# Next Step Infinity

** **Infinity can violate our human intuition, which is based on finite systems, and create perplexing philosophical problems.

ANTHONY AGUIRRE holds a BS (1995) in mathematics and physics from Brown University and a PhD (2000) in astronomy from Harvard University. He is an associate professor of physics at the University of California, Santa Cruz, where he studies a variety of topics in theoretical cosmology, including the early universe and inflation, gravity physics, first stars, the intergalactic medium, galaxy formation, and black holes.

_{Excerpted from Future Science: Essays From The Cutting Edge, Edited by Max Brockman (Vintage Books, 2011)}

**[ANTHONY AGUIRRE:]** The question of whether the world is finite or infinite has bedeviled us for a long time. It was a classic question in ancient Indian philosophy. Aristotle cogently argued that while infinity made sense in the “potential,” the world could not “actually” be infinite. Giordano Bruno declared the world infinite and was burned at the stake. Galileo, more circumspect, had his clever alter ego, Salviati, completely befuddle Simplicio with how paradoxical and slippery infinity is. And Immanuel Kant really threw down the gauntlet, arguing that both an infinite and a finite world were logically impossible: an infinite universe would take an infinite time to be “synthesized” and thus could never at any one time be said to be infinite—but a finite universe must somehow be embedded in a seemingly meaningless “emptiness” that is not part of the universe. Because finite and infinite spaces alike tax our conception of space, and because we, as finite creatures, clearly cannot measure or directly observe an infinite system, it might appear that the question could most conveniently be consigned to the dustheap of purely philosophical inquiries that hard-nosed scientists can safely ignore.

Yet Albert Einstein’s theories of space and time radically reformulated the questions of finite and infinite spaces and times, and the ensuing development of cosmology has brought infinity into the domain of testable physical science. For example, a uniform space can be curved like a sphere—and comprise a universe that is finite in volume without having any “edge” or empty space outside it. Even more impressive are the tricks that relativity can play concerning infinite spaces, which have come to occupy a central place in contemporary cosmology. To tell this story, I will contend in the following four sections of this essay that: