In this article we highlight some of the main contours of the urge towards the infinite in order to focus on the twofold role of infinity in mathematics. Our brief discussion of the discovery of the wholeparts relation explains the switch from infinity as endlessness to infinity turned �inwards�, evinced in the infinite divisibility of (spatial) continuity. The traditional Aristotelian distinction between the potential infinite and the actual infinite constitutes the background of our subsequent analysis which touches upon Zeno's paradoxes and Aristotle's objections to the actual infinite. Since Descartes mathematicians increasingly reverted the relation between the potential infinite and the actual infinite by considering the latter as the basis of the former. The historical dominance of the potential infinite was eventually challenged by Cantor in his transfinite arithmetic. Weyl even portrayed mathematics as the science of the infinite. However, this view prompts us to analyse in more detail what the difference between the potential infinite and the actual infinite really is. Cantor's own definition of these two kinds of infinity serves as the starting-point of our ensuing analysis. It is argued that set theory (while employing the actual infinite) crucially depends upon �borrowing� (imitating) key features from space, namely the just-mentioned whole-parts relation and the spatial (time) order of simultaneity (at once). A spatially deepened account of the nature of real numbers has to consider them as being present at once. Attention is also given to the objections raised by Paul Bernays, the coworker of David Hilbert, regarding the assumed arithmetization of modern mathematics. Bernays argues that it is the totality character of continuity (which is originally a geometrical notion) resisting a complete airthmetization of mathematics. It is striking that the spatial feature of wholeness receives opposing interpretations in the thought of Bernays and Brouwer. The former explores the totality character of the continuum whereas the latter focuses on the whole-parts relation. Ultimately the impossibility to articulate the nature of the at once infinite without (implicitly or explicitly) exploring key elements of space therefore uproots the claims of arithmeticism. Although the potential infinite is purely arithmetical in nature, the actual infinite is not, because no single account of it succeeded in avoiding the above-mentioned key spatial characteristics. Lorenzen aptly points out that arithmetic provides no motif for introducing the at once infinite. Therefore the question posed in the title of this article, namely: �Is infinity purely arithmetical in nature?� should be answered in a twofold way: (i) The potential infinite (successive infinite) is a purely arithmetical concept, whereas (ii) the actual infinite (at once infinite) is not purely numerical in nature. Some of the key elements of the argument is captured in the Figure inserted in paragraph 22.

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