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This article about a caterpillar specialist claims:

There are roughly sixty-five hundred species of mammals, nine thousand species of amphibians, and eleven thousand species of birds. These are what people tend to think of when they picture the world’s biodiversity. But the planet’s real diversity lies mostly beneath our regard. The largest family of beetles, the Curculionidae, commonly known as weevils, contains some sixty thousand described species; another beetle family, the Tenebrionidae, comprises twenty thousand species. It is estimated that in one family of parasitic wasps, the Ichneumonidae, there are nearly a hundred thousand species, which is more than there are of vertebrates of all kinds.

I am suspicious of this claim for two reasons:

First, there is no established single definition of what constitutes a "species." So maybe they are applying one definition to vertebrates that is not reasonably comparable to the definition being used here for insects?

The article seems to say that, at least when it comes to lepidoptera, entomologists have decided that every polymorphism of a single gene (CO1) constitutes a different species, which to me sounds like a fast-and-loose definition.

Whatever the definition, I am skeptical that entomologists can keep track of tens of thousands of samples of "unique species" of a single family of insects in such a way as to ensure there are no duplicates. E.g., 60,000 "descriptions" of beetles they are confident are unique? (Much less so "unique" as to constitute different species?)

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    Meh, I'm not sure this is valid a Q here. You're challenging whether species of insects are defined in the same or comparable way to those for mammals. The quote doesn't explicitly make that claim. If the quote is misleading because the definitions differ, then it's misleading, there's not much else to say. If your question is about the comparability of definitions [and it seems to be], biology.stackexchange.com is probably the better place to ask. Commented May 2, 2023 at 19:13
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    See skeptics.meta.stackexchange.com/questions/2177/… for meta-discussion about Qs revolving around defintions. Commented May 2, 2023 at 19:21
  • @Fizz is it not enough to be skeptical of a claim? If it brings it on topic I would be happy to remove my speculation as to a way the claim might "true" but misleadingly so, and just leave it at, "Are there really more species of weevils than vertebrates?"
    – feetwet
    Commented May 2, 2023 at 19:28
  • There are like at least a dozen of (alternative) definitions for species. Whether the claim/comparison holds under all of those is perhaps interesting, but perhaps not what the authors (or the quote) intended. Commented May 2, 2023 at 19:29
  • Generally speaking, when people make this kind of claim they seem to use the IUCN counts & definitions. Commented May 2, 2023 at 19:55

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[Answer in progress, please edit to improve!]

I checked the Catalogue of Life.

[TODO: evidence for the reliability of this source, and info about taxonomy methods in general.]

Subphylum Vertebrata, as of this writing, includes 70347 species, of which 70302 are listed as "accepted", the rest being "provisionally accepted".

Superfamily Ichneumonoidea, includes 42362 species (42201 accepted). So, not more.

On the other hand, family Curculionidae, the "true weevils", has 73871 species, which is indeed more than subphylum Vertebrata. However only 16151 are "accepted".

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    I am not sure how to address the question's "I am skeptical that entomologists can keep track of tens of thousands of samples of 'unique species' of a single family of insects in such a way as to ensure there are no duplicates." Other than to say, that is the entire point of the field of taxonomy, and an awful lot of scientists have spent an an awful lot of time trying to achieve it. Commented May 2, 2023 at 4:06
  • I rejected that statement on the basis that you don't need to have tracked every single unique species in order to be able to get a reasonable estimate (e.g. by sampling).
    – Oddthinking
    Commented May 2, 2023 at 4:20
  • Wikipedia cites a reference suggesting Ichneumonoidea"is thought to contain as many as 100,000 species, many of which have not yet been described."
    – Oddthinking
    Commented May 2, 2023 at 4:23
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    @Oddthinking: I think COL limits itself to described species, which seems to imply that the description has been published under peer review. But it's certainly plausible that the vast majority of vertebrates have been described, whereas the proportion for insects may be much smaller. Commented May 2, 2023 at 4:34
  • I think that is very likely the case, making your "So, not more." statement unreliable, and the OPs thoughts about individual tracking of species a strawman.
    – Oddthinking
    Commented May 2, 2023 at 8:14

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