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These suffixes do not change the part of speech, so are they inflectional endings?

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    The suffix -ist also doesn't change the part of speech in violin – violinist, nevertheless it is obviously a derivational suffix. The ability to change the part of speech is a sufficient but surely not necessary condition for an affix to be considered derivational.
    – Yellow Sky
    Aug 5 at 13:12
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    While -th is clearly an ordinal suffix, -nd and -rd aren’t. Second is completely suppletive and doesn’t have any suffix (it comes from Latin secundus ‘following, next’, derived from the verb sequor ‘follow’); and in third, the r is part of the number (three), and only the -d is really suffix-like. Aug 5 at 13:35
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    seco- isn't even a word in English, so any attempt to break second apart is doomed to failure. The others are at least polymorphemic although note that the r in third is part of the root not the affix
    – Tristan
    Aug 5 at 13:36
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    And the -st in first is just a superlative, like the ones in last and next.
    – jlawler
    Aug 5 at 13:57

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Introductory textbooks sometimes tout "changes the part of speech" as being the key feature of a derivational affix. But this definition has a lot of flaws, and you've run into one of them.

One, there are some affixes that do change the part of speech, but don't seem to be derivational. "Run" is a verb, but "running" is a noun or an adjective depending how it's used. And that certainly doesn't seem like a derivational change.

And two, there are some affixes that don't change the part of speech, but do seem to be derivational. "Un" is a classic example of this: "do" and "undo" are both verbs, but would you really say they're forms of the same word? In the comments, Yellow Sky also mentions violin → violinist, both nouns.

(Some textbooks also say it's derivational if the meaning changes, but then you have to define "meaning" and explain why "run" and "running" have the same meaning when clearly there's a reason to use one instead of the other.)

So the rule of thumb I tell my students is that derivational affixes make a new word that deserves its own dictionary entry. Native speakers' intuition tends to work well with this rule. More formally, you can look at how regular this process is; I can't think of a single verb in English that can't take "-ing", while very few can take "un-" (it's more of an adjective thing). If you can only use an affix on a particular category of meanings (rather than a part of speech), that's a good sign it's derivational. But the categories of "inflection" and "derivation" aren't actually that strict and clear in the real world, and linguists can (and do!) disagree about what category something can fall into.

For your particular example, I would say "-th" is derivational, because "four" and "fourth" feel like distinct words, and it can't be applied to all adjectives, only the ones that mean numbers (mathematicians say "n-th" all the time, and I've heard "k-th" in computer science, but you can't say *"blue-th" or *"cold-th").

The other ones, like fir-st, seco-nd, thi-rd, are not affixes at all in my opinion. What is a *seco on its own? These are just exceptions to the pattern. The formal name for "first" and "second" is suppletion, when the regular, expected form gets replaced with a form taken from a totally different word, while "third" is fossilized: long ago it was completely regular, but it was commonly-used enough that it didn't change as the language around it changed, and now it's an exception.

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  • Diachronically of course you could say -st is superlative and -nd is the gerundive in Latin, but synchronically that's certainly not how native speakers acquire these words.
    – Draconis
    Aug 5 at 15:54
  • English does have some defective verbs that can’t take -ing (quoth, hight, methinks, etc.), but they are peculiar in other ways as well. But I do think it still makes sense to consider first a superlative synchronically as well; it certainly behaves like one in many ways. We can say ‘the very first/best’, ‘first/best of all’, etc. I’m not sure how you’d test whether native speakers acquire words like first the same way they acquire superlatives, but I’m not sure that’s really relevant. I remember the realisation that first and last are superlatives – made immediate sense. Aug 5 at 16:14
  • >> I can't think of a single verb in English that can't take "-ing" — I'd say I can't think of a single word in English that can't take "-ing", since in English every word can be used as a verb.
    – Yellow Sky
    Aug 5 at 16:24
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    "some defective verbs" en.wiktionary.org/wiki/Appendix:English_modal_verbs
    – devio
    Aug 5 at 16:51
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    @Araucaria-him Good call there. I had been ignoring modal verbs (which are generally Ts rather than Vs), auxiliaries (which are also a different category from V in many models), and archaic ones, but "beware" does seem distinctly verb-like despite its missing forms.
    – Draconis
    Aug 8 at 16:22

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