From: mk_thisisit
Many scientific paradoxes are contrary to human intuition [00:00:02]. While some paradoxes are now understood, such as the twin paradox or the Banach-Tarski paradox, others remain challenging to fully grasp [00:01:34]. The concept of infinity, though seemingly abstract, is a natural construct in mathematics, leading to conclusions that can be counter-intuitive [00:07:32].
Understanding Infinity
Our everyday intuition suggests that we know how to count objects, which works well for finite sets [00:02:48]. However, problems arise when dealing with infinite sets [00:03:00]. For instance, it seems that there are fewer even numbers than all natural numbers, yet mathematically, there are exactly the same number of them [00:00:57]. Similarly, a section from zero to one has the same number of points as the entire straight line [00:01:06]. This highlights the abstract nature of mathematical points, which are distinct from physical objects [00:08:15].
Hilbert’s Hotel Paradox
The great mathematician David Hilbert devised a thought experiment to illustrate the strange behavior of infinite sets [00:03:02].
- Accommodating one new guest: Imagine a hotel with infinitely many rooms, all occupied [00:03:18]. If a new guest arrives, the hotel can still accommodate them [00:03:32]. This is achieved by moving the guest from room number 1 to room number 2, the guest from room number 2 to room number 3, and generally, the guest in room
nmoves to roomn + 1[00:03:43]. This frees up room number 1 for the new guest [00:04:16]. Mathematically, this demonstrates that adding one to infinity still results in infinity [00:04:28]. - Accommodating infinitely many new guests: The paradox extends further; if an infinite number of new guests arrive, they can also be accommodated [00:04:34]. The original guests are asked to move from room
nto room2n[00:05:13]. This places all existing guests in even-numbered rooms, leaving all odd-numbered rooms free for the infinite new customers [00:05:21]. - Infinite hotels: Even if one were to build an infinite number of Hilbert hotels, each with infinitely many rooms, the total number of beds would be the same as in just one Hilbert hotel [00:11:14]. This implies that infinite sets can be combined without increasing their “size” in certain contexts [00:16:06].
Infinity in Physics vs. Mathematics
The concept of infinity is handled differently in physics and mathematics.
- Physics’ struggle with infinity: In physics, the term “infinity” is generally avoided, as physicists often cannot cope with the concept [00:06:13]. The size of the universe, for example, is considered finite, as it is expanding but will eventually stop and shrink in a “big contraction” [00:05:53]. Infinite densities, like those in the center of a black hole, indicate a breakdown of physics [00:10:39].
- Mathematics’ acceptance and exploration: In mathematics, not only is infinity accepted, but there are also “bigger and smaller infinities” [00:06:32]. This concept, initially puzzling [00:06:36], stems from the idea that there is no largest natural number, naturally leading to the concept of infinity [00:07:21].
Equinumerosity and Cardinality
To precisely define how sets have the same number of elements, mathematics uses the concept of equinumerosity [00:12:13]. Two sets are equinumerous if their elements can be put into a one-to-one correspondence, without needing to count them [00:12:27]. For example, if every seat in a cinema hall is occupied by a person without anyone standing, then the number of seats and people is the same [00:12:47]. This principle applies to infinite sets [00:13:01].
While many infinite sets can be shown to be equinumerous (like natural numbers and even numbers), there are indeed different “sizes” of infinity [00:13:19]. For instance, the number of points in a very short section (e.g., from 0 to 1) is “infinitely many times more” than in an infinite Hilbert hotel or even infinitely many such hotels [00:18:41]. This is because a section (representing real numbers) has fundamentally different characteristics than natural numbers or integers [00:18:20].
Banach-Tarski Paradox
The Banach-Tarski paradox is another mathematical abstraction that defies intuition [00:53:31]. It states that a single sphere can be disassembled into five parts which, by reassembling them through rigid motions (rotations and translations) without changing their shape, can form two identical spheres, each the same size as the original [00:54:14].
- Mathematical abstraction vs. physical reality: This paradox is not contrary to physical principles like conservation of mass because it deals with mathematical points, not physical objects composed of atoms [00:54:41]. It exists purely at the level of mathematical abstraction [00:55:27].
- Dimension specificity: This paradox does not exist in one or two dimensions (e.g., it’s impossible to construct a paradoxical circle from which two circles could be formed) [00:55:57]. It is a peculiar feature of three-dimensional space [00:56:15].
Mathematics as a Tool for Understanding the Universe
The interplay between mathematics and physics is evident in how we understand complex phenomena. For instance, quantum mechanics, with its paradoxical nature like a photon being in two places at once, is described through mathematical reasoning [00:20:11]. We cannot use our senses to perceive particles at the quantum scale; instead, mathematics becomes our “sixth sense,” allowing us to predict how the world functions at the nano scale by solving equations on paper [00:24:04]. These mathematical reasonings have led to great discoveries and have been confirmed by countless experiments [00:24:24].
The binary system (0 and 1) is a prime example of this interplay [00:56:36]. This mathematical theory, developed in the 19th century, was later found to be perfectly suited for digital electronics in the 20th century [00:57:02]. While pure mathematics provides a precise, consistent, and elegant and beautiful theory [00:57:34], its application in physics introduces practical “margins,” such as defining “zero” voltage within a range rather than an exact point [00:58:00]. Conversely, physicists sometimes introduce mathematically “absurd” concepts (like the Dirac delta) that later acquire full mathematical meaning [00:58:41].
Ultimately, paradoxes are essential [00:58:57]. They push the boundaries of our understanding and intuition, leading to profound insights into the nature of reality.