First of all, there is the idea to look for the remains of Atlantis where they should be, according to Plato. If we take his description literally, we should look for evidence of the existence of Atlantis at the tip of the Atlantic Ocean, including both Americas, which were allegedly connected to Atlantis by a chain of islands.
The deep conviction in the existence of Atlantis was the main argument of Ignacy Donnelly (1832-1901), an eccentric American politician and writer who, in his book on Atlantis published in 1882, personally brought the topic out of the deep shadows it had been in since the Renaissance. Donnelly listed dozens of parallels between the pre-Columbian civilizations of the New and Old World, from the construction of pyramids and mummification to similar myths and symbols, such as the sign of the cross. All this, he believed, needs to be explained, and the existence of Atlantis provides such an explanation.
The most ancient stone buildings in Europe are multi-chamber tombs and megalithic monuments. Of course, atlantologists consider megaliths to be proof of their rightness, but the problem is that there is nothing like these structures on the other side of the Atlantic.
Even if we shift the dating in Plato's history to the beginning of the era of megalithic construction (4500 BC and beyond) or the construction of pyramids (2700 BC and beyond), these monuments themselves do not say anything about the continent, which died in the waters of the Atlantic. In short, we simply have no archaeological evidence for the theory of transatlantic connection between Europe and Africa on the one hand and North and South America on the other.
All aforesaid does not deny certain similarity between civilizations of the Old and New World. For example, Donnelly was right, pointing out that cultures on both sides of the Atlantic Ocean have strikingly similar myths about the Great Flood. Naturally, there are many other explanations that do not suggest the existence of a continent that has sunk in Atlantic waters. And if any of the transatlantic parallels are really fair (which is quite doubtful), it would be easier and more logical to conclude that this is a consequence of sea voyages, and not to invent the whole continent. Of course, there are not yet open centers of civilization, missing fragments of prehistoric puzzle. However, no historical data indicate that one of these "missing fragments" is on the bottom of the Atlantic Ocean.
From the geological point of view, the theory of the continent sunken in the Atlantic Ocean, too, does not stand up to criticism. Until the beginning of the XX century, many geologists readily accepted the idea of the existence of a land isthmus that once connected Europe and North America, partly because of the similarity of prehistoric flora and fauna on both sides of the Atlantic. Although it was believed that the isthmus disappeared hundreds of thousands of years before the dive of Platonic Atlantis (9600 BC), some geologists believe that it existed for a much longer period of time and was the cause of the legends of Atlantis. This was the opinion of a prominent French geologist Louis de Loñais, published in 1921. However, in those years there was already a revolution of thinking, which later buried the hypothesis of Atlantis.
In 1915, the German meteorologist Alfred Wegener published his theory of "continental drift".
The main evidence in support of this theory is in the middle of the Atlantic Ocean, where a long mountain range with underwater volcanoes stretches in the meridional direction. However, geologists have come to the opposite conclusion: the ridge is made up of much younger rocks formed by the solidification of magma flowing through a series of faults in the oceanic crust. Spreading out from the central ridge, magma stretches the tectonic plates on both sides, moving both America and Europe and Africa even further apart.
Every few million years, the polarity of the Earth's magnetic field has changed: the magnetic North Pole has become southern, and vice versa. Cooling, the melted magma kept a "print" of terrestrial magnetism at present time. It is important to note that the same sequence of magnetic "imprints" was found in rocks on both sides of the Mid-Atlantic Ridge. This is a convincing proof of the expansion, or spreading, of the seabed, continental drift and the relatively young age of the Atlantic oceanic crust.
Strictly speaking, plate tectonics is still only a theoretical model.
Given all the circumstances, the possibility of the existence of a "missing" continent in the Atlantic, which could have served as a home for a highly developed civilization during the last glacial age, is vanishingly small. Strictly speaking, we have no proof of the existence of Atlantis other than a literary and far from realistic description of Plato. Without Platon's writings, the idea of a sunken continent might never have been.