One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted. If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics.
Interpretation
The essence of mathematics goes beyond its language and requires an understanding of the mental processes involved in mathematical thinking.
Brouwer emphasizes that to truly understand mathematics, one must look beyond the symbols and language used to express it. This involves examining the cognitive processes that underlie mathematical thought, suggesting that without this deeper inquiry, one merely engages with mathematics superficially, missing its fundamental essence and operations.
In practice
In a lecture on the philosophy of mathematics, this quote can emphasize the importance of understanding the mental processes behind mathematical concepts.
Who are the militia? They consist now of the whole people, except a few public officers. But I cannot say who will be the militia of the future day. If that paper on the table [the Constitution] gets no alteration, the militia of the future day may not consist of all classes, high and low, and rich and poor.
The greatest incitement to guilt is the hope of sinning with impunity.
As belief shrinks from the world, it is more necessary than ever that someone believe. Wild-eyed men in caves. Nuns in black. Monks who do not speak. We are left to believe. Fools, children. Those who have abandoned belief must still believe in us. They are sure they are right not to believe but they know belief must not fade completely. Hell is when no one believes.
The land on which the cattle grazed was communal property. It was owned by no one. It was nobody's private farm. It was the common property of the people, shared by the people. So the practice of sharing was central to the concept of ownership of property.
The objections to religion are of two sorts - intellectual and moral. The intellectual objection is that there is no reason to suppose any religion true; the moral objection is that religious precepts date from a time when men were more cruel than they are and therefore tend to perpetuate inhumanities which the moral conscience of the age would otherwise outgrow.
It is only the poor who are forbidden to beg.
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