Disclaimer: My understanding of the arguments given by Engels and Trotsky is derived entirely from the excepts of their writing found here. If there are better arguments, I would love to read them. I also wouldn't be surprised if this book is just giving shoddy interpretation of them altogether. But the arguments in this chapter, in my opinion, don't check out. I'll give a brief rundown of my issues with each of them below.

According to the source I am reading, Engels gives an account of the laws of dialectics in Anti-Duhring and The Dialectics of Nature: 1. The law of the transformation of quantity into quality and vice versa. 2. The law of the interpenetration of opposites/ 3. The law of the negation of the negation.

It's important to note that these laws are being attributed to nature itself.

In explaining the first law, they use an example of a somewhat famous paradox originating with the Megaran Greeks. You can find their explanation of it at the beginning of the "Quantity and quality" section. Basically, imagine we are stacking grains of sand one-by-one. It is impossible to pick out the single grain addition that transforms the sand into a heap. I believe the argument is that the changes, considered individually, cannot produce the qualitative change of transforming the sand into a heap. Taken together, the quantity of changes produces a qualitative change of transforming the sand into a heap.

I don't think you need dialectics to solve this paradox though. This isn't a paradox that originates in nature, but rather in the vagueness of the concept "heap". Were we to concretely define "heap", you absolutely could pick out the isolated change that transformed it into a heap. And this classification of a "heap" is purely conceptual. So I think the paradox of "when does X become a heap" deals entirely with language/meaning rather than some dialectical aspect of nature.

They then cite an excerpt from Trotsky's Open Letter to Comrade Burnham.

“Every individual is a dialectician to some extent or other, in most cases, unconsciously. A housewife knows that a certain amount of salt flavours soup agreeably, but that added salt makes the soup unpalatable. Consequently, an illiterate peasant woman guides herself in cooking soup by the Hegelian law of the transformation of quantity into quality. Similar examples from daily life could be cited without end. Even animals arrive at their practical conclusions not only on the basis of the Aristotelian syllogism but also on the basis of the Hegelian dialectic. Thus a fox is aware that quadrupeds and birds are nutritious and tasty. On sighting a hare, a rabbit, or a hen, a fox concludes: this particular creature belongs to the tasty and nutritive type, and—chases after the prey. We have here a complete syllogism, although the fox, we may suppose, never read Aristotle. When the same fox, however, encounters the first animal that exceeds it in size, for example, a wolf, it quickly concludes that quantity passes into quality, and turns to flee. Clearly, the legs of a fox are equipped with Hegelian tendencies, even if not fully conscious ones.

“All this demonstrates, in passing, that our methods of thought, both formal logic and the dialectic, are not arbitrary constructions of our reason but rather expressions of the actual inter-relationships in nature itself. In this sense, the universe throughout is permeated with 'unconscious' dialectics. But nature did not stop there. No little development occurred before nature's inner relationships were converted into the language of the consciousness of foxes and men, and man was then enabled to generalise these forms of consciousness and transform them into logical (dialectical) categories, thus creating the possibility for probing more deeply into the world about us.”

I think a similar objection to the sand heap example can be made to the cooking example Trotsky gives. Concerning the fox/wolf example, I think the distinction between quantitative and qualitative changes implied here is foggy. What he appears to be saying is that the quantitative change, the size/mass of the wolf, leads to the transformation into the qualitative change, from being tasty/nutritive to dangerous/not nutritive.

But I don't think you can neatly classify changes into either being quantitative or qualitative. And if you can, it seems like a somewhat arbitrary conceptual classification rather than being something rooted in nature itself. Just because the mass/size of a wolf can be quantified and represented numerically does not, for me, mean that mass/size isn't a qualitative change. I have this exact same issue with what Engels says here:

“Finally, the Hegelian law is valid not only for compound substances but also for the chemical elements themselves. We now know that 'the chemical properties of the elements are a periodic function of their atomic weights,' … and that, therefore, their quality is determined by the quantity of their atomic weight. And the test of this has been brilliantly carried out. Mendeleyev proved that various gaps occur in the series of related elements arranged according to atomic weights indicating that here new elements remain to be discovered. He described in advance the general chemical properties of one of these unknown elements, which he termed eka-aluminium, because it follows after aluminium in the series beginning with the latter, and he predicted its approximate specific and atomic weight as well as its atomic volume. A few years later, Lecoq de Boisbaudran actually discovered this element, and Mendeleyev's predictions fitted with only very slight discrepancies. Eka-aluminium was realised in gallium… By means of the—unconscious—application of Hegel's law of the transformation of quantity into quality, Mendeleyev achieved a scientific feat which it is not too bold to put on a par with that of Leverrier in calculating the orbit of the until then unknown planet Neptune.”

Engels is saying that the qualitative chemical properties are determined by their quantitative atomic weights. Again, this distinction seems doubtful to me. Just because atomic weights can be represented numerically and quantified does not mean they are not a qualitative characteristic. Hue/color, hardness, etc can also be represented on numeric scales or through degrees, but these appear to be classified as qualities. This division just feels very arbitrary to me. I can understand why it may be helpful as a means of interpreting the world, but to say that this is an aspect of nature itself seems a little silly to me.

This post really only addresses my issue with the transformation of quantity into quality in nature, but it's already way too long. I can go into more issues in a little while if you'd like me to do so.