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What are some similarities between the science of today and the approaches that the pre-socratics took? What are some differences?

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That is rather a large question, given the variety of pre-Socratic philosophers and of modern scientists! However, one could cite Leucippus as having a surprisingly modern approach with his theory of atomism - that all matter was composed of much smaller elements - although he was a bit shaky in the details.

The differences are plenty - ancient thinkers were just that - thinkers. They did not have the resources for investigation that modern scientists have. That was why Anaxagoras could reckon that the Sun was a glowing ball in the sky but only about the size of the Peloponnese peninsula of southern Greece! 
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The similarity is that they are finding the right solution/s to a problem. But then, it is more advance at present than in the past. We cannot deny that technology is more advanced in the current industrial revolution.
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An interesting comment, given that there were plenty of varieties of opinion which suggests that they cannot all have been correct! This would only apply in any case on matters of ethics and social living, whereas most pre-Socratics were concerned primarily with explaining why the observed world was the way it was and how the human mind worked. 
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Socrates and Sophists are often mistaken because they both talk about human affairs and arete, also known as virtue or excellence of humanity. They also use rhetoric, “the art of communication and argument, in technique of persuasion” when they discuss these topics.
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The pre-Socratic Atomists were a group of ancient thinkers who proposed a materialistic theory of the cosmos. Among the first to propose a mechanistic view of the universe, the Atomists argued that the world was composed of atoms. Their work was crucial in the development of ancient philosophy and modern science.
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Some similarities between Socrates' approach to science and that of the pre-Socratics include the emphasis on questioning and critical thinking, seeking knowledge through observation and inquiry, and valuing reason over mythology or superstition.
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