does size limit intelligence?
Yes, but the limits are not where many assume them to be.
Older work on the subject relying on brain size (weight), body to brain weight ratio or complexity (all those little wrinkles & folds) has been superseded by more recent work.
Intelligence appears to be limited by 'total processing power', the number of neurons & connections between neurons (so much as processing power & memory in computers, unsurprising then when you think about it) rather than it's physical size but there are physical limits to how small you can make a neuron, how closely you can pack them & how many connections you can cram in inevitably leading to limits on the physical size of a brain that has 'intelligence' of any specific level.
So you can't just make a brain smaller & smaller & retain the same level of intelligence, eventually you reach a point where you have to begin shedding neurons (& intelligence) to get any smaller.
The smallest brains with the highest intelligence by weight (& the highest neuron count by weight) known in nature are birds (unsurprising as they need to keep weight down for flight, so any tricks there are to achieve more with less for brains will have been heavily selected for by evolutionary pressures), they have more neurons per gram of brain than in mammals
It's not unlikely birds are pretty close to having got as close to as small as you can go for the brains' weight to intelligence ratio given the evolutionary pressures they're under to cut weight for flight & (very rough unscientific) extrapolations from a Ravens 15g brain suggest something in the region of a 45g bird brain 'might' plausible be as intelligent as a human, that's likely as small as you can go & have adult human intelligence (& that of course is just the brain, you need a body around it as well).
But it will also depend on how much of that 45g brain is given over to higher reasoning rather than other brain functions like processing sensory information & motor control of course.
"very rough unscientific" aka, 'mine'.
Could a microorganism achieve human-like intelligence?
No, there is absolutely no way you can achieve that without magic, see above.
They'd have to organise themselves into some sort of colony or modular organism to achieve the necessary neuron count but then you won't really have a microorganism any more.
Not Really
There are a couple of creatures that are small and rather clever. I remember stories about octopuses for example escaping from their tanks, and one the says the octopus snuck into another thing to eat some of the fish and then snuck back into its own tank. I even remember the story about a guy who was training a raven to use a keyboard only for that raven to sneak out of it enclosure, destroy the keyboard, and sneak back into its enclosure in order to play innocent. While it is true that bigger size might allow more room for a bigger brain, never underestimate the cleverness of the little guy.
Microorganisms might have to work together though
I think for a microscopic organism you’re going to have to have a lot of them working together as a colony in order to become sentient. There’s an interesting species from a lovely Stargate episode called Message in a Bottle to give you an idea of what I’m talking about. https://stargate.fandom.com/wiki/A%27t%27trr
As others have written, size matters.
However, it doesn't have to be the size of the individual that counts.
Single ants or bees are stupid. Anthills and beehives are much smarter. Nowhere near human smart, but perhaps in another world they could be.
On Earth, hive minds "think" using pheromones. This is slow. Slooow. You probably want find a faster way for hive mind members to communicate.
We don't know, but an intelligent insect or microorganism is dubious.
The size of the brain scales with the size of the species. It is the brain-to-body-size ratio that matters, more than the absolute size of the brain. This is thought to be because a larger body requires more neurons for sensory and motor control, so only the neurons beyond that minimum count towards abstract "intelligence." So a small species could be intelligent if it had a large brain relative to its size.
Crows are fairly intelligent for animals, while being much smaller than dogs or dolphins who exhibit possibly comparable intelligence. A crow-sized animal with an unusually large brain and human intelligence could be plausible.
Below a certain size, you're going to have to invent new computation mechanisms other than neurons. Fruit flies only have around 100,000 neurons; this isn't enough to remember very much. But if your fictional creature encodes information in molecules like RNA, then it could remember a lot more. I'd still be skeptical about a microorganism having intelligence. Still, I'd be willing to suspend disbelief. It's been done before.
This book provides good physical arguments for the minimum size for an intelligent system based on chemistry. If we rely on molecules for information processing and storage, one bit cannot be smaller than a single atom (0.1 to 0.3 nanometers). From this, the book derives a rough minimum size for living cells based on the amount of information they must store (0.1 to 1 micrometers). This gives the minimum size of a neuron and the minimum size for a complex brain.
Hence, information processing based on chemical compounds is limited by the size of atoms and likely cannot create a complex brain at the microorganism scale. However, there is a regime of physics where the structures are 100,000 times smaller and the timescales are at least that much faster: nuclear physics. We can't really build complex structures from nuclear matter at standard conditions. Just like the organic molecules that make up life on Earth are not stable on the surface of the sun, our world is much too cold for complex nuclear "molecules" (if such things can exist). But maybe in the core of a neutron star, you could build something like a neuron out of nuclear matter that is $10^5$ times smaller. In that case, you could construct something with the complexity of a human brain at the scale of about 1 µm, similar to the size of microorganism. It could also operate much faster. Of course the rules in nuclear matter are much different than in chemistry and we know very little about them. Certainly quantum effects will be more important for the nuclear neurons than for chemical neurons.
Frame challenge: Instead on focusing on the possibilities for "intelligence" (where, after all, "intelligence" is a description of behaviour), perhaps you can focus on what behaviours you want your creatures to have, and how those creatures could achieve them.
For example, emergent behaviours can be (or appear to us) complex while while the components appear simple. As another example, evolution by natural selection has "developed" very complex machinery inside cells, in virus, bacteria, immune systems, and lots more. While not generally considered intelligence, its complex. Maybe your desired behaviours can be achieved by evolution in appropriate conditions.
Of course, if "human-like intelligence" means "do everything humans do" including talk in English, then you probably need [idiosyncratic] humans for that...
