Meaning: The quote "We have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed" by James Newman encapsulates a fundamental shift in the way humanity has come to understand mathematics and its relationship to the human mind. This quote reflects a departure from the traditional view that mathematical truths are objective, universal realities that exist independently of human cognition, towards a recognition of the deeply subjective and constructed nature of mathematical knowledge.

In the history of human thought, the concept of mathematical truths as existing independently of the human mind has been deeply entrenched. This perspective, often referred to as Platonism, posits that mathematical objects and truths have an objective existence in a realm separate from the physical world and are discovered rather than invented by human beings. This view has been influential in shaping the way mathematics has been studied and understood for centuries, with the belief that mathematical knowledge is a reflection of an external reality.

However, the quote by James Newman suggests a departure from this traditional perspective, indicating that we have overcome the notion of mathematical truths having an existence independent of our minds. This shift in perspective can be attributed to developments in the philosophy of mathematics, cognitive science, and the recognition of the role of human subjectivity in shaping mathematical knowledge.

One important influence on this shift in perspective is the work of philosophers and mathematicians such as Ludwig Wittgenstein, who challenged the idea of a transcendent realm of mathematical objects and argued that mathematical language and concepts are deeply embedded in human practices and forms of life. Wittgenstein's ideas contributed to a more pragmatic and language-based approach to mathematics, emphasizing the role of human language and social practices in shaping mathematical knowledge.

Additionally, advances in cognitive science and psychology have shed light on the ways in which human cognition and perception shape mathematical understanding. Research in cognitive psychology has shown that mathematical concepts are constructed and understood through the interaction of the human mind with the external world, and that mathematical knowledge is deeply intertwined with human perception, reasoning, and cultural practices.

Furthermore, the advent of computer science and the study of artificial intelligence have raised questions about the nature of mathematical knowledge and its relationship to human cognition. The development of algorithms, machine learning, and computational approaches to mathematics has highlighted the role of human creativity and intuition in mathematical discovery and problem-solving, challenging the view of mathematics as a purely objective and independent realm of truth.

In contemporary philosophy of mathematics, there is a growing recognition of the human and social dimensions of mathematical knowledge, with scholars exploring the ways in which mathematical concepts are shaped by cultural, historical, and linguistic factors. This shift in perspective has led to new approaches to the study of mathematics, emphasizing the importance of understanding mathematics as a human practice embedded in specific social and cultural contexts.

In conclusion, the quote by James Newman reflects a profound shift in the way we understand the nature of mathematical truths, signaling a departure from the traditional view of mathematics as an objective and independent realm of knowledge. This shift has been influenced by developments in philosophy, cognitive science, and the recognition of the deeply subjective and constructed nature of mathematical knowledge. As we continue to explore the human and social dimensions of mathematics, we are gaining a deeper understanding of the ways in which mathematical truths are shaped by human cognition, perception, and cultural practices. This recognition opens up new avenues for the study of mathematics and invites us to consider the diverse ways in which mathematical knowledge is constructed and understood by human minds.